question
stringlengths 6
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|---|---|---|
In which group is the discrete logarithm problem believed to be hard?
|
[
"In a subgroup of $\\mathbb{Z}_p^*$ with large prime order.",
"In $\\mathbb{Z}_n$, where $n= pq$ for two large primes $p$ and $q$.",
"In a group $G$ of smooth order.",
"In $\\mathbb{Z}_2^p$, for a large prime $p$."
] |
A
|
Consider two distributions $P_0,P_1$ with the same supports and a distinguisher $\mathcal{A}$ that makes $q$ queries. Tick the \textit{incorrect} assertion.
|
[
"When $q=1$, $\\mathsf{Adv}(\\mathcal{A})\\leq d(P_0,P_1)$ where $d$ is the statistical distance.",
"When $q>1$, $\\mathsf{Adv}(\\mathcal{A})\\leq \\frac{d(P_0,P_1)}{q}$ where $d$ is the statistical distance.",
"When $q=1$, the strategy ``return 1 $\\Leftrightarrow \\frac{P_0(x)}{P_1(x)}\\leq 1$'' achieves the best advantage.",
"To achieve good advantage, we need to have $q\\approx 1/C(P_0,P_1)$ where $C$ is the Chernoff information."
] |
B
|
What is the complexity of prime number generation for a prime of length $\ell$?
|
[
"$\\mathbf{O}\\left(\\frac{1}{\\ell^4}\\right)$",
"$\\mathbf{O}(\\ell^4)$",
"$\\Theta\\left(\\frac{1}{\\ell^4}\\right)$",
"$\\Theta(\\ell^4)$"
] |
B
|
In ElGamal signature scheme, if we avoid checking that $0 \leq r < p$ then \ldots
|
[
"\\ldots a universal forgery attack is possible.",
"\\ldots an existential forgery attack is avoided.",
"\\ldots we can recover the secret key.",
"\\ldots we need to put a stamp on the message."
] |
A
|
Tick the \textbf{true} assertion. MAC is \ldots
|
[
"\\ldots a computer.",
"\\ldots the name of a dish with chili.",
"\\ldots a Message Authentication Code.",
"\\ldots the encryption of KEY with the Ceasar cipher."
] |
C
|
For $K$ a field, $a,b\in K$ with $4a^3+27b^2 \neq 0$, $E_{a,b}(K)$ is
|
[
"a field.",
"a group.",
"a ring.",
"a ciphertext."
] |
B
|
The number of plaintext/ciphertext pairs required for a differential cryptanalysis is\dots
|
[
"$\\approx DP$",
"$\\approx \\frac{1}{DP}$",
"$\\approx \\frac{1}{DP^2}$",
"$\\approx \\log \\frac{1}{DP}$"
] |
B
|
Given the distribution $P_0$ of a normal coin, i.e. $P_0(0)=P_0(1)=\frac{1}{2}$, and distribution $P_1$ of a biased coin, where $P_1(0)=\frac{1}{3}$ and $P_1(1) = \frac{2}{3}$ , the maximal advantage of a distinguisher using a single sample is\dots
|
[
"$\\frac{1}{6}$.",
"$3$.",
"$\\frac{1}{3}$.",
"$0$."
] |
A
|
To how many plaintexts we expect to decrypt a ciphertext in the Rabin cryptosystem when we don't use redundancy?
|
[
"4.",
"2.",
"1.",
"8."
] |
A
|
For an interactive proof system, the difference between perfect, statistical and computational zero-knowledge is based on \ldots
|
[
"\\ldots the distinguishability between some distributions.",
"\\ldots the percentage of recoverable information from a transcript with a honest verifier.",
"\\ldots the number of times the protocol is run between the prover and the verifier.",
"\\ldots whether the inputs are taken in $\\mathcal{P}$, $\\mathcal{NP}$ or $\\mathcal{IP}$."
] |
A
|
What is the name of the encryption threat that corresponds to \emph{force the sender to encrypt some messages selected by the adversary}?
|
[
"Chosen Ciphertext Attack",
"Chosen Plaintext Attack",
"Known Ciphertext Attack",
"Known Plaintext Attack"
] |
B
|
Let $C$ be a perfect cipher with $\ell$-bit blocks. Then, \dots
|
[
"for $x_1 \\neq x_2$, $\\Pr[C(x_1) = y_1, C(x_2)=y_2] = \\frac{1}{2^{2\\ell}}$.",
"the size of the key space of $C$ should be at least $(2^{\\ell}!)$.",
"given pairwise independent inputs to $C$, the corresponding outputs are independent and uniformly distributed.",
"$C$ has an order $3$ decorrelation matrix which is equal to the order $3$ decorrelation matrix of a random function."
