Improvements to the number field sieve for non-prime finite fields

2014/08/04 by Razvan Barbulescu, Barbulescu, Razvan, Pierrick Gaudry +5
Computer Science · #Coding theory and cryptography #Cryptography and Data Security #Cryptography and Residue Arithmetic #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1408.0718

openalex publication_date 2014/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose various strategies for improving the computation of discrete logarithms in non-prime fields of medium to large characteristic using the Number Field Sieve. This includes new methods for selecting the polynomials; the use of explicit automorphisms; explicit computations in the number fields; and prediction that some units have a zero virtual logarithm. On the theoretical side, we obtain a new complexity bound of L_pⁿ(1/3,√[3]96/9) in the medium characteristic case. On the practical side, we computed discrete logarithms in F_p² for a prime number p with 80 decimal digits.Warning: This unpublished version contains some inexact statements.

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