Problem Set 5 Due April 29 Stinson Problem 69 P277 On 140592

Problem Set 5due April 291 Stinson Problem 69 P277 Only Decrypt

Decrypt the first three ciphertext elements (3781, 14409), (31552, 3930), and (27214, 15442) from problem 6.9, p.277 of Stinson. Find all generators of Z11. Prove that if g_x ≡ g_y (mod m) for a generator g of Z m, then x ≡ y (mod φ(m)). Let p and q be primes with p = 2q + 1; prove that if α ∈ Z p satisfies that neither α² mod p nor α^q mod p equals 1, then α is a generator of Z p. Use properties of the order of elements and the number of generators to support this proof. Examine the ElGamal signature scheme: if Alice signs two messages with the same ephemeral key k, and gcd(s1 − s2, p − 1) = 1, show how an eavesdropper can efficiently determine k given the messages m1, m2, and their signatures. Given specific parameters (p=31847, α=5, β=25703), and intercepted messages and signatures, find the ephemeral key k. For an extension: show how to find collisions for a hash function H with compression function f, given a collision for h2, and analyze the process assuming a collision exists for H, with particular focus on the structure and sequence of chaining variables.

Paper For Above instruction

The assignment involves multiple cryptographic problems related to decryption, group theory, prime structures, digital signatures, and hash functions. The first task requires decrypting specific ciphertext pairs, which demands applying knowledge of classical or modern cryptosystems, possibly RSA or ElGamal, depending on context. Decrypting these values involves understanding the encryption scheme and relevant keys or factoring techniques. The second problem asks to find all generators of the multiplicative group Z11, which is the set of units modulo 11. Since 11 is prime, Z11 is cyclic of order 10, and its generators (primitive roots) are elements of order 10. The identification involves testing elements for their order or repeatedly checking exponentiation results, as primitive roots modulo a prime p are well-studied. The third problem asks to prove that in a cyclic group generated by g, if g_x ≡ g_y, then x ≡ y mod φ(m). This is a fundamental property of cyclic groups, relying on the fact that powers of a generator are injective modulo the group's order. The proof hinges on properties of the Euler totient function and the uniqueness of discrete logarithms. The fourth problem involves number theory: given p and q primes with p=2q+1, and an element α in Z* p, it is to be proven that if neither α² nor α^q equals 1 mod p, then α must be a generator. This proof uses properties of the order of elements dividing p−1 and the structure of cyclic groups, connecting to primitive root theory and the factorization of p−1. The probability aspect highlighted explains why randomly selecting α yields a generator with roughly 50% chance, emphasizing the efficiency of testing candidate elements based on known suborder conditions.

The fifth problem discusses the cryptanalysis of the ElGamal signature scheme, especially when the same ephemeral key k is reused across different messages. The critical insight is that this reuse leaks enough information to solve for k algebraically, particularly given the signatures and messages involved, through systems of equations and gcd conditions. Specific parameters are provided, and the task is to compute k explicitly. The attack proceeds by leveraging the known equations relating signatures, messages, and the common k, and using the extended Euclidean algorithm or modular inverses to isolate k.

The sixth problem involves constructing collisions in hash functions, relying on known collisions in one part of the scheme to produce collisions for the overall hash. This involves exploring the structure of the Merkle-Damgård construction, exploiting the properties of the underlying compression functions, and demonstrating straightforward methods to find colliding inputs.

Finally, the seventh problem asks to analyze the difficulty of deriving a collision for a compression function based on collisions in the overall hash function H. It involves understanding the chaining process, the structure of inputs, and how collisions at the hash level translate into collisions at the compression level, emphasizing the importance of properties like the length of inputs and the sequence of chaining variables. This analysis underscores the interconnectedness of collision resistance properties at different layers within cryptographic hash functions. Overall, these questions highlight fundamental concepts in cryptographic security, group theory, number theory, and hash function design, each requiring rigorous proofs and computational techniques grounded in theoretical foundations.

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