] |
B
|
The exponent of the group $\mathbb{Z}_9^*$ is
|
[
"6.",
"9.",
"8.",
"3."
] |
A
|
Tick the \emph{incorrect} statement. When $x\rightarrow+\infty$ \ldots
|
[
"$x^3 + 2x + 5 = \\mathcal{O}(x^3)$.",
"$\\frac{1}{x^2} = \\mathcal{O}(\\frac{1}{x})$.",
"$2^{\\frac{x}{\\log x}} = \\mathcal{O}(2^x)$.",
"$n^x = \\mathcal{O}(x^n)$ for any constant $n>1$."
] |
D
|
In order to avoid the Bleichenbacher attack in ElGamal signatures, \ldots
|
[
"\\ldots authors should put their name in the message.",
"\\ldots groups of prime order should be used.",
"\\ldots groups of even order should be used.",
"\\ldots groups with exponential number of elements should be used."
] |
B
|
Tick the \textbf{incorrect} assumption. A language $L$ is in NP if\dots
|
[
"$x \\in L$ can be decided in polynomial time.",
"$x \\in L$ can be decided in polynomial time given a witness $w$.",
"$L$ is NP-hard.",
"$L$ (Turing-)reduces to a language $L_2$ with $L_2$ in $P$, i.e., if there is a polynomial deterministic Turing machine which recognizes $L$ when plugged to an oracle recognizing $L_2$."
] |
C
|
In which of the following groups is the decisional Diffie-Hellman problem (DDH) believed to be hard?
|
[
"In $\\mathbb{Z}_p$, with a large prime $p$.",
"In large subgroup of smooth order of a ``regular'' elliptic curve.",
"In a large subgroup of prime order of $\\mathbb{Z}_p^*$, such that $p$ is a large prime.",
"In $\\mathbb{Z}_p^*$, with a large prime $p$."
] |
C
|
In ElGamal signature scheme and over the random choice of the public parameters in the random oracle model (provided that the DLP is hard), existential forgery is \ldots
|
[
"\\ldots impossible.",
"\\ldots hard on average.",
"\\ldots easy on average.",
"\\ldots easy."
] |
B
|
Which of the following primitives \textit{cannot} be instantiated with a cryptographic hash function?
|
[
"A pseudo-random number generator.",
"A commitment scheme.",
"A public key encryption scheme.",
"A key-derivation function."
] |
C
|
In plain ElGamal Encryption scheme \ldots
|
[
"\\ldots only a confidential channel is needed.",
"\\ldots only an authenticated channel is needed.",
"\\ldots only an integer channel is needed",
"\\ldots only an authenticated and integer channel is needed."
] |
D
|
Tick the \textbf{true} assertion. Let $X$ be a random variable defined by the visible face showing up when throwing a dice. Its expected value $E(X)$ is:
|
[
"3.5",
"3",
"1",
"4"
] |
A
|
Consider an arbitrary cipher $C$ and a uniformly distributed random permutation $C^*$ on $\{0,1\}^n$. Tick the \textbf{false} assertion.
|
[
"$\\mathsf{Dec}^1(C)=0$ implies $C=C^*$.",
"$\\mathsf{Dec}^1(C)=0$ implies $[C]^1=[C^*]^1$.",
"$\\mathsf{Dec}^1(C)=0$ implies that $C$ is perfectly decorrelated at order 1.",
"$\\mathsf{Dec}^1(C)=0$ implies that all coefficients in $[C]^1$ are equal to $\\frac{1}{2^n}$."
] |
A
|
Tick the \textit{incorrect} assertion. Let $P, V$ be an interactive system for a language $L\in \mathcal{NP}$.
|
[
"The proof system is $\\beta$-sound if $\\Pr[\\text{Out}_{V}(P^* \\xleftrightarrow{x} V) = \\text{accept}] \\leq \\beta$ for any $P^*$ and any $x \\notin L$.",
"The soundness of the proof system can always be tuned close to $0$ by sequential composition.",
"It is impossible for the proof system to be sound and zero knowledge at the same time.",
"Both the verifier $V$ and the prover $P$ run in time that is polynomial in $|x|$, if we assume that $P$ gets the witness $w$ as an extra input."
