Question: Consider a public-key cryptosystem. Let $K_p$, $K_s$, $X$, and $Y$ be respectively the public key, private key, plaintext and ciphertext. Which assertion is \emph{always true}?

To analyze the assertions regarding the public-key cryptosystem, we need to clarify the roles of each key and the operations involved:

- \( K_p \) is the public key, which can be shared with everyone.
- \( K_s \) is the private key, which is kept secret by the owner.
- \( X \) is the plaintext, the original message we want to protect or send.
- \( Y \) is the ciphertext, the encrypted message.

In a public-key cryptosystem, the typical operations are:
- **Encryption** using the public key: \( Enc_{K_p}(X) \) produces ciphertext \( Y \).
- **Decryption** using the private key: \( Dec_{K_s}(Y) \) recovers the plaintext \( X \).

Now, let’s analyze each option:

### Option 1: \( Enc_{K_p}(Dec_{K_s}(X))=X \)
This is incorrect. The operation \( Dec_{K_s}(X) \) does not make sense because \( X \) is the plaintext and not ciphertext. The decryption function expects ciphertext as input. Therefore, this assertion is not always true.

### Option 2: \( Enc_{K_s}(Dec_{K_p}(Y))=Y \)
This is also incorrect. The operation \( Dec_{K_p}(Y) \) does not make sense because \( Y \) is ciphertext and should be decrypted using the private key \( K_s \), not the public key. Hence, this assertion is not always true.

### Option 3: \( Dec_{K_p}(Enc_{K_s}(Y))=Y \)
This assertion is incorrect. The operation \( Enc_{K_s}(Y) \) does not make sense in this context. The encryption operation is typically performed on plaintext, not ciphertext. Therefore, this assertion is not always true.

### Option 4: \( Dec_{K_s}(Enc_{K_p}(X))=X \)
This assertion is correct. Here, we first encrypt the plaintext \( X \) using the public key \( K_p \), resulting in ciphertext \( Y = Enc_{K_p}(X) \). When we then apply the decryption operation using the private key \( K_s \), we recover the original plaintext \( X \). This is the fundamental feature of public-key cryptography, guaranteeing the correct decryption of the encrypted message.

### Conclusion
The correct assertion is **Option 4**: \( Dec_{K_s}(Enc_{K_p}(X))=X \).

### Rationale
The correctness of Option 4 is based on the definition of public-key cryptography, where encryption with a public key can only be reversed by the corresponding private key. The operations are designed to ensure that after encrypting a plaintext with the public key, it can always be decrypted back to the original plaintext using the private key.

### Summary of Incorrect Options
- **Option 1** is incorrect because it misapplies the decryption operation to plaintext.
- **Option 2** is incorrect because it incorrectly applies decryption on ciphertext with the wrong key.
- **Option 3** is incorrect as it attempts to encrypt ciphertext, which is not a valid operation in this context.


Question: Implement MAP score

The Mean Average Precision (MAP) score is calculated by first determining the Average Precision (AP) for each query, which involves averaging the precision values at ranks where relevant documents are found. Then, the MAP is the mean of these AP scores over all queries, providing a holistic measure of a retrieval system's performance across ranked results. This metric is particularly useful in assessing the quality of ranking algorithms in retrieving relevant information.


Question: Estimate the 95% confidence intervals of the geometric mean and the arithmetic mean of pageviews using bootstrap resampling. The data is given in a pandas.DataFrame called df and the respective column is called "pageviews". You can use the scipy.stats python library.

### Understanding Confidence Intervals

Confidence intervals (CIs) provide a range of values that are likely to contain the population parameter (such as the mean) with a specified level of confidence, typically 95%. When using bootstrap methods, we can estimate these intervals without making strong assumptions about the underlying distribution of the data.

### Bootstrap Resampling

Bootstrap resampling is a statistical technique that involves repeatedly sampling from the observed data, with replacement, to create a distribution of a statistic (such as the mean). The idea is to mimic the process of obtaining new samples to estimate the variability of the statistic.

The steps to perform bootstrap resampling are as follows:

1. **Draw Samples**: Randomly draw samples from the dataset with replacement. This means some observations may be repeated in a single bootstrap sample while others may not be included.

2. **Compute Statistic**: For each bootstrap sample, compute the desired statistic (in this case, the arithmetic mean and the geometric mean).

3. **Repeat**: Repeat the process many times (often thousands of times) to create a distribution of the statistic.

4. **Calculate Confidence Intervals**: From the distribution of the computed statistics, determine the percentiles corresponding to the desired confidence level (for a 95% CI, this typically involves the 2.5th and 97.5th percentiles).

### Arithmetic Mean and Geometric Mean

- **Arithmetic Mean**: The arithmetic mean is calculated as:
  \[
  \text{Mean} = \frac{1}{n} \sum_{i=1}^{n} x_i
  \]
  where \(x_i\) are the values in the dataset and \(n\) is the number of observations.

- **Geometric Mean**: The geometric mean is calculated as:
  \[
  \text{Geometric Mean} = (x_1 \cdot x_2 \cdot \ldots \cdot x_n)^{1/n}
  \]
  It is particularly useful for data that varies exponentially or multiplicatively.

### Implementation in Python

Using the `scipy.stats` library, we can perform bootstrap resampling easily. The following steps outline the implementation:

1. **Prepare the Data**: Ensure that the data is in a format that can be processed by the bootstrap function.

2. **Bootstrap for Arithmetic Mean**: Use `scipy.stats.bootstrap` to compute the bootstrap distribution for the arithmetic mean.

3. **Bootstrap for Geometric Mean**: Similarly, compute the bootstrap distribution for the geometric mean using the appropriate statistic function.

4. **Extract Confidence Intervals**: From the results of the bootstrap method, extract the confidence intervals for both means.

### Code Explanation

Here’s the provided code with added clarity:

```python
import scipy.stats as stats
import numpy as np
from scipy import stats as mstats

# Perform bootstrap resampling for the arithmetic mean
amean = stats.bootstrap((df.pageviews.values,), statistic=np.mean)

# Perform bootstrap resampling for the geometric mean
gmean = stats.bootstrap((df.pageviews.values,), statistic=mstats.gmean)

# Print the resulting confidence intervals
print("Arith. mean 95%CI:", amean.confidence_interval.low, amean.confidence_interval.high)
print("Geom. mean  95%CI:", gmean.confidence_interval.low, gmean.confidence_interval.high)
```

- Here, `df.pageviews.values` retrieves the pageview data from the DataFrame.
- The `stats.bootstrap` function takes this data, computes the specified statistic (arithmetic or geometric mean), and returns an object containing the confidence intervals.
- Finally, the confidence intervals are extracted and printed.

### Conclusion

The use of bootstrap resampling allows for a robust estimation of confidence intervals for both the arithmetic and geometric means without requiring normality or specific distributional assumptions. This technique leverages the power of repeated sampling to reflect the uncertainty inherent in statistical estimates and is particularly useful when dealing with real-world data that may not meet classical statistical assumptions.


Question: (Neural networks) Training only the first layer of a deep neural network using the logistic loss is equivalent to training a logistic regression over a transformed feature space.

To analyze the claim that training only the first layer of a deep neural network using the logistic loss is equivalent to training a logistic regression over a transformed feature space, we need to break down what each component means and how they relate to each other.

First, let’s define what a deep neural network (DNN) is. A DNN consists of multiple layers, where each layer transforms its input through a function, often involving weights, biases, and a non-linear activation function. For simplicity, consider a DNN with one input layer, one hidden layer, and one output layer. The transformation can be represented mathematically as follows:

1. Let \( x \) be the input vector.
2. The output of the first layer (hidden layer) can be computed as:
   \[
   h = \sigma(W_1 x + b_1)
   \]
   where \( W_1 \) are the weights of the first layer, \( b_1 \) is the bias, and \( \sigma \) is a non-linear activation function (like ReLU, sigmoid, etc.).

3. The output layer then computes the prediction based on the hidden layer output:
   \[
   \hat{y} = \sigma(W_2 h + b_2)
   \]
   where \( W_2 \) and \( b_2 \) are the weights and bias of the output layer.

Now, when we talk about training only the first layer using logistic loss, we mean that we are updating the weights \( W_1 \) and biases \( b_1 \) while keeping \( W_2 \) and \( b_2 \) fixed. The logistic loss (or binary cross-entropy loss) for a single instance can be expressed as:
\[
L(y, \hat{y}) = -y \log(\hat{y}) - (1-y) \log(1 - \hat{y})
\]
where \( \hat{y} \) is the predicted probability of the positive class, and \( y \) is the true label.

Now, let’s consider what training a logistic regression model involves. Logistic regression is a linear model that can be viewed as a single layer neural network without any hidden layers. It can be represented mathematically as:
\[
\hat{y} = \sigma(W x + b)
\]
where \( W \) and \( b \) are the parameters of the logistic regression model.

If we only train the first layer of the DNN, we are effectively transforming the original input \( x \) into a new space defined by \( h \) (i.e., the output of the first layer). However, in standard logistic regression, there is no non-linear transformation of the input features unless we explicitly add them to the model design.

The important point here is that training only the first layer of a deep neural network does not correspond to training a logistic regression model in the transformed feature space of the hidden layer output. Instead, the logistic regression would directly operate on the original input features \( x \) without any transformation. The non-linear transformation applied by the activation function in the hidden layer is crucial and fundamentally alters the relationship between input and output.

Thus, training a DNN by only adjusting the first layer effectively creates a model that captures more complex relationships than a simple logistic regression, which is inherently linear with respect to its features. The adjustment of the weights in a DNN layer using logistic loss does not equate to a logistic regression over a transformed space in the traditional sense.

Therefore, the claim that "Training only the first layer of a deep neural network using the logistic loss is equivalent to training a logistic regression over a transformed feature space" is false.

The correct answer is **False**.


Question:
Assume you are working on a mobile application. Users complain that your app's image gallery uses too much of their mobile data.
In one sentence, explain the first step towards improving this:

One effective approach to address users' concerns about high mobile data usage is to implement image streaming, which allows the app to download only the images currently in view instead of preloading all images at once.


Question: The term frequency of a term is normalized

The term frequency of a term is normalized by the maximal frequency of the term in the document collection. This approach ensures that the term's impact is adjusted relative to its occurrence across all documents, providing a more balanced representation of its importance.


Question: What is the communication complexity of the FloodSet algorithm in number of bits?

The communication complexity of the FloodSet algorithm is \(O(n)\) bits, where \(n\) is the size of the input set, because each participant must send their individual set information to ensure all participants learn the final union of all sets.


Question: Which of the following statements are true about command injection?

Let's evaluate each statement regarding command injection:

1. **The root cause of command injection is the lack of distinction between data plane and control/code plane.**
   - This statement is somewhat true, as command injection often occurs when user input is improperly handled, leading to a lack of separation between data and control commands. However, it's not the most precise way to describe the root cause.

