The Discrete Log Problem

Given a large prime number p, a large intgeger a, and a number g such that every number b coprime to p is congruent to a power of g mod p (i.e. g is a primite root mod p), then g^a mod p = s is very easy to compute using the repeated squaring algorithm. However, determining the value of a if we are given g, p and s is extremely diffucult, and thus discrete log is considered a one way function. Several important public-key cryptographic systems base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. These include Diffie-Hellman Key Exchange, ElGamal encryption, and the Digital Signature Algorithm.

While the discrete log problem is considered cryptographically sound, this repository provides three algorithms that provide significant improvement over a brute-force tactic. They include the Pohlig-Hellman Algorithm (practically the fastest), the Baby Step Giant Step Algorithm and the Rho Algorithm.

Java code

Containts a Spring Boot application with four endpoints, one that calculates the discrete log problem using all three algorithms, and then one endpoint for each algorithm.

generator^solution mod prime = base

http://localhost:8080/discrete-log?generator=5&base=222137199848&prime=4889427811007

http://localhost:8080/pohlig-hellman?generator=5&base=222137199848&prime=4889427811007
http://localhost:8080/rho?generator=5&base=222137199848&prime=4889427811007
http://localhost:8080/baby-giant?generator=5&base=222137199848&prime=4889427811007

Python code

import the discrete_log module and then you have the three algorithms available to you

import discrete_log

rho_metadata = discrete_log.rho.solve(5, 222137199848, 4889427811007)
print(rho_metadata.solution, rho_metadata.secondsTaken)

pohlig_hellman_metadata = discrete_log.pohlig_hellman.solve(5, 222137199848, 4889427811007)
print(pohlig_hellman_metadata.solution, pohlig_hellman_metadata.secondsTaken)

baby_giant_metadata = discrete_log.baby_giant.solve(5, 222137199848, 4889427811007)
print(baby_giant_metadata.solution, baby_giant_metadata.secondsTaken)