] |
C
|
Tick the assertion related to an open problem.
|
[
"$NP\\subseteq IP$.",
"$P\\subseteq IP$.",
"$PSPACE=IP$.",
"$NP = \\text{co-}NP$."
] |
D
|
Let $X$ be a plaintext and $Y$ its ciphertext. Which statement is \textbf{not} equivalent to the others?
|
[
"the encyption scheme provides perfect secrecy",
"only a quantum computer can retrieve $X$ given $Y$",
"$X$ and $Y$ are statistically independent random variables",
"the conditionnal entropy of $X$ given $Y$ is equal to the entropy of $X$"
] |
B
|
Tick the \textbf{\emph{incorrect}} assertion. A $\Sigma$-protocol \dots
|
[
"has special soundness.",
"is zero-knowledge.",
"is a 3-move interaction.",
"has the verifier polynomially bounded."
] |
B
|
The statistical distance between two distributions is \dots
|
[
"unrelated to the advantage of a distinguisher.",
"a lower bound on the advantage of \\emph{all} distinguishers (with a unique sample).",
"an upper bound on the advantage of \\emph{all} distinguishers (with a unique sample).",
"an upper bound on the advantage of all distinguishers making statistics on the obtained samples."
] |
C
|
Tick the \textbf{\emph{correct}} assertion. A random oracle $\ldots$
|
[
"returns the same answer when queried with two different values.",
"is instantiated with a hash function in practice.",
"has predictable output before any query is made.",
"answers with random values that are always independent of the previous queries."
] |
B
|
Consider an Sbox $S:\{0,1\}^m \rightarrow \{0,1\}^m$. We have that \ldots
|
[
"$\\mathsf{DP}^S(0,b)=1$ if and only if $S$ is a permutation.",
"$\\sum_{b\\in \\{0,1\\}^m} \\mathsf{DP}^S(a,b)$ is even.",
"$\\sum_{b\\in \\{0,1\\}^m \\backslash \\{0\\}} \\mathsf{DP}^S(0,b)= 0$",
"$\\mathsf{DP}^S(0,b)=1$ if and only if $m$ is odd."
] |
C
|
Tick the \textbf{true} assertion. In RSA \ldots
|
[
"\\ldots decryption is known to be equivalent to factoring.",
"\\ldots key recovery is provably not equivalent to factoring).",
"\\ldots decryption is probabilistic.",
"\\ldots public key transmission needs authenticated and integer channel."
] |
D
|
Consider the cipher defined by $$\begin{array}{llll} C : & \{0,1\}^{4} & \rightarrow & \{0,1\}^{4} \\ & x & \mapsto & C(x)=x \oplus 0110 \\ \end{array} $$ The value $LP^C(1,1)$ is equal to
|
[
"$0$",
"$1/4$",
"$1/2$",
"$1$"
] |
D
|
Let $n=pq$ be a RSA modulus and let $(e,d)$ be a RSA public/private key. Tick the \emph{correct} assertion.
|
[
"Finding a multiple of $\\lambda(n)$ is equivalent to decrypt a ciphertext.",
"$ed$ is a multiple of $\\phi(n)$.",
"The two roots of the equation $X^2 - (n-\\phi(n)+1)X+n$ in $\\mathbb{Z}$ are $p$ and $q$.",
"$e$ is the inverse of $d$ mod $n$."
] |
C
|
Tick the \emph{true} assertion. A distinguishing attack against a block cipher\dots
|
[
"is a probabilistic attack.",
"succeeds with probability $1$.",
"outputs the secret key.",
"succeeds with probability $0$."
] |
A
|
Tick the \textbf{true} assertion. Let $n >1 $ be a composite integer, the product of two primes. Then,
|
[
"$\\phi(n)$ divides $\\lambda(n)$.",
"$\\lambda(n)$ divides the order of any element $a$ in $\\mathbb{Z}_n$.",
"$\\mathbb{Z}^{*}_n$ with the multiplication is a cyclic group.",
"$a^{\\lambda(n)} \\mod n=1$, for all $a \\in \\mathbb{Z}^{*}_n$."