2. **Command injection allows the attacker to launch new processes or invoke internal shell commands.**
   - This statement is true. Command injection vulnerabilities allow attackers to execute arbitrary commands on the host operating system, which can include launching new processes or invoking existing commands.

3. **To mitigate command injection, it is best to replace powerful shell commands (e.g., `system()`) with less privileged alternatives such as `read_file()`.**
   - This statement is partially true. While replacing powerful commands with less vulnerable alternatives can help mitigate risks, the best practice is to avoid executing shell commands with user input altogether, using safer alternatives like parameterized APIs.

4. **Command injection can be mitigated, in general, by prohibiting the `";"` character.**
   - This statement is misleading and not entirely true. While prohibiting certain characters can help reduce the risk, it's not a comprehensive solution. Attackers can still find ways to exploit command injection vulnerabilities using other techniques.

5. **Command injection is unique to PHP web applications.**
   - This statement is false. Command injection vulnerabilities can occur in any programming environment or language that allows for system command execution, not just PHP.

Based on this analysis, the two correct statements are:

- **Command injection allows the attacker to launch new processes or invoke internal shell commands.**
- **The root cause of command injection is the lack of distinction between data plane and control/code plane.** (though this one is more nuanced)

So, the two options that can be considered correct are:
- Option 2
- Option 1 (though it's important to note that its phrasing is not as clear-cut as option 2)


Question: Show that P is the weakest failure detector for Group Membership.

### Step 1: Define Key Concepts

To demonstrate that P is the weakest failure detector for Group Membership, we begin by defining the critical concepts involved:

- **Failure Detector**: This is an abstraction in distributed systems that provides information regarding the status of processes, specifically whether they are alive or have failed.
- **Group Membership**: This refers to the dynamic management of processes in a distributed group, ensuring all members have a consistent view of which processes are active.

### Step 2: Introduce P Failure Detector

The P failure detector allows processes to suspect other processes, meaning that a process can indicate that it thinks another has failed, but no guarantees are given on the consistency of these suspicions across all processes. This characteristic makes it a weak failure detector.

### Step 3: Establish Properties of Group Membership

For a failure detector to effectively support group membership, it must satisfy two essential properties:

1. **Completeness**: If a process has failed, it must eventually be suspected by some other process.
2. **Weak Accuracy**: If a process is not suspected, it must be alive. However, it allows for the possibility that some processes may not suspect a failed process immediately.

### Step 4: Show that P Meets Group Membership Requirements

The P failure detector meets the properties required for group membership in the following ways:

1. **Completeness**: The P detector ensures that any process that has failed is eventually suspected by at least one other process. This aligns with the requirement that failed processes should not remain in the group.

2. **Weak Accuracy**: The P detector allows for inconsistencies in suspicion. It does not require all processes to have the same view, which means that some might not suspect a failed process right away. This is acceptable for group membership, as long as the eventual suspicion of failures is guaranteed.

### Step 5: Conclusion

The P failure detector serves as the weakest possible failure detector for group membership because it fulfills the necessary requirements of completeness and weak accuracy while allowing for inconsistencies in membership views. This flexibility is particularly useful in environments where failures are common, making it suitable for maintaining a dynamic view of group membership without imposing strict synchronization requirements. Thus, P is a fitting choice for supporting group membership in distributed systems.


Question: Which one of these is \emph{not} a hard computational problem?

To determine which of the options is \emph{not} a hard computational problem, we need to analyze each one in terms of computational complexity.

1. **Factoring**: Factoring a large integer into its prime components is considered a hard problem, particularly when the integer is the product of two large primes. This difficulty is the basis for the security of many cryptographic systems, such as RSA.

2. **Extracting square roots**: Extracting square roots (specifically, computing the square root of a number) is a straightforward operation that can be performed efficiently using algorithms like the Newton-Raphson method or even simple arithmetic for small numbers. Hence, this is not a hard computational problem.

3. **Computing the Jacobi symbol**: The Jacobi symbol is a generalization of the Legendre symbol and can be computed efficiently using algorithms that run in polynomial time. Therefore, it is not considered a hard problem in the same sense as factoring or computing the discrete logarithm.

4. **Computing the discrete log**: Computing the discrete logarithm, which involves finding the exponent in the equation \( g^x \equiv y \mod p \) for given \( g \), \( y \), and \( p \), is also a well-known hard problem in computational complexity and forms the basis for various cryptographic schemes, such as Diffie-Hellman and ElGamal.

Given this analysis, the only option that is not considered a hard computational problem is:

**Extracting square roots.**

This option has a straightforward and efficient solution, while the others (factoring, computing the Jacobi symbol, and computing the discrete log) are associated with more complex algorithms and are considered hard problems, especially in the context of cryptography.


Question: Implement a function that computes the confidence for a given set of rules and their respective support.   You can use the following formula:  $$\mathrm{conf}(X \Rightarrow Y) = \mathrm{supp}(X \cup Y) / \mathrm{supp}(X)$$

To compute the confidence for a given set of rules based on their respective support, we can follow a structured approach. Below are the steps involved, along with an implementation in Python.

### Explanation of the Process

1. **Understand Inputs**:
    - **freqSet**: A list representing the frequent itemset of size \( N \).
    - **H**: A list of candidate elements \( Y_1, Y_2, \ldots \) that are part of the frequent itemset.
    - **supportData**: A dictionary that stores the support values for each itemset.
    - **rules**: An array to store the generated rules.
    - **min_confidence**: A threshold to prune rules with confidence below this value.

2. **Calculate Confidence**:
   The confidence for a rule \( X \Rightarrow Y \) is calculated using:
   \[
   \mathrm{conf}(X \Rightarrow Y) = \frac{\mathrm{supp}(X \cup Y)}{\mathrm{supp}(X)}
   \]
   Here, \( \mathrm{supp}(X \cup Y) \) is the support of the union of \( X \) and \( Y \), and \( \mathrm{supp}(X) \) is the support of \( X \).

3. **Iterate through Candidate Elements**:
   For each candidate element in \( H \), compute the confidence of the rule formed with \( freqSet \).

4. **Prune Rules Based on Minimum Confidence**:
   If the computed confidence is greater than or equal to \( min_confidence \), store the rule.

### Implementation

Here’s a Python function that implements this process:

```python
def compute_confidence(freqSet, H, supportData, min_confidence):
    """
    Compute the confidence for rules generated from a frequent itemset.

    Parameters:
    - freqSet (set): A frequent itemset.
    - H (list): List of candidate elements.
    - supportData (dict): Dictionary storing itemset support values.
    - min_confidence (float): Minimum confidence threshold.

    Returns:
    - rules (list): List of rules with sufficient confidence.
    """
    rules = []

    # Support of the frequent itemset
    freqSet_support = supportData[frozenset(freqSet)]

    for candidate in H:
        # Create the rule X => Y
        rule = (freqSet, candidate)

        # Calculate the union of freqSet and candidate
        union_set = freqSet.union(frozenset([candidate]))

        # Get the support of the union
        union_support = supportData[frozenset(union_set)]

        # Calculate confidence
        confidence = union_support / freqSet_support

        # Check if the confidence meets the minimum threshold
        if confidence >= min_confidence:
            rules.append((rule, confidence))

    return rules
```

### Example Usage

Here’s how you can use the function:

```python
# Sample data
freqSet = frozenset(['A', 'B'])
H = ['C', 'D']
supportData = {
    frozenset(['A', 'B']): 0.4,
    frozenset(['A', 'B', 'C']): 0.3,
    frozenset(['A', 'B', 'D']): 0.2,
    frozenset(['C']): 0.5,
    frozenset(['D']): 0.6
}
min_confidence = 0.5

# Compute rules
rules = compute_confidence(freqSet, H, supportData, min_confidence)
print("Generated Rules with Confidence:", rules)
```

### Conclusion
This function will compute the confidence for each candidate in \( H \) and return the rules that meet or exceed the specified minimum confidence threshold. The output will include the valid rules along with their confidence values.


Question: Given an odd prime $p$, for any $a \in \mathbb{Z}_p$ the equation

**

To solve the equation \( x^2 - a = 0 \) in a special set of numbers called \( \mathbb{Z}_p \), where \( p \) is an odd prime number, we need to understand how many solutions it can have based on the value of \( a \).

1. First, let's rewrite the equation to \( x^2 = a \). This means we're looking for numbers \( x \) that, when squared, give us \( a \).
2. Now, whether \( a \) has solutions depends on whether it is a "quadratic residue." This is just a fancy way of saying that there exists some number when squared that equals \( a \). If \( a \) is a quadratic residue, we will find two different numbers that satisfy the equation.
3. If \( a \) is not a quadratic residue, there are no solutions at all.
4. There’s also a special case when \( a = 0 \). In that situation, the equation becomes \( x^2 = 0 \), which only has one solution: \( x = 0 \).

So, in summary, the equation \( x^2 - a = 0 \) can have:
- No solutions if \( a \) is not a quadratic residue,
- One solution if \( a = 0\),
- Two solutions if \( a \) is a quadratic residue.

Given those options, the best choice is that the equation **has at most two solutions**.

**


Question: Assume you work in a team that is developing a weather application that brings together data from several sources. One of your colleagues is responsible for creating a client for a weather service that returns data in JSON format. Your colleague suggests creating a weather client interface that returns the weather as a string, then a class that gets (fetches from the weather service) and returns the JSON, and a decorator that extracts the weather prediction from that JSON and returns it. What do you think about this approach?

Answer:**
The proposed approach of using a weather client interface, a class for fetching data, and a decorator for extracting weather predictions is reasonable. This design allows for clear separation of responsibilities, enhances flexibility, and supports future extensibility.

**Explanation:**
The reasoning highlights that the use of decorators can provide valuable benefits in terms of flexibility and separation of concerns. By encapsulating the extraction logic within a decorator, the data-fetching class can remain focused on its primary role of retrieving data from the weather service. This modular approach allows for the extraction logic to be easily modified or extended without needing to change the data-fetching implementation.

Moreover, this design supports better testing practices, as each component can be tested independently, promoting reusability of the extraction logic across different contexts. While there may be some initial complexity introduced by using a decorator, the long-term benefits in terms of maintainability, clarity, and adaptability to changing requirements make this approach a viable solution for developing a weather application.


Question: The little Fermat theorem states that for a prime $n$ and any $b\in \mathbb{Z}_n ^\star$ we have\dots

The little Fermat theorem states that for a prime \( n \) and any integer \( b \) such that \( b \) is coprime to \( n \) (which means \( b \in \mathbb{Z}_n^\star \)), we have:

\[
b^{n-1} \equiv 1 \mod n.
\]

Thus, the correct answer is:

- \( b^{n-1} \mod n = 1 \).

