Nontrivial solutions to the cubic sieve congruence problem x^3=y^2 z (mod p)

Abstract

In this paper we discuss the problem of finding nontrivial solutions to the Cubic Sieve Congruence probem, that is, solutions of x2 = y2z (mod p), where x,y,z < p1/2 and x3 = y2z. The solutions to this problem are useful in solving the Discrete Log Problem or factorization by index calculus method.. Apart from the cryptographic interest, this problem is motivatin by itself from a number theoretic point of view. Though we could not solve the problem completely, we could identify certain subclasses of primes where the problem can be solved in time polynomial in log p. Further we could extend the idea of Reynert's sieve and identify some cases where the problem can even be solved in constant tiem. Designers of ctyptosystems should avoid all primes contained in our detected cases.