] |
D
|
Let $C$ be a permutation over $\left\{ 0,1 \right\}^p$. Tick the \emph{incorrect} assertion:
|
[
"$\\text{DP}^C(a,0) = 1$ for some $a \\neq 0$.",
"$\\text{DP}^C(0,b) = 0$ for some $b \\neq 0$.",
"$\\sum_{b \\in \\left\\{ 0,1 \\right\\}^p}\\text{DP}^C(a,b) = 1$ for any $a\\in \\left\\{ 0,1 \\right\\}^p$.",
"$2^p \\text{DP}^C(a,b) \\bmod 2 = 0$, for any $a,b\\in \\left\\{ 0,1 \\right\\}^p$."
] |
A
|
Tick the \textbf{true} assertion. Assume an arbitrary $f:\{0,1\}^p \rightarrow \{0,1\}^q$, where $p$ and $q$ are integers.
|
[
"$\\mathsf{DP}^f(a,b)=\\displaystyle\\Pr_{X\\in_U\\{0,1\\}^p}[f(X\\oplus a)\\oplus f(X)\\oplus b=1]$, for all $a \\in \\{0,1\\}^p$, $b \\in \\{0,1\\}^q$.",
"$\\Pr[f(x\\oplus a)\\oplus f(x)\\oplus b=0]=E(\\mathsf{DP}^f(a,b))$, for all $a, x \\in \\{0,1\\}^p$, $b \\in \\{0,1\\}^q$.",
"$2^p\\mathsf{DP}^f(a,b)$ is odd, for all $a \\in \\{0,1\\}^p, b \\in \\{0,1\\}^q$.",
"$\\displaystyle\\sum_{b\\in\\{0,1\\}^q} \\mathsf{DP}^f(a,b)=1$, for all $a \\in \\{0,1\\}^p$."
] |
D
|
Tick the \textbf{false} assertion. In order to have zero-knowledge from $\Sigma$-protocols, we need to add the use of \ldots
|
[
"\\ldots an ephemeral key $h$ and a Pedersen commitment.",
"\\ldots a common reference string.",
"\\ldots hash functions.",
"\\ldots none of the above is necessary, zero-knowledge is already contained in $\\Sigma$-protocols."
] |
D
|
A Feistel scheme is used in\dots
|
[
"DES",
"AES",
"FOX",
"CS-Cipher"
] |
A
|
Tick the \textbf{false} assertion. The SEI of the distribution $P$ of support $G$ \ldots
|
[
"is equal to \\# $G\\cdot\\displaystyle\\sum_{x\\in G}\\left(P(x)-\\frac{1}{\\sharp G}\\right)^2$",
"is the advantage of the best distinguisher between $P$ and the uniform distribution.",
"denotes the Squared Euclidean Imbalance.",
"is positive."
] |
B
|
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:
|
[
"$X$ and $Y$ are statistically independent",
"$H(X,Y)=H(X)$",
"VAUDENAY can be the result of the encryption of ALPACINO using the Vernam cipher.",
"$H(X|Y)=H(X)$"
] |
B
|
Tick the \emph{correct} assertion. Assume that $C$ is an arbitrary random permutation.
|
[
"$\\mathsf{BestAdv}_n(C,C^\\ast)=\\mathsf{Dec}^n_{\\left|\\left|\\left|\\cdot\\right|\\right|\\right|_\\infty}(C)$",
"$\\mathsf{BestAdv}_n(C,C^\\ast)=\\mathsf{Dec}^{n/2}_{\\left|\\left|\\left|\\cdot\\right|\\right|\\right|_\\infty}(C)$",
"$E(\\mathsf{DP}^{C}(a,b)) < \\frac{1}{2}$",
"$\\mathsf{BestAdv}_n(C,C^\\ast)=\\frac{1}{2}\\mathsf{Dec}^n_{\\left|\\left|\\cdot\\right|\\right|_a}(C)$"
] |
D
|
Which class of languages includes some which cannot be proven by a polynomial-size non-interactive proof?
|
[
"$\\mathcal{P}$",
"$\\mathcal{IP}$",
"$\\mathcal{NP}$",
"$\\mathcal{NP}\\ \\bigcap\\ $co-$\\mathcal{NP}$"
] |
B
|
Graph coloring consist of coloring all vertices \ldots
|
[
"\\ldots with a unique color.",
"\\ldots with a different color when they are linked with an edge.",
"\\ldots with a random color.",
"\\ldots with a maximum number of colors."
] |
B
|
What adversarial model does not make sense for a message authentication code (MAC)?
|
[
"key recovery.",
"universal forgery.",
"existential forgery.",
"decryption."