This theorem is fundamental in number theory and has applications in cryptography, as it provides a way to simplify calculations involving powers of integers modulo prime numbers.


Question: In order to summarize the degree distribution in a single number, would you recommend using the average degree? Why, or why not? If not, what alternatives can you think of? Please elaborate!

Using the average degree may not be ideal for summarizing degree distribution due to its sensitivity to outliers, which can skew the result. Alternatives include the median degree, which provides a better central tendency in skewed distributions, and the mode, which highlights the most common degree. Additionally, measures like variance or standard deviation can provide insights into the spread of degrees.

No, I would not recommend using the average degree due to its sensitivity to outliers; alternatives include the median, mode, or measures of spread like variance.


Question: Consider a binary classification problem with classifier $f(\mathbf{x})$ given by $$ f(\mathbf{x})= \begin{cases}1, & g(\mathbf{x}) \geq 0 \\ -1, & g(\mathbf{x})<0\end{cases} $$ and $\mathbf{x} \in \mathbb{R}^{6}$. Consider a specific pair $(\mathbf{x}, y=1)$ and assume that $g(\mathbf{x})=8$. In particular this means that this point is classified correctly by $f$. Assume further that we have computed the gradient of $g$ at $\mathbf{x}$ to be $\nabla_{\mathbf{x}} g(\mathbf{x})=(+1,-2,+3,-4,+5,-6)$. You are allowed to make one step in order to (hopefully) find an adversarial example. In the following four questions, assume $\epsilon=1$. Which offset $\delta$ with $\|\delta\|_{\infty} \leq 1$ yields the smallest value for $g(\mathbf{x}+\delta)$, assuming that $g$ is (locally) linear?

To determine which offset \(\delta\) yields the smallest value for \(g(\mathbf{x}+\delta)\), we first need to understand how \(g(\mathbf{x})\) behaves in the vicinity of the point \(\mathbf{x}\). Given that \(g\) is locally linear, we can use the first-order Taylor expansion for \(g\):

\[
g(\mathbf{x} + \delta) \approx g(\mathbf{x}) + \nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta
\]

We know from the problem statement that \(g(\mathbf{x}) = 8\) and \(\nabla_{\mathbf{x}} g(\mathbf{x}) = (+1,-2,+3,-4,+5,-6)\). Thus, we can rewrite the expression for \(g(\mathbf{x} + \delta)\) as:

\[
g(\mathbf{x} + \delta) \approx 8 + \nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta
\]

To minimize \(g(\mathbf{x} + \delta)\), we need to minimize the term \(\nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta\). The dot product can be calculated explicitly as:

\[
\nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta = 1\delta_1 - 2\delta_2 + 3\delta_3 - 4\delta_4 + 5\delta_5 - 6\delta_6
\]

Here, \(\delta = (\delta_1, \delta_2, \delta_3, \delta_4, \delta_5, \delta_6)\) where \(\|\delta\|_{\infty} \leq 1\), meaning each component of \(\delta\) must lie within the range \([-1, 1]\).

To minimize this expression, we should choose \(\delta_i\) values such that they will have the most negative contribution according to the coefficients in the gradient. Specifically:

- For \(\delta_1\) (coefficient \(+1\)), we should set \(\delta_1 = -1\) to minimize its contribution.
- For \(\delta_2\) (coefficient \(-2\)), we should set \(\delta_2 = +1\) to maximize the negative impact.
- For \(\delta_3\) (coefficient \(+3\)), we should set \(\delta_3 = -1\) to minimize its contribution.
- For \(\delta_4\) (coefficient \(-4\)), we should set \(\delta_4 = +1\) to maximize the negative impact.
- For \(\delta_5\) (coefficient \(+5\)), we should set \(\delta_5 = -1\) to minimize its contribution.
- For \(\delta_6\) (coefficient \(-6\)), we should set \(\delta_6 = +1\) to maximize its negative impact.

Combining these choices, we find that the optimal \(\delta\) is:

\[
\delta = (-1, +1, -1, +1, -1, +1)
\]

This corresponds to the choice labeled as option 6.

Now we will compute \(\nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta\) with our chosen \(\delta\):

\[
\nabla_{\mathbf{x}} g(\mathbf{x}) \cdot \delta = 1(-1) - 2(1) + 3(-1) - 4(1) + 5(-1) - 6(1)
\]
\[
= -1 - 2 - 3 - 4 - 5 - 6 = -21
\]

Therefore,

\[
g(\mathbf{x} + \delta) \approx 8 - 21 = -13
\]

This is the minimum value we can achieve given our constraints.

Thus, the offset \(\delta\) that yields the smallest value for \(g(\mathbf{x}+\delta)\) is:

\((-1, +1, -1, +1, -1, +1)\)

However, upon checking the options provided, this corresponds to option 6, confirming that \( (-1, +1, -1, +1, -1, +1) \) is indeed the correct answer.

So the final answer is:

**6.** \((-1,+1,-1,+1,-1,+1)\)


Question: Which of the following is correct regarding prediction models?

To evaluate the statements regarding prediction models systematically, we need to understand key concepts such as training error, test error, overfitting, underfitting, model complexity, bias, and variance. Let's break down each statement:

1. **Training error being less than test error means overfitting**:
   - **True.** When you train a model, you compute the training error (the model's performance on the training dataset). If this error is significantly lower than the test error (the model's performance on a separate validation/test dataset), it often indicates that the model has learned the noise in the training data rather than the underlying patterns. This phenomenon is known as overfitting, where the model performs well on the training data but poorly on unseen data.

2. **Training error being less than test error means underfitting**:
   - **False.** Underfitting occurs when a model is too simple to capture the underlying patterns in the data, resulting in high training error and also high test error. In this case, the training error would not be less than the test error; rather, both would be relatively high. So, the statement is incorrect.

3. **Complex models tend to overfit, unless we feed them with more data**:
   - **True.** Complex models (like deep neural networks with many layers) have a greater capacity to learn intricate patterns, but they also have a higher risk of capturing noise in the training data leading to overfitting. However, providing more data can help mitigate this risk because larger datasets can better represent the underlying distribution, allowing the model to generalize better. Thus, while complex models are prone to overfitting, sufficient data can help balance this issue.

4. **Simple models have lower bias than complex models**:
   - **False.** In the bias-variance tradeoff, simple models typically have higher bias and lower variance. They make strong assumptions about the data (like linearity in linear regression) and may fail to capture the underlying relationships, leading to systematic errors (high bias). In contrast, complex models, while potentially overfitting, can adapt more flexibly to the data, thus reducing bias. Therefore, this statement is incorrect.

In summary, the correct statements regarding prediction models are:
- **Training error being less than test error means overfitting**.
- **Complex models tend to overfit, unless we feed them with more data**.

These insights illustrate the balance that must be struck in model selection and training, as understanding the trade-offs between bias and variance is crucial for developing robust predictive models.


Question: The primal formulation of the soft-margin SVM is NOT equivalent to $\ell_2$ adversarial training for a linear model trained with the hinge loss ($\ell(z) = \max\{0, 1 - z\}$).

True.

The primal formulation of the soft-margin Support Vector Machine (SVM) is specifically designed to find hyperplanes that separate the classes while allowing for some misclassifications, controlled by a regularization parameter. The soft-margin SVM uses hinge loss, which penalizes misclassified points.

On the other hand, adversarial training typically involves generating adversarial examples (perturbations of the input data) to improve the robustness of a model against these perturbations. While both methods may use hinge loss, the contexts and objectives can differ significantly.

In adversarial training, the focus is on creating a model that can withstand adversarial attacks, while soft-margin SVM focuses on separating classes with a margin while allowing for some errors. Therefore, the two are not equivalent in their objectives and formulations.

Thus, the statement is **True**.


Question: A passive adversary can \ldots

A passive adversary can:

2. only listen to communications.

This means the adversary can observe and analyze the data being transmitted but cannot alter or interfere with the communication.


Question:
What is the mean squared error of $f$ for a sample, where $\textbf{x}$ is an input, $y$ a target and $f(\textbf{x},W)$ the mapping function  ?
	(One answer)

The correct answer is: **$||y - f(\textbf{x},W)||^2$**.

### Explanation:

The mean squared error (MSE) is a commonly used metric to evaluate the performance of a model by measuring the average of the squares of the errors—that is, the average squared difference between the estimated values (the model's predictions) and the actual values (the target values).

In this context:

- **$y$** represents the actual target values (the true outputs).
- **$f(\textbf{x}, W)$** is the output of the model given the input data **$\textbf{x}$** and the parameters **$W$**.

The expression **$||y - f(\textbf{x}, W)||^2$** calculates the squared difference between the actual target **$y$** and the predicted value **$f(\textbf{x}, W)$**. The notation **$||\cdot||$** typically refers to the norm (or length) of a vector, and in this case, taking the square of this norm corresponds to summing the squared differences across all data points, which aligns with the definition of MSE.

The other options represent different concepts:

- **$||y - f(\textbf{x},W)||$** is the L2 norm of the error but does not square the differences, so it does not represent mean squared error.
- **$-\log(P(y=i | \textbf{x}))$** and **$P(y=i | \textbf{x}) = \frac{e^{\textbf{f}_i(\textbf{x},W)}}{\sum_j e^{\textbf{f}_j(\textbf{x},W)}}$** are related to probabilistic models and log-likelihood rather than mean squared error.

Thus, when asked about the mean squared error of the mapping function **$f$**, the correct expression is indeed **$||y - f(\textbf{x}, W)||^2$**.


Question: Tick the \textbf{true} assertion. Let $n >1 $ be a composite integer, the product of two primes. Then,

To determine which assertions are true, we need to analyze each option with respect to the properties of composite integers that are products of two distinct primes, denoted as \( n = pq \) where \( p \) and \( q \) are primes.

1. **Assertion (1):** \( \phi(n) \) divides \( \lambda(n) \).
   - Here, \( \phi(n) = (p-1)(q-1) \) and \( \lambda(n) = \text{lcm}(p-1, q-1) \).
   - Generally, \( \phi(n) \) does not divide \( \lambda(n) \) since \( \lambda(n) \) can be equal to or greater than \( \phi(n) \) depending on the values of \( p \) and \( q \).
   - Thus, this assertion is **false**.

2. **Assertion (2):** \( \lambda(n) \) divides the order of any element \( a \) in \( \mathbb{Z}_n^* \).
   - The group \( \mathbb{Z}_n^* \) consists of integers coprime to \( n \).
   - The order of any element \( a \in \mathbb{Z}_n^* \) divides \( \phi(n) \), and \( \lambda(n) \) is a divisor of \( \phi(n) \) only under certain conditions, but this does not hold for all cases.
   - Therefore, this assertion is **false**.

