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COMP0025: Introduction to Cryptography
创建日期：2024-05-25 17:40:00
标签： COMP0025
Introduction to Cryptography

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COMP0025: Introduction to Cryptography

Practice Exam
Question 1: Hash Functions [25 marks]
(a) Consider the hash function H : {0, 1}256` → {0, 1}256 defined by H(m1, . . . ,m`) = m1⊕. . .⊕m`.
Describe an adversary that on input m = (m1,m2) performs a successful second-preimage attack.
[5 marks]
(b) SnakeOil Ltd is suggesting a super fast hash function H : {0, 1}∗ → {0, 1}100 defined by H(x) =
7x+5 mod 2100. Describe an attack that finds a second-preimage for any message x. [5 marks]
(c) Given a collision-resistant compression function f : {0, 1}512 → {0, 1}256, we define a hash
function H : {0, 1}∗ → {0, 1}256 as follows:
(a) On input x pad it with 0 . . . 0 such that it has length 256n.
(b) Let xi be the ith 256-bit block of the padded x ‖ 0 . . . 0.
(c) Let z0 = 0
256 and zi = f(zi−1 ‖ xi) for i = 1, . . . , n.
(d) Return zn as the output H(x).
Is this construction collision-resistant? Why or why not? [5 marks]
(d) Let p, q be two large prime numbers such that p = 2q + 1 and let g, h be two random group
elements in Z∗p. The function H : Zp−1 × Zp−1 → Z∗p defined by
H(x, y) = gxhy mod p
is a hash function mapping (p− 1)2 possible inputs to p− 1 possible outputs. Unfortunately, H
is not collision-resistant. Argue why not. [5 marks]
Hint: You may use the fact that half of the elements in Z∗p have order q.
(e) Let p, q bet two large prime numbers such that p = 2q + 1 and let g, h ∈ Zp bet two random
generators for the order q subgroup G ⊆ Z∗p. Define the hash function H : Zq × Zq → G by
H(x, y) = gxhy mod p
Show that H is collision-resistant if the discrete logarithm is hard in G. [5 marks]
1
Question 2: Message Authentication Codes [25 marks]
(a) Consider the following one-time message authentication code for 10-bit messages.
• Key generation: The secret key is sk = (a, b) where a, b $←− Z1031 are picked at random
(1031 is a prime).
• Authentication: To authenticate a message m compute the authentication tag t = am+
b mod 1031.
• Verification: To verify an authentication tag t on a message m check whether t = am +
b mod 1031.
Consider an adversary that does not know anything about the secret key but does learn one pair
(m, t) of an arbitrary message and its corresponding authentication tag. What is the probability
that the adversary can find a different message m′ and its corresponding tag t′? [5 marks]
(b) Consider the following message authentication code for 128-bit messages built from AES encryp-
tion function E.
• Key generation: Pick a 128-bit secret key k.
• Authentication: Given a message m compute the authentication tag t = Ek(m).
• Verification: To verify an authentication tag t on message m check whether t = Ek(m).
Assume Ek were a truly random permutation of 128-bit strings. Suppose the adversary learns
pairs of messages and authentication tags (m1, t1), . . . , (m1024, t1024) for 1024 different messages
of its own choosing. What is the probability that it can find a new message m /∈ {m1, . . . ,m1024}
and guess the corresponding message authentication tag t? [5 marks]
(c) Recall that NMAC and HMAC are message authentication codes built from a Merkle-Damgard
type hash function such as for instance SHA256. HMAC computes the message authentication
code as HMAC(k,m) = H(k⊕ opad ‖ H(k⊕ ipad) ‖ m) whereas NMAC computes the message
authentication code as NMAC(k,m) = Hk1(Hk2(m)) with k = (k1, k2). What is an advantage
of NMAC over HMAC and an advantage of HMAC over NMAC? [5 marks]
(d) Consider CBC-MAC for 128`-bit messages with fixed `. We use AES as the underlying block
cipher with 128-bit keys. Let E be the AES encryption function.
• Key generation: Pick a 128-bit secret key k.
• Authentication: Given a message m = m1 ‖ · · · ‖ m` where m1, . . . ,m` are 128-bit
blocks define t0 = 0 and compute t1, . . . , t` recursively as ti = Ek(ti−1 ⊕mi). Output the
authentication tag t = t`.