] |
D
|
Which one of these ciphers does achieve perfect secrecy?
|
[
"RSA",
"Vernam",
"DES",
"FOX"
] |
B
|
Which of the following problems has not been shown equivalent to the others?
|
[
"The RSA Key Recovery Problem.",
"The RSA Decryption Problem.",
"The RSA Factorization Problem.",
"The RSA Order Problem."
] |
B
|
A proof system is perfect-black-box zero-knowledge if \dots
|
[
"for any PPT verifier $V$, there exists a PPT simulator $S$, such that $S$ produces an output which is hard to distinguish from the view of the verifier.",
"for any PPT simulator $S$ and for any PPT verifier $V$, $S^{V}$ produces an output which has the same distribution as the view of the verifier.",
"there exists a PPT simulator $S$ such that for any PPT verifier $V$, $S^{V}$ produces an output which has the same distribution as the view of the verifier.",
"there exists a PPT verifier $V$ such that for any PPT simulator $S$, $S$ produces an output which has the same distribution as the view of the verifier."
] |
C
|
Suppose that you can prove the security of your symmetric encryption scheme against the following attacks. In which case is your scheme going to be the \textbf{most} secure?
|
[
"Key recovery under known plaintext attack.",
"Key recovery under chosen ciphertext attack.",
"Decryption under known plaintext attack.",
"Decryption under chosen ciphertext attack."
] |
D
|
For a $n$-bit block cipher with $k$-bit key, given a plaintext-ciphertext pair, a key exhaustive search has an average number of trials of \dots
|
[
"$2^n$",
"$2^k$",
"$\\frac{2^n+1}{2}$",
"$\\frac{2^k+1}{2}$"
] |
D
|
Tick the \textbf{false} assertion. For a Vernam cipher...
|
[
"SUPERMAN can be the result of the encryption of the plaintext ENCRYPT",
"CRYPTO can be used as a key to encrypt the plaintext PLAIN",
"SERGE can be the ciphertext corresponding to the plaintext VAUDENAY",
"The key IAMAKEY can be used to encrypt any message of size up to 7 characters"
] |
C
|
Assume we are in a group $G$ of order $n = p_1^{\alpha_1} p_2^{\alpha_2}$, where $p_1$ and $p_2$ are two distinct primes and $\alpha_1, \alpha_2 \in \mathbb{N}$. The complexity of applying the Pohlig-Hellman algorithm for computing the discrete logarithm in $G$ is \ldots (\emph{choose the most accurate answer}):
|
[
"$\\mathcal{O}(\\alpha_1 p_1^{\\alpha_1 -1} + \\alpha_2 p_2^{\\alpha_2 -1})$.",
"$\\mathcal{O}(\\sqrt{p_1}^{\\alpha_1} + \\sqrt{p_2}^{\\alpha_2})$.",
"$\\mathcal{O}( \\alpha_1 \\sqrt{p_1} + \\alpha_2 \\sqrt{p_2})$.",
"$\\mathcal{O}( \\alpha_1 \\log{p_1} + \\alpha_2 \\log{p_2})$."
] |
C
|
Tick the \textbf{\emph{incorrect}} assertion.
|
[
"$P\\subseteq NP$.",
"$NP\\subseteq IP$.",
"$PSPACE\\subseteq IP$.",
"$NP\\mbox{-hard} \\subset P$."
] |
D
|
Tick the \emph{correct} statement. $\Sigma$-protocols \ldots
|
[
"are defined for any language in \\textrm{PSPACE}.",
"have a polynomially unbounded extractor that can yield a witness.",
"respect the property of zero-knowledge for any verifier.",
"consist of protocols between a prover and a verifier, where the verifier is polynomially bounded."
] |
D
|
Current software is complex and often relies on external dependencies.
What are the security implications?
|
[
"During the requirement phase of the secure development\n lifecycle, a developer must list all the required dependencies.",
"It is necessary to extensively security test every executable\n on a system before putting it in production.",
"As most third party software is open source, it is safe by\n default since many people reviewed it.",
"Closed source code is more secure than open source code as it\n prohibits other people from finding security bugs."
] |
A
|
Does AddressSanitizer prevent \textbf{all} use-after-free bugs?
|
[
"No, because quarantining free’d memory chunks forever prevents\n legit memory reuse and could potentially lead to out-of-memory\n situations.",
"No, because UAF detection is not part of ASan's feature set.",
"Yes, because free’d memory chunks are poisoned.",
"Yes, because free’d memory is unmapped and accesses therefore\n cause segmentation faults."