3. **Assertion (3):** \( \mathbb{Z}^{*}_n \) with multiplication is a cyclic group.
   - The group \( \mathbb{Z}_n^* \) is cyclic if \( n \) is of the form \( p^k \) or \( 2p^k \) where \( p \) is an odd prime and \( k \geq 1 \). For \( n = pq \) where \( p \) and \( q \) are distinct odd primes, \( \mathbb{Z}_n^* \) is not guaranteed to be cyclic.
   - Thus, this assertion is **false**.

4. **Assertion (4):** \( a^{\lambda(n)} \mod n = 1 \) for all \( a \in \mathbb{Z}^{*}_n \).
   - By definition, \( \lambda(n) \) is the least common multiple of the orders of the group elements, and for any \( a \in \mathbb{Z}_n^* \), \( a^{\phi(n)} \equiv 1 \mod n \) holds, and since \( \lambda(n) \) divides \( \phi(n) \), it follows that \( a^{\lambda(n)} \equiv 1 \mod n \) for all \( a \in \mathbb{Z}_n^* \).
   - Therefore, this assertion is **true**.

After analyzing all options, the only true assertion is option (4).

Answer: [4]


Question: Which of the following elements belongs to $\mathbb{Z}_{78}^*$?

To determine which elements belong to \(\mathbb{Z}_{78}^*\), we must identify which of the provided options are coprime to 78. An element \(a\) is in \(\mathbb{Z}_{78}^*\) if \(\gcd(a, 78) = 1\), meaning that \(a\) shares no common factors with 78 other than 1.

First, we need to factor 78 to understand its prime factors:
\[
78 = 2 \times 3 \times 13
\]

Now let's analyze each option one by one:

**A) 46**
- Calculate \(\gcd(46, 78)\):
  - The prime factorization of 46 is \(2 \times 23\).
  - The common factor with 78 is 2.
  - Thus, \(\gcd(46, 78) = 2 \neq 1\).
  - **Conclusion**: 46 is not in \(\mathbb{Z}_{78}^*\).

**B) 35**
- Calculate \(\gcd(35, 78)\):
  - The prime factorization of 35 is \(5 \times 7\).
  - There are no common factors with 78.
  - Thus, \(\gcd(35, 78) = 1\).
  - **Conclusion**: 35 is in \(\mathbb{Z}_{78}^*\).

**C) 21**
- Calculate \(\gcd(21, 78)\):
  - The prime factorization of 21 is \(3 \times 7\).
  - The common factor with 78 is 3.
  - Thus, \(\gcd(21, 78) = 3 \neq 1\).
  - **Conclusion**: 21 is not in \(\mathbb{Z}_{78}^*\).

**D) 65**
- Calculate \(\gcd(65, 78)\):
  - The prime factorization of 65 is \(5 \times 13\).
  - The common factor with 78 is 13.
  - Thus, \(\gcd(65, 78) = 13 \neq 1\).
  - **Conclusion**: 65 is not in \(\mathbb{Z}_{78}^*\).

Now to summarize the findings:
- **46**: Not in \(\mathbb{Z}_{78}^*\).
- **35**: In \(\mathbb{Z}_{78}^*\).
- **21**: Not in \(\mathbb{Z}_{78}^*\).
- **65**: Not in \(\mathbb{Z}_{78}^*\).

The only element from the provided options that belongs to \(\mathbb{Z}_{78}^*\) is **B) 35**.


Question: Consider two fully connected networks, A and B, with a constant width for all layers, inputs and outputs. Network A has depth $3L$ and width $H$, network B has depth $L$ and width $2H$. Everything else is identical for the two networks and both $L$ and $H$ are large. In this case, performing a single iteration of backpropagation requires fewer scalar multiplications for network A than for network B.

To determine whether performing a single iteration of backpropagation requires fewer scalar multiplications for network A than for network B, we will analyze the computational complexity of both networks based on their depth and width.

1. **Definitions**:
   - **Depth**: The number of layers in the neural network.
   - **Width**: The number of neurons in each layer.
   - **Scalar Multiplications**: The primary operation in neural networks during forward and backward propagation, which involves multiplying weights by inputs.

2. **Network A**:
   - Depth: \(3L\)
   - Width: \(H\)

3. **Network B**:
   - Depth: \(L\)
   - Width: \(2H\)

4. **Computational Cost Analysis**:
   - For a fully connected layer, the number of scalar multiplications required to compute the output of one layer is given by the product of the number of neurons in the previous layer and the number of neurons in the current layer.
   - Assuming each layer has a constant width and that all layers are fully connected, we can express the total number of scalar multiplications for each network.

5. **Calculating Scalar Multiplications for Network A**:
   - The first layer connects inputs to the first hidden layer (assuming \(N\) inputs): \(N \times H\) multiplications.
   - Each subsequent layer (for \(3L-1\) layers) connects \(H\) neurons to \(H\) neurons: \((3L-1) \times H \times H\).
   - Total multiplications for Network A:
     \[
     \text{Total}_A = N \times H + (3L-1) \times H^2
     \]

6. **Calculating Scalar Multiplications for Network B**:
   - The first layer connects inputs to the first hidden layer: \(N \times 2H\) multiplications.
   - Each subsequent layer (for \(L-1\) layers) connects \(2H\) neurons to \(2H\) neurons: \((L-1) \times 2H \times 2H\).
   - Total multiplications for Network B:
     \[
     \text{Total}_B = N \times 2H + (L-1) \times (2H)^2
     \]

7. **Comparing Total Multiplications**:
   - For large \(L\) and \(H\), we can simplify the expressions:
     \[
     \text{Total}_A \approx 3L \times H^2
     \]
     \[
     \text{Total}_B \approx 4L \times H^2
     \]
   - Thus, we can see that:
     \[
     \text{Total}_A < \text{Total}_B
     \]

8. **Conclusion**:
Since the total number of scalar multiplications required for network A is less than that for network B, it follows that performing a single iteration of backpropagation requires fewer scalar multiplications for network A than for network B.

Therefore, the statement provided in the question is **True**.

Answer: A) True


Question:
Does the disparity in class proportions hurt the model? If yes, how can you fix it? If not, justify the reasons behind your choice.
Hint: The learning objective of a classifier can be modified by altering the importance of each class in the computation of the loss function.

Based you answer on the following confusion matrix:

|precision |   recall | f1-score  | support|
|-|-|-|-|
 |          0  |    0.973  |   0.997   |  0.985  |     330|
 |         1    |  0.750  |   0.250  |   0.375   |     12|

1. **Identify the Disparity**: The confusion matrix shows a significant class imbalance, with class 0 (330 instances) vastly outnumbering class 1 (12 instances). This disparity can lead the model to be biased towards predicting the majority class, resulting in high accuracy but poor performance on the minority class.

2. **Modify Class Weights**: To address this imbalance, we can adjust the loss function by incorporating class weights. For instance, if using a binary cross-entropy loss, we can assign a higher weight to the minority class (class 1) while assigning a lower weight to the majority class (class 0). This can be expressed mathematically as:
   \[
   L(y, \hat{y}) = -\frac{1}{N} \sum_{i=1}^{N} \left( w_0 y_i \log(\hat{y}_i) + w_1 (1 - y_i) \log(1 - \hat{y}_i) \right)
   \]
   where \(w_0\) and \(w_1\) are the weights for classes 0 and 1 respectively.

3. **Use Resampling Techniques**: Another approach is to apply resampling techniques such as oversampling the minority class or undersampling the majority class. For instance, techniques like SMOTE (Synthetic Minority Over-sampling Technique) can generate synthetic samples for class 1 to balance the dataset.

4. **Evaluate Model Performance**: After implementing adjustments, it’s crucial to evaluate model performance using metrics that account for class imbalance, such as F1-score, precision, and recall. Monitoring these metrics will help ensure that improvements in performance on class 1 do not come at the expense of class 0.

5. **Iterate and Optimize**: Finally, model tuning may require iterative adjustments of hyperparameters and weights based on evaluation results. Techniques like cross-validation can be utilized to ensure that the model generalizes well across different subsets of data.

In conclusion, the disparity in class proportions does hurt the model’s performance on the minority class. By modifying class weights in the loss function and employing resampling techniques, we can enhance classification performance across both classes.


Question: To constrain an object of an RDF statement from being of an atomic type (e.g., String), one has to use the following RDF/RDFS property:

To understand which RDF/RDFS property can be used to constrain an object of an RDF statement from being of an atomic type, let's analyze each option with examples and reasoning.

### A. rdf:type
The `rdf:type` property is used to state that a resource is an instance of a class. It does not inherently constrain the type of the object itself; instead, it relates a subject to a class. For example:

```turtle
:John rdf:type :Person .
```

This states that John is a Person but does not impose constraints on what types of objects can be used in statements involving John.

### B. rdfs:range
The `rdfs:range` property specifies the datatype or class of the objects that a property can have. If we want to restrict the values that can be used with a certain predicate, we would use `rdfs:range`. For instance:

```turtle
:hasAge rdfs:range xsd:integer .
```

This means that any object of the property `:hasAge` must be of type `xsd:integer`, which is an atomic type. If we want to ensure that an object cannot be just a String, we could define the range to be something other than String. Thus, `rdfs:range` is indeed useful for constraining the object type.

### C. rdfs:domain
The `rdfs:domain` property specifies the class to which subjects of a property must belong. Like `rdf:type`, it doesn't constrain the object type, but rather indicates what subjects are allowed for a particular predicate. For example:

```turtle
:hasAge rdfs:domain :Person .
```

This means that only instances of `:Person` can have the property `:hasAge`, but it does not affect the type of value that can be assigned to `:hasAge`.

### D. rdfs:subClassOf
The `rdfs:subClassOf` property is used to indicate that one class is a subclass of another. While this establishes a hierarchical relationship between classes, it does not constrain the datatype of objects directly. For example:

```turtle
:Employee rdfs:subClassOf :Person .
```

This merely states that all Employees are Persons but does not place constraints on what types of values can be associated with properties related to either class.

### Conclusion
The correct option for constraining an object of an RDF statement from being of an atomic type is **B. rdfs:range**. This property allows us to specify the allowed types for objects linked by a given predicate, effectively controlling what values can be assigned and preventing atomic types like String if desired.

In summary:
- **A (rdf:type)** - relates subjects to classes, does not constrain objects.
- **B (rdfs:range)** - constrains the type of objects allowed for a property, making it correct.
- **C (rdfs:domain)** - relates subjects to classes, does not constrain objects.
- **D (rdfs:subClassOf)** - establishes hierarchical relationships between classes, does not constrain objects.

Thus, **the answer is B**.


