Trial multiplication is not optimal but... On the symmetry of finite cyclic groups (Z/pZ)∗

The Discrete Logarithm Problem is at the base of the famous Diffie Hellman key agreement algorithm and many others. The key idea behind Diffie Helmann is the usage of the Discrete Logarithm function in (Z/pZ)∗ as a trap door function. The Discrete Logarithm function output in (Z/pZ)∗ seems to escape to any attempt of finding some sort of pattern. Nevertheless some new characterization will be introduced together with a novel and more efficient trial multiplication algorithm. The Discrete Logarithm Problem The Discrete Logarithm Problem (DLP) is at the base of many cryptographic techniques. This paper will focus only on DLP for finite cyclic group G in (Z/pZ)∗ (prime moduli) of order p − 1 with generator g. Having a finite cyclic group G and a generator g, the DLP of α, denoted dloggα, is the unique integer x, 0 ≤ x ≤ (p − 1), such that α = gx. The finite cyclic group can also be denoted by G = {g0,g1,...,gp−2} where gi = gi (mod p) for 0 ≤ i ≤ p − 2. Using as example g = 2 and p = 53 we can try to plot G = {1, 2, 4, 8, 16, 32, 11, 22, 44, 35, 17,....,27} having Cartesian coordinates xi = i and yi=gi for 0 ≤ i ≤ (p − 2). This draws the plot: Figure 1: xi = i and yi=gi for 0 ≤ i ≤ (p − 2)