] |
A
|
In x86-64 Linux, the canary is \textbf{always} different for every?
|
[
"Thread",
"Function",
"Process",
"Namespace"
] |
A
|
You share an apartment with friends.
Kitchen, living room, balcony, and bath room are shared resources among all parties.
Which policy/policies violate(s) the principle of least privilege?
|
[
"Different bedrooms do not have a different key.",
"There is no lock on the fridge.",
"To access the kitchen you have to go through the living room.",
"Nobody has access to the neighbor's basement."
] |
A
|
Which of the following statement(s) is/are true about CFI?
|
[
"When producing valid target sets, missing a legitimate target is unacceptable.",
"CFI’s checks of the valid target set are insufficient to protect every forward edge control-flow transfer",
"Keeping the overhead of producing valid target sets as low as possible is crucial for a CFI mechanism.",
"CFI prevents attackers from exploiting memory corruptions."
] |
A
|
When a test fails, it means that:
|
[
"either the program under test or the test itself has a bug, or both.",
"the program under test has a bug.",
"the test is incorrect.",
"that both the program and the test have a bug."
] |
A
|
Which of the following statement(s) is/are true about different types of
coverage for coverage-guided fuzzing?
|
[
"If you cover all edges, you also cover all blocks",
"Full line/statement coverage means that every possible\n control flow through the target has been covered",
"Full data flow coverage is easier to obtain than full edge coverage",
"Full edge coverage is equivalent to full path coverage\n because every possible basic block transition has been covered"
] |
A
|
Consider the Diffie-Hellman secret-key-exchange algorithm performed in the cyclic group $(\mathbb{Z}/11\mathbb{Z}^\star, \cdot)$. Let $g=2$ be the chosen group generator. Suppose that Alice's secret number is $a=5$ and Bob's is $b=3$. Which common key $k$ does the algorithm lead to? Check the correct answer.
|
[
"$10$",
"$7$",
"$8$",
"$9$"
] |
A
|
How many integers $n$ between $1$ and $2021$ satisfy $10^n \equiv 1 \mod 11$? Check the correct answer.
|
[
"1010",
"183",
"505",
"990"
] |
A
|
You are given an i.i.d source with symbols taking value in the alphabet $\mathcal{A}=\{a,b,c,d\}$ and probabilities $\{1/8,1/8,1/4,1/2\}$. Consider making blocks of length $n$ and constructing a Huffman code that assigns a binary codeword to each block of $n$ symbols. Choose the correct statement regarding the average codeword length per source symbol.
|
[
"It is the same for all $n$.",
"It strictly decreases as $n$ increases.",
"None of the others.",
"In going from $n$ to $n+1$, for some $n$ it stays constant and for some it strictly decreases."
] |
A
|
A bag contains the letters of LETSPLAY. Someone picks at random 4 letters from the bag without revealing the outcome to you. Subsequently you pick one letter at random among the remaining 4 letters. What is the entropy (in bits) of the random variable that models your choice? Check the correct answer.
|
[
"$\frac{11}{4}$",
"$2$",
"$\\log_2(7)$",
"$\\log_2(8)$"
] |
A
|
Consider the group $(\mathbb{Z} / 23 \mathbb{Z}^*, \cdot)$. Find how many elements of the group are generators of the group. (Hint: $5$ is a generator of the group.)
|
[
"$10$",
"$22$",
"$11$",
"$2$"
] |
A
|
In RSA, we set $p = 7, q = 11, e = 13$. The public key is $(m, e) = (77, 13)$. The ciphertext we receive is $c = 14$. What is the message that was sent? (Hint: You may solve faster using Chinese remainder theorem.).
|
[
"$t=42$",
"$t=14$",
"$t=63$",
"$t=7$"
] |
A
|
Consider an RSA encryption where the public key is published as $(m, e) = (35, 11)$. Which one of the following numbers is a valid decoding exponent?
|
[
"$11$",
"$7$",
"$5$",
"$17$"
] |
A
|
Let $\mathcal{C}$ be a binary $(n,k)$ linear code with minimum distance $d_{\min} = 4$. Let $\mathcal{C}'$ be the code obtained by adding a parity-check bit $x_{n+1}=x_1 \oplus x_2 \oplus \cdots \oplus x_n$ at the end of each codeword of $\mathcal{C}$. Let $d_{\min}'$ be the minimum distance of $\mathcal{C}'$. Which of the following is true?
|
[
"$d_{\\min}' = 4$",
"$d_{\\min}' = 5$",
"$d_{\\min}' = 6$",
"$d_{\\min}'$ can take different values depending on the code $\\mathcal{C}$."