Question: In this problem we design an LSH for points in $\mathbb{R}^d$ with the $\epsilonll_1$ distance, i.e. $$d(p,q) =\sum_{i=1}^d |p_i - q_i|.$$ Define a class of hash functions as follows: Fix a positive number $w$. Each hash function is defined via a choice of $d$ independently selected random real numbers $s_1,s_2,\dots,s_d$, each uniform in $[0,w)$. The hash function associated with this random set of choices is $$h(x_1,\dots ,x_d) = \left(\left\lfloor \frac{x_1 - s_1}{w}\right\rfloor ,\left\lfloor \frac{x_2 - s_2}{w}\right\rfloor,\dots,\left\lfloor \frac{x_d - s_d}{w}\right\rfloor\right).$$ Let $\alpha_i = |p_i - q_i|$. What is the probability that $h(p) = h(q)$, in terms of the $\alpha_i$ values?  It may be easier to first think of the case when $w=1$.  Try to also simplify your expression if $w$ is much larger than $\alpha_i$'s, using that $(1-x) \approx e^{-x}$ for small values of $x\geq 0$.

To determine the probability that \( h(p) = h(q) \) given \( d(p,q) = \sum_{i=1}^d |p_i - q_i| = \sum_{i=1}^d \alpha_i \), we need to analyze the components of the hash function. For \( h(p) \) to equal \( h(q) \), it is necessary that for each \( i \), the difference \( |p_i - q_i| \) falls within the same "bin" defined by the random shift \( s_i \). Specifically, we need:

\[
\left\lfloor \frac{p_i - s_i}{w} \right\rfloor = \left\lfloor \frac{q_i - s_i}{w} \right\rfloor
\]

This condition holds if:

\[
\frac{p_i - s_i}{w} \text{ and } \frac{q_i - s_i}{w} \text{ are in the same interval of width 1, i.e., } |p_i - q_i| < w.
\]

Now, since \( \alpha_i = |p_i - q_i| \), the probability that \( h(p) = h(q) \) for a given dimension \( i \) is:

\[
P(h(p_i) = h(q_i)) = 1 - \frac{\alpha_i}{w} \quad \text{(if } \alpha_i < w\text{)}.
\]

Thus, for all \( d \) dimensions, assuming independence, the overall probability is given by:

\[
P(h(p) = h(q)) = \prod_{i=1}^d \left( 1 - \frac{\alpha_i}{w} \right).
\]

If \( w \) is much larger than the \( \alpha_i \)'s, we can use the approximation \( 1 - x \approx e^{-x} \) for small \( x \), leading to:

\[
P(h(p) = h(q)) \approx e^{-\sum_{i=1}^d \frac{\alpha_i}{w}} = e^{-\frac{d(p,q)}{w}}.
\]


Question:
You want to create an application that allows users to manage their e-books. These books will be stored in a local database, with attributes like name, file, etc. In addition, your application will allow to add notes on books, which will be stored separately in the database, and to send a book with its notes by e-mail to friends, who can import the book and the notes in the app.
What modules would you define?

Creating an application for managing e-books with an integrated note-taking feature involves careful consideration of its architecture and the various modules required to ensure a smooth user experience. Below are some essential modules to define for your application, along with their responsibilities:

### 1. **User Management Module**
   - **Authentication**: Handle user registration, login, and password recovery.
   - **User Profiles**: Allow users to manage their profiles and preferences.
   - **Permissions**: Define access levels for users, especially if there are shared features.

### 2. **E-Book Management Module**
   - **E-Book Upload/Import**: Functionality for users to upload or import e-books, which could include file type validation (e.g., PDF, EPUB).
   - **E-Book Metadata**: Store and manage attributes like title, author, genre, publication date, and file path.
   - **E-Book Library**: A central repository for all uploaded books, with sorting, filtering, and search functionalities.

### 3. **Note Management Module**
   - **Note Creation**: Allow users to create, edit, and delete notes associated with specific e-books.
   - **Note Organization**: Enable users to categorize notes (e.g., by topic or chapter).
   - **Note Retrieval**: Efficient retrieval of notes when accessing respective e-books.

### 4. **Sharing Module**
   - **Export Functionality**: Allow users to package e-books and their associated notes into a shareable format (e.g., ZIP file or custom format).
   - **Email Integration**: Facilitate the sending of e-books with notes via email, possibly using an SMTP service for delivery.
   - **Import Functionality**: Allow recipients to easily import shared e-books and notes into their own application instance.

### 5. **Database Management Module**
   - **Local Database**: Design and manage a local database (e.g., SQLite) for storing e-books and notes, ensuring efficient data retrieval and storage.
   - **Data Synchronization**: If applicable, manage data synchronization mechanisms for users who may want to access their data on multiple devices.

### 6. **User Interface Module**
   - **Main Interface**: Design the primary UI for navigating the library, viewing e-books, and accessing notes.
   - **E-Book Reader**: Implement a simple e-book reader within the app for users to read their books.
   - **Note Viewer/Editor**: Provide a user-friendly interface for creating and viewing notes.

### 7. **Settings and Preferences Module**
   - **Application Settings**: Allow users to customize their app experience (e.g., changing themes, font sizes, etc.).
   - **Backup and Restore**: Implement options for users to back up their data and restore it if necessary.

### 8. **Notification Module**
   - **Reminders**: Allow users to set reminders for reading or reviewing notes.
   - **Alerts**: Notify users of successful sharing, updates, or other relevant activities.

### 9. **Reporting and Analytics Module (Optional)**
   - **Usage Statistics**: Track user behavior, such as reading time, most read books, and note-taking frequency.
   - **Feedback Collection**: Allow users to provide feedback on their experience with the app.

### 10. **Integration Module (Optional)**
   - **Third-Party Services**: Integrate with cloud storage services (e.g., Google Drive, Dropbox) for e-book backups.
   - **APIs for E-Book Metadata**: Use APIs from online databases (like Goodreads) to fetch metadata for books.

### Considerations for Implementation:
- **Data Security**: Ensure that user data, especially notes, are securely stored and transmitted.
- **User Experience**: Focus on intuitive navigation and a clean design to make the app user-friendly.
- **Scalability**: Design modules in a way that makes it easy to add features in the future.
- **Cross-Platform Compatibility**: If applicable, consider how your app will function on different operating systems or devices.

By carefully defining these modules and considering their interdependencies, you can create a robust application that meets user needs while providing opportunities for future expansion and enhancement.


Question: Let $X$, $Y$, and $K$ be respectively the plaintext, ciphertext, and key distributions. $H$ denotes the Shannon entropy. Considering that the cipher achieves \emph{perfect secrecy}, tick the \textbf{false} assertion:

To determine which assertion is false regarding a cipher that achieves perfect secrecy, we must consider the key properties and implications of perfect secrecy.

### Key Properties of Perfect Secrecy:
1. **Perfect Secrecy Definition**: A cipher achieves perfect secrecy if the ciphertext \( Y \) provides no information about the plaintext \( X \). Mathematically, this is represented as:
   \[
   H(X|Y) = H(X)
   \]
   This indicates that knowing the ciphertext \( Y \) does not reduce uncertainty about the plaintext \( X \).

2. **Statistical Independence**: Under perfect secrecy, the plaintext \( X \) and ciphertext \( Y \) are statistically independent. This means that \( P(X|Y) = P(X) \), implying that the knowledge of \( Y \) gives no information about \( X \).

3. **Joint Entropy**: For two random variables \( X \) and \( Y \):
   \[
   H(X,Y) = H(X) + H(Y|X)
   \]
   Since \( Y \) is independent of \( X \) in the context of perfect secrecy, we also have:
   \[
   H(Y|X) = H(Y)
   \]
   Therefore, the joint entropy can be expressed as:
   \[
   H(X,Y) = H(X) + H(Y)
   \]
   This means that \( H(X,Y) \) is not equal to \( H(X) \) alone, but rather the sum of the entropies of \( X \) and \( Y \).

### Analyzing the Assertions:
Now let's evaluate the assertions presented:

1. **Option 1**: \( X \) and \( Y \) are statistically independent.
   **True**: This statement is a direct consequence of perfect secrecy.

2. **Option 2**: \( H(X,Y) = H(X) \).
   **False**: This statement is incorrect. Under perfect secrecy, \( H(X,Y) = H(X) + H(Y) \), so it cannot equal \( H(X) \) alone.

3. **Option 3**: VAUDENAY can be the result of the encryption of ALPACINO using the Vernam cipher.
   **True**: This is true provided that the key used is as long as the plaintext and is truly random, which aligns with the properties of the Vernam cipher.

4. **Option 4**: \( H(X|Y) = H(X) \).
   **True**: This is indeed true and is a defining property of perfect secrecy.

### Conclusion:
The false assertion among the options provided is:

\[
\boxed{H(X,Y) = H(X)}
\]

This answer provides a clear and comprehensive understanding of perfect secrecy and evaluates the assertions based on the established properties of such ciphers.


Question: One of the following ciphers is a \emph{block} cipher. Which one?

Option 1: AES is the correct choice because it is a well-known block cipher that processes fixed-size blocks of data, while the other options are either stream ciphers or a hash function.


Question: In an interactive proof system for a language $L$, having $\beta$-soundness means that\dots

The answer is the first option: if we run the protocol with input \( x \not\in L \), with a malicious prover, and a honest verifier, the probability that the protocol succeeds is upper-bounded by \( \beta \). This aligns with the definition of soundness in interactive proof systems.


Question: Following the notation used in class, let us denote the set of terms by $T=\{k_i|i=1,...,m\}$, the set of documents by $D=\{d_j |j=1,...,n\}$, and let $d_i=(w_{1j},w_{2j},...,w_{mj})$. We are also given a query  $q=(w_{1q},w_{2q},...,w_{mq})$. In the lecture we studied that,   $sim(q,d_j) = \sum^m_{i=1} \frac{w_{ij}}{|d_j|}\frac{w_{iq}}{|q|}$ .  (1)  Another way of looking at the information retrieval problem is using a probabilistic approach. The probabilistic view of information retrieval consists of determining the conditional probability $P(q|d_j)$ that for a given document $d_j$ the query by the user is $q$. So, practically in probabilistic retrieval when a query $q$ is given, for each document it is evaluated how probable it is that the query is indeed relevant for the document, which results in a ranking of the documents.  In order to relate vector space retrieval to a probabilistic view of information retrieval, we interpret the weights in Equation (1) as follows:  -  $w_{ij}/|d_j|$ can be interpreted as the conditional probability $P(k_i|d_j)$ that for a given document $d_j$ the term $k_i$ is important (to characterize the document $d_j$).  -  $w_{iq}/|q|$ can be interpreted as the conditional probability $P(q|k_i)$ that for a given term $k_i$ the query posed by the user is $q$. Intuitively, $P(q|k_i)$ gives the amount of importance given to a particular term while querying.  With this interpretation you can rewrite Equation (1) as follows:  Show that indeed with the probabilistic interpretation of weights of vector space retrieval, as given in Equation (2), the similarity computation in vector space retrieval results exactly in the probabilistic interpretation of information retrieval, i.e., $sim(q,d_j)= P(q|d_j)$. Given that $d_j$ and $q$ are conditionally independent, i.e., $P(d_j \cap q|ki) = P(d_j|k_i)P(q|k_i)$. You can assume existence of joint probability density functions wherever required. (Hint: You might need to use Bayes theorem)

To derive \( P(q | d_j) \), we start with the definition of conditional probability:\n\n1. **Definition**: \n   \[\n   P(q | d_j) = \frac{P(q \cap d_j)}{P(d_j)}\n   \]\n\n2. **Apply Total Probability**:\n   \[\n   P(q \cap d_j) = \sum_{i=1}^m P(q \cap d_j | k_i) P(k_i)\n   \]\n\n3. **Use Conditional Independence**:\n   \[\n   = \sum_{i=1}^m P(q | k_i) P(d_j | k_i) P(k_i)\n   \]\n\n4. **Rearrange with Bayes' Theorem**:\n   \[\n   P(q | d_j) = \sum_{i=1}^m P(k_i | d_j) P(q | k_i)\n   \]\n\nThis reflects how students might misinterpret independence or the application of Bayes' theorem, leading to incorrect formulations.


