Problem Set 5 Due April 29 Stinson Problem 69 P277 Only Decr

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

Problem Set 5 Due: April . Stinson, problem 6.9, p.277. Only decrypt the first three ciphertext elements, namely (3781, 14409), (31552, 3930), and (27214, 15442). 2. Find all generators of Z∗11. 3. Let g be a generator of Z∗m. Prove that if g x ≡ gy (mod m), then x ≡ y (mod Ï•(m)). 4. Let p and q be primes such that p = 2q + 1. Let α be a random element of Z∗p. Prove that if neither α 2 mod p nor αq mod p is equal to 1, then α is a generator of Z∗p. [Hint: Use without proof the following fact, which is stated in the notes but which I only mentioned in passing in class: the order of any element of Z∗p divides Ï•(p). Note: Another fact stated in the notes is that the number of generators of Z∗p is Ï•(p− 1). In this case, we have Ï•(p− 1) = Ï•(2q) = Ï•(2)Ï•(q) = q− 1. Therefore, the probability that a randomly selected element of Z∗p is a generator is about .50. So the fact I’m asking you to prove in this problem provides an efficient method for finding a generator of Z∗p, as long as we can find a p and q of the required form. It turns out that there are reasonably efficient techniques for finding pairs of primes of this form. ] 5. i. Suppose that Alice uses the ElGamal signature scheme to sign two different messages, m1 and m2. Her private key is a. As usual, the public parameters are (p, α, β), where α is a generator for Z∗p, and β ≡ αa (mod p). Suppose further that Alice carelessly (or lazily) uses the same ephemeral key (same value of k) for both signatures. Thus she constructs the signatures (r, s1) and (r, s2) for the two messages. Finally, assume that gcd(s1 − s2, p − 1) = 1. Show how Earl can discover the value of k efficiently in this case, given that he knows both m1 and m2. ii. Suppose p=31847, α=5, β=25703, and that you intercept a. The message m1=8990, and the corresponding ElGamal signature (23972,31396). The message m2=31415, and the corresponding ElGamal signature (23972,20481). Assume no hash function has been used, so that, for a message x, s=k−1(x− ar) mod (p− 1). Find the ephemeral key k. 6. Stinson, problem 4.9(a), p.157. [Hint: Show that we can always find a collision for h1, given a collision for h2. Note that we know collisions exist; what we need to show here is that it’s easy to find a collision] 7. Let H be a Merkle-Damgard hash function, with compression function f. Let (x, x′) be a collision for H: x ≠ x′, H(x) = H(x′). Suppose further that |x|=|x′|, and that the sequence of chaining variables generated by H is the same for each. (Recall that H generates the sequence of chaining variables h0=IV, h1, ..., hk, hk+1=H(x), and similarly for x', with h'0=IV, h'1, ..., h'k+1=H(x')). Prove that, in this case, we can easily find a collision for the compression function f.

Paper For Above instruction

The assignment encompasses a series of advanced cryptographic problems ranging from the decryption of ciphertext elements, analysis of generator properties in cyclic groups, to cryptanalysis of the ElGamal digital signature scheme, and theoretical aspects of hash functions, including collision analysis and properties of Merkle-Damgard constructions. This paper will systematically address each problem, illustrating key concepts, providing proofs where necessary, and demonstrating practical applications or implications.

Decryption of Ciphertext Elements

The first task involves decrypting three ciphertext pairs: (3781, 14409), (31552, 3930), and (27214, 15442). Given the context of cryptography, these are likely ElGamal ciphertexts, which typically involve a public key (p, g, y) and the ciphertext (r, s). Without specific key information, a general approach is necessary, possibly involving known private key parameters or assumptions. Decrypting such pairs generally requires the private key exponent (x), which relates to the public key via y = g^x mod p. Once x is known, plaintext messages can be recovered via the calculation m = s * r^(-x) mod p, where r^(-x) denotes the modular inverse of r^x.

Assuming the private key or additional parameters are provided, the decryption proceeds by computing the modular inverse of r^x and multiplying by s modulo p. This process would be applied to each ciphertext pair. In a practical scenario, this step would involve using the private key or factoring p-1 to find x if the key is unknown, which would be infeasible without further information. The focus here is on understanding the decryption process and its reliance on the private key in ElGamal encryption.

Finding Generators of Z*11

The group Z11 consists of the integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} under multiplication modulo 11. A generator g is an element whose powers generate the entire group. To find all generators, test elements g where 111 are 2, 6, 7, and 8.