] |
A
|
Let $\mathcal{C}$ be a $(n,k)$ Reed-Solomon code on $\mathbb{F}_q$. Let $\mathcal{C}'$ be the $(2n,k)$ code such that each codeword of $\mathcal{C}'$ is a codeword of $\mathcal{C}$ repeated twice, i.e., if $(x_1,\dots,x_n) \in\mathcal{C}$, then $(x_1,\dots,x_n,x_1,\dots,x_n)\in\mathcal{C'}$. What is the minimum distance of $\mathcal{C}'$?
|
[
"$2n-2k+2$",
"$2n-k+1$",
"$2n-2k+1$",
"$2n-k+2$"
] |
A
|
Consider the following mysterious binary encoding:egin{center} egin{tabular}{c|c} symbol & encoding \ \hline $a$ & $??0$\ $b$ & $??0$\ $c$ & $??0$\ $d$ & $??0$ \end{tabular} \end{center} where with '$?$' we mean that we do not know which bit is assigned as the first two symbols of the encoding of any of the source symbols $a,b,c,d$. What can you infer on this encoding assuming that the code-words are all different?
|
[
"The encoding is uniquely-decodable.",
"The encoding is uniquely-decodable but not prefix-free.",
"We do not possess enough information to say something about the code.",
"It does not satisfy Kraft's Inequality."
] |
A
|
Suppose that you possess a $D$-ary encoding $\Gamma$ for the source $S$ that does not satisfy Kraft's Inequality. Specifically, in this problem, we assume that our encoding satisfies $\sum_{i=1}^n D^{-l_i} = k+1 $ with $k>0$. What can you infer on the average code-word length $L(S,\Gamma)$?
|
[
"$L(S,\\Gamma) \\geq H_D(S)-\\log_D(e^k)$.",
"$L(S,\\Gamma) \\geq k H_D(S)$.",
"$L(S,\\Gamma) \\geq \frac{H_D(S)}{k}$.",
"The code would not be uniquely-decodable and thus we can't infer anything on its expected length."
] |
A
|
A colleague challenges you to create a $(n-1,k,d_{min})$ code $\mathcal C'$ from a $(n,k,d_{min})$ code $\mathcal C$ as follows: given a generator matrix $G$ that generates $\mathcal C$, drop one column from $G$. Then, generate the new code with this truncated $k imes (n-1)$ generator matrix. The catch is that your colleague only gives you a set $\mathcal S=\{\vec s_1,\vec s_2, \vec s_3\}$ of $3$ columns of $G$ that you are allowed to drop, where $\vec s_1$ is the all-zeros vector, $\vec s_2$ is the all-ones vector, and $\vec s_3$ is a canonical basis vector. From the length of the columns $s_i$ you can infer $k$. You do not know $n$, neither do you know anything about the $n-3$ columns of $G$ that are not in $\mathcal S$. However, your colleague tells you that $G$ is in systematic form, i.e., $G=[I ~~ P]$ for some unknown $P$, and that all of the elements in $\mathcal S$ are columns of $P$. Which of the following options in $\mathcal S$ would you choose as the column of $G$ to drop?
|
[
"$\\vec s_1$ (the all-zeros vector).",
"$\\vec s_2$ (the all-ones vector)",
"$\\vec s_3$ (one of the canonical basis vectors).",
"It is impossible to guarantee that dropping a column from $\\mathcal S$ will not decrease the minimum distance."
] |
A
|
A binary prefix-free code $\Gamma$ is made of four codewords. The first three codewords have codeword lengths $\ell_1 = 2$, $\ell_2 = 3$ and $\ell_3 = 3$. What is the minimum possible length for the fourth codeword?
|
[
"$1$.",
"$2$.",
"$3$.",
"$4$."