Question: Let $N=3\cdot5\cdot13$. Then a quadratic residue in $\mathbb{Z}_N^*$ has

To analyze the number of square roots of a quadratic residue in \(\mathbb{Z}_N^*\) where \(N = 3 \cdot 5 \cdot 13\), we can use the fact that \(N\) is a product of distinct primes. The number of square roots of a quadratic residue in \(\mathbb{Z}_N^*\) is given by \(2^k\), where \(k\) is the number of distinct prime factors of \(N\).

In this case, \(N\) has three distinct prime factors: \(3\), \(5\), and \(13\). Therefore, \(k = 3\), and the number of square roots of a quadratic residue in \(\mathbb{Z}_N^*\) is \(2^3 = 8\).

Thus, the answer is:

**Correct Option: D. 8 square roots.**

Explanation: Since \(N\) has 3 distinct prime factors, a quadratic residue in \(\mathbb{Z}_N^*\) has \(2^3 = 8\) square roots.


Question: Diffie-Hellman refers to \ldots

The correct answer is **"a key-agreement protocol."**

### Rationale for the Correct Option

The Diffie-Hellman key exchange protocol is specifically designed for two parties to securely establish a shared secret key over an insecure communication channel. This protocol allows the two parties to generate a symmetric key that can be used for subsequent encryption of messages. The fundamental mechanism relies on the mathematical properties of modular exponentiation and the difficulty of the discrete logarithm problem, which ensures that even if an eavesdropper observes the exchanged values, they cannot easily deduce the shared secret.

The process operates as follows:
1. Both parties agree on a large prime number \( p \) and a base \( g \).
2. Each party selects a private key (a secret integer) — let’s call them \( a \) and \( b \).
3. They compute their public keys using the formula \( A = g^a \mod p \) and \( B = g^b \mod p \) and exchange these public keys.
4. Finally, each party computes the shared secret key by raising the received public key to their own private key, resulting in:
   - Party 1 computes \( s = B^a \mod p \)
   - Party 2 computes \( s = A^b \mod p \)

Both computations yield the same shared secret \( s \), which can then be used for encryption.

### Why the Other Options Are Incorrect

1. **"a signature scheme."**
   - A signature scheme is a cryptographic mechanism that allows a user to sign a message, providing authenticity and integrity to the message. Diffie-Hellman does not involve signing messages; instead, it focuses solely on establishing a shared secret for encryption. Therefore, this option is incorrect.

2. **"a public-key cryptosystem."**
   - While Diffie-Hellman involves public keys, it is not a public-key cryptosystem in the traditional sense like RSA or ElGamal, where encryption and decryption can both occur using a public key and a private key. Instead, Diffie-Hellman is primarily a key exchange mechanism. It does not encrypt or decrypt messages but rather facilitates the secure exchange of a symmetric key. Thus, this option does not accurately describe Diffie-Hellman.

3. **"the inventors of the RSA cryptosystem."**
   - This option is factually incorrect as it misrepresents the identities associated with different cryptographic schemes. The RSA cryptosystem was developed by Ron Rivest, Adi Shamir, and Leonard Adleman. Diffie-Hellman was created by Whitfield Diffie and Martin Hellman, who focused on key exchange rather than on the public-key cryptosystem itself. Therefore, this option is not related to the Diffie-Hellman protocol.

In summary, the Diffie-Hellman protocol is accurately characterized as a "key-agreement protocol," focusing on the secure exchange of keys rather than on signatures, public-key encryption, or authorship of the RSA scheme.


Question:
Assume you are part of a team developing a mobile app using Scrum. At the last sprint planning, you were assigned the task of adding a new authentication method. However, a customer representative just sent you an email:
"the representative believes authentication is less important than support for right-to-left languages, and would like you to work on that instead."
Explain in 1 sentence what you should do:

In response to the customer representative's request, you should continue focusing on the assigned authentication method for the current sprint and inform the Product Owner about the request for possible inclusion in the product backlog.

**Step-by-step breakdown:**

1. **Prioritize Sprint Goals**: Focus on completing the tasks committed to in the sprint backlog to maintain workflow.

2. **Acknowledge the Request**: Understand the significance of the customer’s feedback regarding right-to-left language support.

3. **Document the Feedback**: Note the customer’s request for accurate communication later.

4. **Inform the Product Owner**: Notify the Product Owner about this feedback, as they prioritize items based on customer needs.

5. **Continue Development**: Keep working on the authentication task to fulfill your sprint commitment.

This way, you adhere to Scrum practices while ensuring customer input is considered for future iterations.


Question: Suppose that you are given an insertion only stream of items. For every $k\geq 1$, give an algorithm that at each point in the stream maintains $k$ uniformly random elements from the prefix of the stream sampled without replacement. Your algorithm must use $O(k\log n)$ space.

To solve the problem of maintaining \( k \) uniformly random elements from an insertion-only stream while using \( O(k \log n) \) space, we can use a method based on reservoir sampling, but we need to adapt it to ensure we sample without replacement.

### Steps of the Algorithm:

1. **Initialization**:
   - We maintain a list \( S \) that will store our \( k \) sampled elements. Initially, \( S \) is empty.

2. **Processing the Stream**:
   - As each new item arrives in the stream (let's call the current item \( x \)), we will keep track of the position \( i \) (starting from 1 for the first item).
   - If \( |S| < k \) (i.e., we haven't yet filled our sample size), we simply add \( x \) to \( S \) and increment \( i \).

3. **Sampling Condition**:
   - If \( |S| = k \), we need to decide whether to replace an element in \( S \) with \( x \). To do this, generate a random integer \( j \) uniformly from the range \( [1, i] \).
   - If \( j \leq k \), replace the \( j \)-th element of \( S \) with \( x \).

4. **Uniformity Guarantee**:
   - This method ensures that each item in the stream has an equal probability of being included in the sample because:
     - Each of the first \( k \) items is included with certainty.
     - For subsequent items, they have a \( \frac{k}{i} \) chance of replacing an existing item in the sample, maintaining uniform probability across all items.

### Space Complexity:
- The space used for the list \( S \) to store \( k \) elements is \( O(k) \).
- We also need to maintain a counter \( i \) for the index of the stream, and a random number generator for selecting \( j \). These additional components use \( O(\log n) \) space (for the counter and random number generation).
- Thus, the total space complexity is \( O(k + \log n) \), which simplifies to \( O(k \log n) \).

### Conclusion:
This algorithm efficiently maintains \( k \) uniformly random samples from an insertion-only stream with a space complexity that meets the requirements. It guarantees that every element in the stream has an equal chance of being included in the sample without replacement.


Question: For a user that has not done any ratings, which method can make a prediction?

To determine which recommendation method can make a prediction for a user who has not provided any ratings, let's analyze each option:

**A) User-based collaborative RS**
- User-based collaborative filtering relies on the ratings and preferences of similar users to make predictions. If a user has not provided any ratings, there are no data points to find similar users, which means this method cannot make predictions. **Not applicable.**

**B) Item-based collaborative RS**
- Item-based collaborative filtering focuses on the relationships between items based on the ratings given by users. It looks at the similarity between items rather than users. However, to make predictions, it still requires some ratings from the target user to find relevant items that are similar to those already rated by other users. Since the user has no ratings, this method also cannot make predictions. **Not applicable.**

**C) Content-based RS**
- Content-based recommendation systems make predictions based on the characteristics of the items themselves rather than relying on user ratings. If the system has information about the features of items (e.g., genre, author, or attributes), it can suggest items based on the user's profile or preferences. However, for a completely new user with no ratings or profile, there might be insufficient information to generate recommendations. Still, if the system can analyze item features, it might provide recommendations based on general popularity or item attributes. **Potentially applicable, depending on system design.**

**D) None of the above**
- This option suggests that none of the recommended systems can make predictions. Since content-based systems can sometimes offer recommendations based on item features, this option is not entirely accurate.

**Conclusion:**
In scenarios where a user has not provided any ratings, **C) Content-based RS** is the most viable option since it can potentially leverage item attributes to make predictions, even if it lacks user-specific data. Therefore, the correct answer is:

**C) Content-based RS**.


Question: Tick the \textbf{false} assertion. In Differential Cryptanalysis, the corresponding differential circuit of \ldots

To address the question systematically, let’s review each assertion and analyze their validity within the context of differential cryptanalysis, which focuses on how differences in input can affect differences in output through various types of gates in a circuit.

1. **Linear Circuit ($Y=M\times X$)**:
   - The assertion states that if the input difference is \(\Delta X=a\), then the output difference is given by \(\Delta Y=^tM\times a\).
   - This assertion is **true**. In linear circuits, the relationship between the input and output differences can be expressed linearly, and the transpose of the matrix \(M\) is used to relate the differences appropriately.

2. **Duplicate Gate ($X=Y=Z$)**:
   - Here, the assertion claims that if the input difference is \(\Delta X=a\), then both outputs \(\Delta Y\) and \(\Delta Z\) will also equal \(a\).
   - This assertion is **true**. Since the output is merely a duplicate of the input, any change (or difference) in the input will be reflected directly in the outputs.

3. **XOR Gate ($X \oplus Y = Z$)**:
   - The assertion states that if there are input differences \(\Delta X=a\) and \(\Delta Y=b\), then the output difference is \(\Delta Z = a \oplus b\).
   - This assertion is **true**. The XOR operation’s nature is such that it combines the differences of the two inputs, and the output difference is indeed the XOR of the two input differences.

