On the coefficients of the polynomial in the number field sieve

Polynomial selection is very important in number field sieve. If the yield of a pair of polynomials is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first. This can speed up the selection of good polynomials. In this paper, we aim to study the correlation between the polynomial coefficients and the yield of the polynomials. By theoretical analysis and experiments, we find that a polynomial with the ending coefficient containing more small primes is usually better in yield than the one whose ending coefficient contains less. One advantage of the ending coefficient over the leading coefficient is that the ending coefficient is bigger and can contain more small primes in root optimizing stage. Using the complete discrimination system, we also analyze the condition on coefficients to obtain more real roots

Root optimization of polynomials in the number field sieve

International audienceThe general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties

Rotations and translations of number field sieve polynomials

Abstract. We present an algorithm that finds polynomials with many roots modulo many primes by rotating candidate Number Field Sieve polynomials using the Chinese Remainder Theorem. We also present an algorithm that finds a polynomial with small coefficients among all integral translations of X of a given polynomial in ZZ[X]. These algorithms can be used to produce promising candidate Number Field Sieve polynomials.

Reflections on the number field sieve

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