Efficiently Solving the Discrete Logarithm Problem over Large Primes for ElGamal Cryptanalysis

Determine whether there exists an algorithm that efficiently solves the discrete logarithm problem modulo a large prime number—specifically, given a public key (p, a, y) with y = a^x mod p, efficiently compute the private key x—in a manner that makes cryptanalysis of the ElGamal cryptosystem feasible.

Background

The paper relies on the security of the ElGamal cryptosystem, which is predicated on the computational hardness of the discrete logarithm problem in groups modulo a large prime number. Given the public parameters (p, a, y) and the relation y = a^x mod p, recovering the private key x corresponds to solving the discrete logarithm.

The authors explicitly state that, to date, no method is known that efficiently solves this problem at a scale that would make cryptanalysis of ElGamal feasible. This assertion underpins their security claims for the proposed modified ElGamal cryptosystem used to encrypt QR codes prior to steganographic embedding.