4. **XOR to Constant Gate ($Y=X \oplus K$)**:
   - The assertion claims that if the input difference is \(\Delta X = a\), then the output difference \(\Delta Y\) also equals \(a\).
   - This assertion is **false**. The reason is that when you XOR an input \(X\) with a constant \(K\), the output difference does not directly equal the input difference. Instead, we need to consider how the constant \(K\) affects the output. If \(K\) is constant, then \(\Delta Y\) will equal \(\Delta X\) only if \(K\) does not contribute any difference, which is not guaranteed. In fact, if \(K\) is such that it introduces a change, then \(\Delta Y\) does not equal \(a\).

### Conclusion
Thus, the **false assertion** is the one related to the XOR to constant gate: "\(\Delta X = a \Rightarrow \Delta Y = a\)." This misrepresents the behavior of the XOR operation with a constant, illustrating the importance of carefully considering how constants can influence differences in cryptographic operations.


Question:
Assume that you are part of a team developing a mobile app using Scrum.
When using the app, you identified multiple bugs and features which you think should be implemented, and took some notes. You want to
share these with the Product Owner. Your backlog of tasks includes the following task:
- [ ] [Bug] When I click on the home button, the app doesn't redirect me there (tested from multiple locations in the app).
Is this item suitable to be submitted to the Product Backlog? Why?

Yes, the item is suitable for submission to the Product Backlog because it clearly describes a bug in the app's functionality, indicating the observed issue and suggesting a necessary improvement for the user experience. It would be beneficial to include steps to reproduce the bug and the expected behavior for further clarity, but the current description is a solid start.


Question: We apply a Gaussian Mixture Model made of $K$ isotropic Gaussians (invariant to rotation around its center) to $N$ vectors of dimension $D$. What is the number of \emph{free} parameters of this model?

To find the number of free parameters in a Gaussian Mixture Model (GMM) made of \( K \) isotropic Gaussians applied to \( N \) vectors of dimension \( D \), we need to consider the parameters for the mixture model itself.

### Parameters of a GMM:

1. **Means**: Each of the \( K \) Gaussian components has a mean vector in \( \mathbb{R}^D \). Thus, the total number of parameters for the means is:
   \[
   K \times D
   \]

2. **Covariances**: Since we are dealing with isotropic Gaussians, each Gaussian has a single variance parameter (which is the same in all directions) instead of a full covariance matrix. Therefore, for \( K \) isotropic Gaussians, we have \( K \) variance parameters.

3. **Mixing Weights**: The GMM has \( K \) mixing weights that define the proportion of each Gaussian in the mixture. These weights must sum to 1, which introduces a constraint. Therefore, we have \( K - 1 \) free parameters for the mixing weights.

### Total Number of Free Parameters:

Combining these contributions, we have:
- \( K \times D \) (mean parameters)
- \( K \) (variance parameters)
- \( K - 1 \) (mixing weights)

The total number of free parameters \( P \) in the model is:
\[
P = KD + K + (K - 1)
\]
Simplifying this, we get:
\[
P = KD + K + K - 1 = KD + 2K - 1
\]

### Conclusion:

Thus, the number of free parameters in the GMM is:

\[
\boxed{KD + 2K - 1}
\]


Question:
Which of the following statement(s) is/are true about different types of
    coverage for coverage-guided fuzzing?

To analyze the statements regarding coverage types in coverage-guided fuzzing, let's evaluate each option carefully:

**Option A: "If you cover all edges, you also cover all blocks."**
- **Evaluation**: This statement is **true**. In a control flow graph (CFG), edges represent transitions between blocks (or nodes). If all edges are covered, it means that every possible transition from one block to another has been executed. Since every block has at least one outgoing edge (if it is not a terminal block), covering all edges ensures that all blocks have been reached. Therefore, if all edges are covered, all blocks must also be covered.

**Option B: "Full line/statement coverage means that every possible control flow through the target has been covered."**
- **Evaluation**: This statement is **false**. Full line or statement coverage only ensures that every line of code in the program has been executed at least once. It does not guarantee that all possible paths or control flows through the program have been tested. There could be branches and conditions that are not exercised, which means some control flows may remain untested even if every line is executed.

**Option C: "Full data flow coverage is easier to obtain than full edge coverage."**
- **Evaluation**: This statement is **false**. Full data flow coverage requires that all definitions and uses of variables must be exercised correctly, which can be quite complex, especially in programs with many variables and complex data dependencies. Full edge coverage, while also challenging, generally requires executing all paths through the control flow graph, which can be more straightforward in terms of identifying which edges need to be traversed. Thus, obtaining full edge coverage is typically considered a more feasible goal than achieving full data flow coverage.

**Option D: "Full edge coverage is equivalent to full path coverage because every possible basic block transition has been covered."**
- **Evaluation**: This statement is **false**. Full edge coverage ensures that all edges in a control flow graph are traversed, but this does not imply that every possible path has been executed. A path can consist of multiple edges and may include various branches and loops that could be traversed in different ways. Full path coverage requires executing every possible path from entry to exit, which is a much stronger condition and often impractical for non-trivial programs due to the exponential number of paths.

### Summary
Based on the evaluations above, the correct answer is **Option A**: "If you cover all edges, you also cover all blocks." This statement is accurate as it reflects the fundamental relationship between edges and blocks in control flow graphs. The other options misrepresent the relationships and complexities involved in different types of code coverage.


Question: If the top 100 documents contain 50 relevant documents

Let's analyze each option step-by-step based on the definitions and the given scenario:

1. **Precision at K** is calculated as the number of relevant documents among the top K results divided by K.

2. **Recall** is calculated as the number of relevant documents retrieved divided by the total number of relevant documents in the entire collection.

From the problem statement, we know:
- The top 100 documents contain **50 relevant documents**.
- We are not given the total number of relevant documents in the entire collection, which is crucial for calculating recall.

Now, let's analyze each option:

**Option 1: The precision of the system at 50 is 0.25**
- Precision at 50 would be calculated as the number of relevant documents in the top 50 divided by 50.
- We do not know how many relevant documents are in the top 50; the statement does not provide this information.
- Therefore, we cannot confirm that the precision at 50 is 0.25, making this option **incorrect**.

**Option 2: The precision of the system at 100 is 0.5**
- Precision at 100 is calculated as:
  \[
  \text{Precision at 100} = \frac{\text{Number of relevant documents in top 100}}{100} = \frac{50}{100} = 0.5
  \]
- This calculation is correct based on the information provided, making this option **correct**.

**Option 3: The recall of the system is 0.5**
- Recall is calculated as:
  \[
  \text{Recall} = \frac{\text{Number of relevant documents retrieved}}{\text{Total number of relevant documents in the collection}}
  \]
- We know that 50 relevant documents are retrieved (from the top 100), but we do not know the total number of relevant documents in the entire collection. Hence, we cannot say that the recall is 0.5 without that information. This option is therefore **incorrect**.

**Option 4: All of the above**
- Since we have established that Option 1 is incorrect, and Option 3 is also incorrect while only Option 2 is correct, this option is **incorrect**.

In conclusion, the correct option is **Option 2: the precision of the system at 100 is 0.5**. The other options are incorrect due to either lack of information or incorrect calculations.


Question: Many general evaluation metrics can be considered for various NLP tasks. The simplest one is accuracy. Give several examples of NLP tasks for which accuracy can be used as an evaluation metric. Justify why. In general, what property(ies) must an NLP task satisfy in order to be evaluable through accuracy?

Accuracy can be used as an evaluation metric for tasks like Part-of-Speech tagging, Information Retrieval, Text Classification, and Information Extraction because these tasks involve clear correct and incorrect predictions that can be compared against a reference. For accuracy to be meaningful, there must be a defined standard to determine what counts as correct or incorrect in the predictions made by the model.


Question: The statistical distance between two distributions is \dots

The statistical distance between two distributions is a measure that quantifies how distinguishable the two distributions are from each other in terms of the probabilities they assign to the same events. Mathematically, it is often represented as the total variation distance, which captures the maximum difference in probabilities for any event.

In the context of distinguishers, which are algorithms designed to differentiate between two distributions (often a target distribution and a uniform or random distribution), the statistical distance provides critical information about the performance of these distinguishers:

1. A small statistical distance indicates that the distributions are very similar, meaning that any distinguisher will have a limited ability to differentiate between them. This results in a low advantage for distinguishers.
2. A larger statistical distance indicates a greater divergence between the distributions, which allows for a more significant advantage for distinguishers, making them better at identifying which distribution a sample comes from.

Importantly, the statistical distance serves as a theoretical bound on the ability of all possible distinguishers to tell apart the two distributions. Specifically, when we consider the case of making decisions based on a single sample from each distribution, the statistical distance can be viewed as:

**an upper bound on the advantage of all distinguishers making statistics on the obtained samples.**

This means that no distinguisher can achieve an advantage greater than this statistical distance, encapsulating the maximum potential of distinguishers in distinguishing between the two distributions based on a single sample.


Question: Design and analyze a polynomial time algorithm for the following problem: \begin{description} \item[INPUT:] An undirected graph $G=(V,E)$. \item[OUTPUT:] A non-negative vertex potential $p(v)\geq 0$ for each vertex $v\in V$ such that \begin{align*} \sum_{v\in S} p(v) \leq |E(S, \bar S)| \quad \mbox{for every $\emptyset \neq S \subsetneq V$ \quad and \quad $\sum_{v\in V} p(v)$ is maximized.} \end{align*} \end{description} {\small (Recall that $E(S, \bar S)$ denotes the set of edges that cross the cut defined by $S$, i.e., $E(S, \bar S) = \{e\in E: |e\cap S| = |e\cap \bar S| = 1\}$.)} \\[1mm] \noindent Hint: formulate the problem as a large linear program (LP) and then show that the LP can be solved in polynomial time.  \\[1mm] {\em (In this problem you are asked to (i) design the algorithm, (ii) show that it returns a correct solution and that it runs in polynomial time.   Recall that you are allowed to refer to material covered in the course.) }

To solve the problem of finding a non-negative vertex potential \( p(v) \geq 0 \) for each vertex \( v \in V \) in an undirected graph \( G = (V, E) \) such that:

\[
\sum_{v \in S} p(v) \leq |E(S, \bar{S})| \quad \text{for every } \emptyset \neq S \subsetneq V,
\]

and maximizing \( \sum_{v \in V} p(v) \), we can leverage the framework of submodular function minimization.

### Step 1: Understanding the Problem

The goal is to assign non-negative potentials to vertices while satisfying the edge-cut constraints for all non-empty proper subsets \( S \). The function we want to maximize, \( \sum_{v \in V} p(v) \), can be viewed as a submodular function.

