Structural Properties of the Mapping g → gx

This paper studies the mapping g → g 2 (mod p) by showing that it inherits its structure from the mapping x → x (mod ordp(g)). We first explore the context in which this mapping is important; specifically, we consider the Diffie-Hellman protocol, Diffie-Hellman problem, and the various problems that have spawned from these including the discrete logarithm and square exponent problems. We then demonstrate that we may infer the structure of g → g 2 (mod p) from x→ x (mod ordp(g)) and translate some results proved by Somer and Kroger about the latter to the former case. 1 The Diffie-Hellman Problem 1.1 The Diffie-Hellman Protocol The motivation for our research stems from what is known as the Diffie-Hellman protocol. First published publicly by Whitfield Diffie and Martin Hellman in 1976, Diffie-Hellman is a secure protocol for public-key cryptography. Alice and Bob both desire to share a secret key using public channels. This means that everyone, Eve, can intercept any messages sent between the two. Alice and Bob agree on a prime p and a generator g ∈ Zp∗ . They choose respectively a and b ∈ [1, p−1]. Alice calculates g, Bob calculates g and they exchange. Alice now has g and a; Bob likewise has g and b. From this information they can both efficiently calculate g — their shared secret key. What makes this protocol secure? The discrepancy between the amount of computation time that it takes Eve to calculate g and the time that it takes Alice or Bob. Eve has g, g, and g; to obtain Alice and Bob’s key, Eve must solve what is known as the Diffie-Hellman problem. That is: given g and g, calculate g. While Alice and Bob can calculate g in O(log x), there exists no known algorithm to efficiently solve this problem in a general finite group. Thus, the Diffie-Hellman protocol is thought to be secure and the problem remains open.