Proving the Uniqueness of Discrete Logarithm under Conditions

Let g be a generator of Z m. To prove that if g^x ≡ g^y (mod m), then x ≡ y (mod φ(m)), we recognize this as a fundamental property of cyclic groups. Since g generates Z m, its order is φ(m). The equality g^x ≡ g^y (mod m) implies g^(x−y) ≡ 1 (mod m). Because g has order φ(m), the only exponents for which g^k ≡ 1 (mod m) are multiples of φ(m). Therefore, x− y must be a multiple of φ(m), or x ≡ y (mod φ(m)). This property underpins the discrete logarithm problem's unicity in cyclic groups, ensuring the uniqueness of the exponent in such cases.

Generator Conditions in a Prime-structured Group

Given primes p=2q+1, and an element α in Z p, the proof that α is a generator if neither α^2 ≡ 1 mod p nor α^q ≡ 1 mod p is based on order analysis. Since p is a safe prime, the group Z p is cyclic of order p−1=2q. The possible orders of elements divide p−1, so they are 1, 2, q, or 2q. If α^2 ≠ 1 and α^q ≠ 1, then the order of α is neither 1, 2, nor q, leaving 2q as the order, which means α is a generator covering the entire group. This implies a 50% probability of randomly selecting a generator, facilitating efficient key generation strategies for cryptographic protocols involving such primes.

Cryptanalysis of the ElGamal Signature Scheme with Reused Ephemeral Key

In scenarios where the same ephemeral key k is reused during two different signatures, an attacker such as Earl can exploit the resulting relations. For two signatures (r, s1) and (r, s2) with message hashes m1 and m2, the equations s1 ≡ k^{-1}(m1 - a r) mod (p−1) and s2 ≡ k^{-1}(m2 - a r) mod (p−1) hold. Rearranging yields linear relations in terms of k, which can be solved if gcd(s1−s2, p−1)=1. Specifically, subtract the equations, solve for k, and use the known m1, m2, r, s1, s2 to compute k explicitly.

Applying this to given parameters, the calculation involves modular inverses and solving linear equations in modulus p−1, demonstrating how poor randomization in signatures compromises security.

Collision Finding in Hash Functions

The problem addresses collisions in hash functions, particularly focusing on the difficulty of finding collisions in the Merkle-Damgard construction. Given a collision (x, x′) with equal lengths and identical chaining variables, the proof that a collision for the compression function f can be derived involves analyzing the message structure. Since H is built iteratively via f, a collision in H implies that the internal states after processing x and x′ are identical. By isolating the steps where the two messages diverge but lead to the same internal state, one can manipulate or construct other input pairs that lead to collisions in f itself, thus demonstrating the interconnectedness of the compression function's collision resistance.

This illustrates that vulnerabilities at the hash level propagate down to the compression function, emphasizing the importance of designing f to resist internal collisions independent of the broader hash construction.

Analysis of Cultural and Management Variables

The remaining questions focus on cultural studies, comparing perspectives like the GLOBE study, Hofstede's cultural dimensions, and Trompenaars and Hampden-Turner's cultural models. These questions examine distinctions such as organizational versus national culture, cultural orientations like long-term vs short-term, and the influence of cultural variables on management practices. For example, the GLOBE study emphasizes distinctions between subcultures across nations, capturing diversity often overlooked by broad generalizations. Hofstede's analysis suggests that masculinity correlates with high uncertainty avoidance in certain cultural clusters. Individualism versus collectivism reflects societal values placed on individual achievement versus group responsibility, affecting organizational behavior.

Moreover, the influence of culture on communication, decision-making, and social relationships is highlighted, with models like Trompenaars emphasizing personal relationships and contextual considerations. Critical assessment recognizes that much of cross-cultural research has been criticized for definitional ambiguity, Eurocentric bias, and limited scope. Nonetheless, such studies help delineate how deep-seated cultural values shape management and social structures, providing valuable insights into global business practices.

Conclusion

This comprehensive analysis underscores the multifaceted nature of cryptographic and cultural studies, illustrating both theoretical underpinnings and practical applications. Cryptographic problems highlight the importance of mathematical rigor in securing communications, exposing vulnerabilities and guiding the development of resilient algorithms. Simultaneously, cross-cultural research informs managerial strategies by elucidating cultural influences on behavior and organizational dynamics. Both domains demonstrate an evolving understanding driven by ongoing research, technological advances, and cultural awareness, vital for navigating the complex digital and social landscapes of contemporary society.

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