### Step 2: Define the Submodular Function

We can define a submodular function \( f: 2^V \to \mathbb{R} \) as follows:

\[
f(S) = |E(S, \bar{S})| - \sum_{v \in S} p(v).
\]

This function \( f(S) \) captures the relationship of the vertex potentials with respect to the edges crossing the cut defined by \( S \). The function is submodular because it satisfies the diminishing returns property, which means adding an element to a smaller set provides at least as much benefit (or less cost) than adding it to a larger set.

### Step 3: Setting Up the Optimization Problem

We need to maximize the sum of potentials \( \sum_{v \in V} p(v) \) subject to the constraints derived from the submodular function. The constraints can be expressed as:

\[
\sum_{v \in S} p(v) \leq |E(S, \bar{S})| \quad \text{for all } \emptyset \neq S \subsetneq V.
\]

### Step 4: Solving the Submodular Function Minimization

Submodular function minimization is known to be solvable in polynomial time. The approach involves using specialized algorithms such as the **Greedy algorithm** or the **Bundle method**, which efficiently find the minimum of a submodular function while satisfying the given constraints.

### Step 5: Evaluating the Submodular Function

We can evaluate our submodular function \( f(S) \) in polynomial time since it relies on counting the edges in the graph, which can be done in linear time relative to the number of edges and vertices in the graph.

### Step 6: Separation Problem

To ensure that the solution satisfies the constraints, we can formulate a separation problem that checks whether the current potential assignment violates any of the constraints. If it does, we can efficiently find a violating subset \( S \) that helps adjust the potentials accordingly. The separation problem can also be solved in polynomial time due to the nature of submodular functions.

### Step 7: Conclusion

By combining submodular function minimization with efficient evaluation of the function and its separation, we can construct a polynomial-time algorithm that finds the required vertex potentials \( p(v) \). The algorithm guarantees that all constraints are satisfied and maximizes the total potential across the vertices.

In summary, the steps to solve the problem are:
1. Define the submodular function based on the edge-cut constraints and vertex potentials.
2. Set up the optimization problem to maximize the sum of potentials with the defined constraints.
3. Use polynomial-time algorithms for submodular function minimization.
4. Evaluate the submodular function efficiently.
5. Solve the separation problem to maintain feasibility.

Thus, we have a clear and efficient method to determine the non-negative vertex potentials that meet the specified requirements.


Question: If there are {t} PoS tags, what is the maximum number of (not necessarily free) parameters the probabilistic model needs to consider to determine the best possible PoS tag sequence given a word sequence of length {w}, subjected to the limited lexical conditioning and limited scope for syntactic dependencies (1 neighbor) hypotheses.Give your answer as a numerical value (not as a formula).

To determine the maximum number of parameters required for a probabilistic model that predicts the best possible Part-of-Speech (PoS) tag sequence given a word sequence, we will consider the following aspects:

1. **Word Sequence Length (w)**: We have a sequence of words of length \( w \).

2. **PoS Tags (t)**: There are \( t \) different PoS tags available for assignment to the words in the sequence.

3. **Limited Lexical Conditioning**: This implies that the PoS tag for each word can depend on the word itself and possibly its neighboring words. In this case, since we have a limited scope for syntactic dependencies (1 neighbor), each word can depend on itself and its immediate neighbor (either before or after it).

4. **Parameters Calculation**:
   - Each word in the sequence can independently take on any of the \( t \) tags.
   - For each position in the word sequence, we can have \( t \) possible tags, and since we consider the neighboring dependency (1 neighbor), we need to consider the transitions between tags.
   - The transitions can be modeled as a Markov model, where the current state (tag) depends only on the previous state (tag).

Given these points, the parameters can be thought of in terms of:
- The number of possible tag assignments for each word.
- The number of transitions between tags for neighboring words.

5. **Calculating Parameters**:
   - For each of the \( w \) words, there are \( t \) possible tags.
   - Thus, the number of parameters for the emissions (tag for each word) is \( w \times t \).
   - For the transitions, since each tag can transition to any of the \( t \) tags based on its neighbor, the number of transition parameters is \( t^2 \) (each tag can go to any of the \( t \) tags).

Combining both emission and transition parameters:
- Emission parameters: \( w \times t \)
- Transition parameters: \( t^2 \)

Total parameters = Emission parameters + Transition parameters = \( w \times t + t^2 \).

6. **Maximizing Parameters**:
To find the maximum number of parameters, we need to sum the contributions from both emissions and transitions.

Thus, the total number of parameters is \( w \times t + t^2 \).

Since \( w \) and \( t \) are not specified in the problem, we will consider the maximum number of parameters based on the fact that \( w \) is the length of the sequence which is a variable.

Assuming we are looking for the maximum number of parameters given the constraints and considering both the emission probabilities and the transition probabilities.

Thus, the answer can be summarized as follows:

```json
{
  "result": {
    "reasoning": "The maximum number of parameters required for the probabilistic model is calculated based on the emissions (w * t) and transitions (t^2) for the tag sequences given the word sequence of length w and t different PoS tags.",
    "answer": "wt + t^2"
  }
}
```

If we were to specify an example with numerical values for \( w \) and \( t \), we could substitute those values to give a concrete numerical answer. However, in this case, the expression \( wt + t^2 \) serves as the answer reflecting the relationship between parameters and the given constraints.


Question: In which of the following groups is the decisional Diffie-Hellman problem (DDH) believed to be hard?

The decisional Diffie-Hellman problem (DDH) is a foundational problem in cryptography, particularly in public key systems. It is believed to be hard in certain groups, making them suitable for secure cryptographic applications. The groups where DDH is typically considered hard include those where the underlying mathematical structures resist efficient algorithms for solving the problem.

Among the given options, the groups can be analyzed as follows:
1. In $\mathbb{Z}_p$, with a large prime $p$: DDH is generally considered hard in this group.
2. In large subgroup of smooth order of a "regular" elliptic curve: DDH is also hard here due to the properties of elliptic curves.
3. In a large subgroup of prime order of $\mathbb{Z}_p^*$: DDH is typically hard in this structure as well.
4. In $\mathbb{Z}_p^*$, with a large prime $p$: While it may seem feasible, DDH is less secure in this group compared to the others listed.

Thus, the most secure and commonly accepted hard groups for DDH are options 1, 2, and 3. However, option 3 specifically refers to a large subgroup of prime order, which is crucial for ensuring hardness.

Therefore, the answer is option 3.

3


Question: Assume your colleague wants to wait until the next minor release to include a major bugfix instead of making a bugfix release. Explain why this is not a good idea.

Answer: Waiting for the next minor release to include a major bugfix can lead to prolonged exposure to the bug for users, potentially causing significant issues or dissatisfaction. Additionally, bundling critical fixes with new features may complicate the release process, increasing the risk of introducing new bugs and delaying the resolution of the existing problem. It is generally better to prioritize critical fixes in a timely manner to maintain user trust and system stability.


Question: Select the \emph{incorrect} statement. The discrete logarithm

A. The discrete logarithm can be solved by a polynomial algorithm is incorrect because, in general scenarios, especially in cryptography, there is no known polynomial time algorithm to solve this problem. I chose option (A) because it incorrectly asserts that the discrete logarithm can be solved by a polynomial algorithm. In the context of cryptography, particularly in the multiplicative group \(\mathbb{Z}_p^*\), the discrete logarithm problem is widely recognized as being computationally hard, and no polynomial time algorithm has been discovered for solving it efficiently. This characteristic is fundamental to the security of various cryptographic protocols. Therefore, option (A) stands out as the incorrect statement among the provided options.


Question: Briefly describe the specific objectives of the morphological module in the general perspective of automated Natural Language Processing.

The morphological module in automated Natural Language Processing (NLP) serves several specific objectives that are crucial for effective language understanding and generation.

1. **Word Structure Analysis**: One of the primary objectives is to analyze the internal structure of words. This involves breaking down words into their constituent morphemes, which are the smallest units of meaning. By identifying roots, prefixes, and suffixes, the module can understand how words are formed and how their meanings change with different morphological modifications.

2. **Lexical Normalization**: The morphological module also aims to normalize words to their base or dictionary forms, known as lemmas. This process is essential for reducing the complexity of language data, allowing subsequent NLP tasks (like parsing or semantic analysis) to operate on a standardized representation of words.

3. **Inflection and Derivation Handling**: Another objective is to manage inflected forms of words (such as verb conjugations, noun plurals, etc.) and derived forms (such as adjectives from nouns). This handling is crucial for ensuring that NLP systems can recognize and process various forms of a word consistently, which is especially important in languages with rich morphological systems.

4. **Part-of-Speech Tagging**: The morphological module contributes to part-of-speech (POS) tagging by providing insights into the grammatical function of words based on their morphological characteristics. This is vital for understanding sentence structure and meaning.

5. **Cross-Linguistic Compatibility**: Given the diversity of languages and their morphological rules, this module helps in creating models that can adapt to different languages' morphological systems. This objective is particularly important for building multilingual NLP applications.

6. **Error Detection and Correction**: Lastly, the morphological module can help in identifying and correcting morphological errors in text, such as incorrect verb forms or noun-adjective agreement. This enhances the overall quality of text processing and understanding in NLP systems.

In summary, the morphological module plays a foundational role in NLP by analyzing, normalizing, and processing the complex structures of words, enabling more accurate and nuanced language understanding and generation.


Question:
Consider the following sentence:
High-energy pulsed laser beams are used in soft-tissue surgery.
Using a 1-gram language model and a tokenizer that splits on whitespaces and punctuation (including hyphens (-)), assume that the tokenization is now enhanced with Named Entity Recognition (NER) specialized on technical and medical terms.
What would be the advantage of doing so? What would be the major drawback? Justify your answers.

**Advantage:** Enhancing the tokenization with Named Entity Recognition (NER) improves probability estimation and inference by treating technical and medical terms as single entities rather than independent tokens. This allows the model to capture contextual relationships between words, leading to more accurate predictions of term co-occurrences and a better understanding of domain-specific language. If sufficient training data is available, this can significantly enhance performance.

**Drawback:** The major drawback is that NER can introduce errors in entity recognition, which may misclassify or overlook important terms. Making decisions based on potentially incorrect NER outputs could propagate these errors through subsequent processing stages. Thus, it may be more beneficial to retain all possible interpretations for later analysis rather than prematurely narrowing down options based on potentially flawed recognitions.


Question: Your friend Charlie was training a neural net, and observed something very curious. During training, the same network, with the same exact weights, doesn't always give the same prediction for a given example. What can cause this behavior?

**Batch normalization, Dropout**

Batch normalization and dropout can introduce randomness during training, which can lead to different predictions even with the same weights. Batch normalization normalizes the input of each layer based on the current batch, which can vary from one training iteration to another. Dropout randomly ignores some neurons during training, adding another layer of randomness that can cause variations in predictions.