New discrete logarithm computation for the medium prime case using the function field sieve

Mukhopadhyay, Madhurima; Sarkar, Palash; Singh, Shashank; Thomé, Emmanuel

doi:10.3934/amc.2020119

Cited by 1 publication

(4 citation statements)

References 20 publications

(66 reference statements)

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“…The main idea presented in [Gui19] for the splitting step in large and medium characteristic finite fields is to substitute the target by another one that has smaller coefficients. In small characteristic finite fields of composite extension degree, [MS20] replaces the target by another one with a smaller degree. The method we propose allows the advantages of both worlds, supplanting the target with candidates with smaller coefficients and smaller degrees.…”

Section: Sublattices For Smaller Norms In the Number Fieldmentioning

confidence: 99%

“…These methods iteratively generate a pair of polynomials and tests both of them for Bsmoothness, for a given bound B. Guillevic [Gui19] exploits the proper subfields of the target finite field, resulting in an algorithm that gives much more smooth decomposition of the target in the initial splitting step. Besides, Mukhopadhyay and Sarkar's method [MS20] deals with at the splitting step for small characteristic finite fields with composite extension degrees. [MS20] is dedicated to the Function Field Sieve and is not applicable in our context.…”

Section: Introductionmentioning

confidence: 99%

“…Besides, Mukhopadhyay and Sarkar's method [MS20] deals with at the splitting step for small characteristic finite fields with composite extension degrees. [MS20] is dedicated to the Function Field Sieve and is not applicable in our context.…”

Section: Introductionmentioning

confidence: 99%

“…Our work. In this article, we improve on the current state-of-the-art Guillevic's algorithm dedicated to the initial splitting step for composite n. While still exploiting the proper subfields of the target finite field, and running a reduction algorithm on a well-defined lattice as in [Gui19], we manage to control the degree of the candidates for the B-smoothness test as in [MS20]. The key idea is to use sublattices of the original lattice of [Gui19] by removing some rows and columns.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Individual discrete logarithm with sublattice reduction

Al Aswad,

Pierrot

2023

Des. Codes Cryptogr.

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The Number Field Sieve and its numerous variants is the best algorithm to compute discrete logarithms in medium and large characteristic finite fields. When the extension degree n is composite and the characteristic p is of medium size, the Tower variant (TNFS) is asymptotically the most efficient one. Our work deals with the last main step, namely the individual logarithm step, that computes a smooth decomposition of a given target T in the finite field thanks to two distinct phases: an initial splitting step, and a descent tree. In this article, we improve on the current state-of-the-art Guillevic's algorithm dedicated to the initial splitting step for composite n. While still exploiting the proper subfields of the target finite field, we modify the lattice reduction subroutine that creates a lift in a number field of the target T . Our algorithm returns lifted elements with lower degrees and coefficients, resulting in lower norms in the number field. The lifted elements are not only much likely to be smooth because they have smaller norms, but it permits to set a smaller smoothness bound for the descent tree. Asymptotically, our algorithm is faster and works for a larger area of finite fields than Guillevic's algorithm, being now relevant even when the medium characteristic p is such that Lpn (1/3) ≤ p < Lpn (1/2). In practice, we conduct experiments on 500-bit to 2048-bit composite finite fields: Our method becomes more efficient as the largest non trivial divisor of n grows, being thus particularly adapted to even extension degrees.

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Section: Sublattices For Smaller Norms In the Number Fieldmentioning

confidence: 99%