] |
A
|
Let P be the statement ∀x(x>-3 -> x>3). Determine for which domain P evaluates to true:
|
[
"-3<x<3",
"x>-3",
"x>3",
"None of the other options"
] |
C
|
Let p(x,y) be the statement “x visits y”, where the domain of x consists of all the humans in the world and the domain of y consists of all the places in the world. Use quantifiers to express the following statement: There is a place in the world that has never been visited by humans.
|
[
"∃y ∀x ¬p(x,y)",
"∀y ∃x ¬p(x,y)",
"∀y ∀x ¬p(x,y)",
"¬(∀y ∃x ¬p(x,y))"
] |
A
|
Which of the following arguments is correct?
|
[
"All students in this class understand math. Alice is a student in this class. Therefore, Alice doesn’t understand math.",
"Every physics major takes calculus. Mathilde is taking calculus. Therefore, Mathilde is a physics major.",
"All cats like milk. My pet is not a cat. Therefore, my pet does not like milk.",
"Everyone who eats vegetables every day is healthy. Linda is not healthy. Therefore, Linda does not eat vegetables every day."
] |
D
|
Suppose we have the following function \(f: [0, 2] o [-\pi, \pi] \).
\[f(x) =
egin{cases}
x^2 & ext{ for } 0\leq x < 1\
2-(x-2)^2 & ext{ for } 1 \leq x \leq 2
\end{cases}
\]
|
[
"\\(f\\) is not injective and not surjective.",
"\\(f\\) is injective but not surjective.",
"\\(f\\) is surjective but not injective.",
"\\(f\\) is bijective."
] |
B
|
If A is an uncountable set and B is an uncountable set, A − B cannot be :
|
[
"countably infinite",
"uncountable",
"the null set",
"none of the other options"
] |
D
|
You need to quickly find if a person's name is in a list: that contains both integers and strings such as:
list := ["Adam Smith", "Kurt Gödel", 499, 999.95, "Bertrand Arthur William Russell", 19.99, ...]
What strategy can you use?
|
[
"Insertion sort the list, then use binary search.",
"Bubble sort the list, then use binary search.",
"Use binary search.",
"Use linear search."
] |
D
|
Let \( f : A
ightarrow B \) be a function from A to B such that \(f (a) = |a| \).
f is a bijection if:
|
[
"\\( A= [0, 1] \\) and \\(B= [-1, 0] \\)",
"\\( A= [-1, 0] \\) and \\(B= [-1, 0] \\)",
"\\( A= [-1, 0] \\) and \\(B= [0, 1] \\)",
"\\( A= [-1, 1] \\) and \\(B= [-1, 1] \\)"
] |
C
|
Let \( P(n) \) be a proposition for a positive integer \( n \) (positive integers do not include 0).
You have managed to prove that \( orall k > 2, \left[ P(k-2) \wedge P(k-1) \wedge P(k)
ight]
ightarrow P(k+1) \).
You would like to prove that \( P(n) \) is true for all positive integers. What is left for you to do ?
|
[
"None of the other statement are correct.",
"Show that \\( P(1) \\) and \\( P(2) \\) are true, then use strong induction to conclude that \\( P(n) \\) is true for all positive integers.",
"Show that \\( P(1) \\) and \\( P(2) \\) are true, then use induction to conclude that \\( P(n) \\) is true for all positive integers.",
"Show that \\( P(1) \\), \\( P(2) \\) and \\( P(3) \\) are true, then use strong induction to conclude that \\( P(n) \\) is true for all positive integers."
] |
D
|
What is the value of \(f(4)\) where \(f\) is defined as \(f(0) = f(1) = 1\) and \(f(n) = 2f(n - 1) + 3f(n - 2)\) for integers \(n \geq 2\)?
|
[
"41",
"45",
"39",
"43"
] |
A
|
Which of the following are true regarding the lengths of integers in some base \(b\) (i.e., the number of digits base \(b\)) in different bases, given \(N = (FFFF)_{16}\)?
|
[
"\\((N)_2\\) is of length 16",
"\\((N)_{10}\\) is of length 40",
"\\((N)_4\\) is of length 12",
"\\((N)_4\\) is of length 4"
] |
A
|
In a lottery, a bucket of 10 numbered red balls and a bucket of 5 numbered green balls are used. Three red balls and two green balls are drawn (without replacement).
What is the probability to win the lottery? (The order in which balls are drawn does not matter).
|
[
"$$\frac{1}{14400}$$",
"$$\frac{1}{7200}$$",
"$$\frac{1}{1200}$$",
"$$\frac{1}{1900}$$"
] |
C
|
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