• Verification: To verify an authentication tag t on a 128`-bit message m compute t` as
above and accept if t = t`.
Suppose we modify CBC-MAC to allow for both 128-bit messages and 256-bit messages by
outputting t = t1 or t = t2. Show that it is possible to forge an authentication tag on a 256-
bit message by combining the authentication tags for two adaptively chosen 128-bit messages.
[5 marks]
(e) Give a formal definition of existential unforgeability for message authentication codes under adap-
tive chosen message attacks. Explain in your own words what the definition means. [5 marks]
2
Question 3: Digital Signatures [25 marks]
(a) A digital signature scheme (Gen,Sign,Verify) is said to be secure if for all probabilistic polynomial
time adversaries A we have
Pr[(vk, sk)← Gen(1λ); (m,σ)← ASignsk(·)(vk) : m /∈ Q ∧ Verifyvk(m,σ) = 1] = negl(λ)
where Q is the set of messages that has been queried to the signing oracle Signsk(·). What is the
name of the security definition? Explain in non-mathematical terms what the goal of the adversary
is in the security definition. Explain in non-mathematical terms what access the adversary has to
the signature scheme in the security definition. [5 marks]
(b) We say p is a safe prime if p = 2p′ + 1 and both p and p′ are primes. Consider an RSA modulus
N = pq where p = 2p′+1 and q = 2q′+1 are two different safe primes. The group of quadratic
residues QRN are the elements y ∈ Z∗N for which there exists a z such that y = z2 mod N . The
size of QRN is p′q′. What is the size of Z∗N? Show that Z∗N is not cyclic. Show that QRN is
cyclic. [5 marks]
Hint: You may want to use the Chinese Remainder Theorem and the fact that Z∗p and Z∗q are
cyclic when p and q are prime.
(c) Camenisch and Lysysanskaya proposed a signature scheme in 2002. A variant of the CL signature
scheme for `m-bit messages works as follows (for suitable choices of `e > 2λ and `r defined in
their paper):
• Gen(1λ): Pick two random λ-bit safe primes p and q. Let N = pq and pick at random
a, b, c ∈ QRN . Return the verification key vk = (N, a, b, c) and the secret key sk = (p, q).
• Sign(sk,m): To sign an `m-bit message m pick at random a prime 2`e < e < 2`e+1
and additional randomness r
$←− {0, 1}`r . Compute a value s ∈ QRN such that se =
abmcr mod N . Return the signature σ = (e, r, s).
• Verify(vk,m, σ): To verify a signature σ on an `m-bit message m check that e > 2`e is a
prime, check that r ∈ {0, 1}`r and check that se = abmcr mod N .
Let vk = (77, 4, 9, 25) be a CL signature scheme verification key. Find the secret key and compute
a CL signature on m = 3 using randomness (e, r) = (103, 8). [5 marks]
Hint: To speed up the calculations, do them modulo p and modulo q and use the Chinese
Remainder Theorem.
(d) Show that if you can guess in advance the number of signing queries t that the adversary will
make, then you can simulate the verification key and simulate the signatures without knowing
the factorization of N . [5 marks]
Hint: Given N pick at random y = z2 mod N . Generate t random primes e1, . . . , et ∈
{2`e , . . . , 2`e+1}. Define E = ∏tj=1 ej and Ei = E/ei. Pick at random α, β $←− {0, 1}3λ.
Let a = yαE mod N , b = yβE mod N , and c = yE mod N .
(e) The strong RSA assumption says that given a safe prime RSA modulus N and a random element
y ∈ QRN , it is hard to find an x ∈ Z∗N and an exponent e > 1 such that y = xe mod N .
Consider an adversary At that obtains exactly t signatures (e1, s1, r1), . . . , (et, st, rt) on messages
m1, . . . ,mt in adaptive chosen message attacks and then forges a CL signature σ = (e, s, r) on
a message m /∈ {m1, . . . ,mt} using a prime e /∈ {e1, . . . , et}. Show that such an adversary can
be used to break the strong RSA assumption. [5 marks]
Hint: Write α = α′p′q′ + [α mod p′q′] over the integers. Observe that no information about α′
is leaked when simulating signatures given that gcd(e, α+ βm+ r) = 1 with high probability.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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