A quadratic sieve on the n-dimensional cube

Abstract

Let N be a large odd integer. We show how to produce a long sequence (formula presented) of integers modulo N which satisfy (formula presented) modulo N, where (formula presented) and (formula presented). Our sequence corresponds to a Hamiltonian path on the u-dimensional hypercube C'n, where n is Θ(log N/ log log N). One application of these techniques is that, at each vertex of the hypercube, it is possible to search for equations of the form U2 ≡ V modulo N with V a product of small primes. The search is as in the quadratic sieve algorithm and therefore very fast. This yields a faster way of changing polynomials in the Multiple Polynomial Quadratic Sieve algorithm, since moving along the hypercube turns out to be very cheap.