69 research outputs found
A kilobit hidden SNFS discrete logarithm computation
We perform a special number field sieve discrete logarithm computation in a
1024-bit prime field. To our knowledge, this is the first kilobit-sized
discrete logarithm computation ever reported for prime fields. This computation
took a little over two months of calendar time on an academic cluster using the
open-source CADO-NFS software. Our chosen prime looks random, and
has a 160-bit prime factor, in line with recommended parameters for the Digital
Signature Algorithm. However, our p has been trapdoored in such a way that the
special number field sieve can be used to compute discrete logarithms in
, yet detecting that p has this trapdoor seems out of reach.
Twenty-five years ago, there was considerable controversy around the
possibility of back-doored parameters for DSA. Our computations show that
trapdoored primes are entirely feasible with current computing technology. We
also describe special number field sieve discrete log computations carried out
for multiple weak primes found in use in the wild. As can be expected from a
trapdoor mechanism which we say is hard to detect, our research did not reveal
any trapdoored prime in wide use. The only way for a user to defend against a
hypothetical trapdoor of this kind is to require verifiably random primes
Root optimization of polynomials in the number field sieve
The 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.Comment: 16 pages, 18 reference
Number Field Sieve with Provable Complexity
In this thesis we give an in-depth introduction to the General Number Field
Sieve, as it was used by Buhler, Lenstra, and Pomerance, before looking at one
of the modern developments of this algorithm: A randomized version with
provable complexity. This version was posited in 2017 by Lee and Venkatesan and
will be preceded by ample material from both algebraic and analytic number
theory, Galois theory, and probability theory.Comment: MSc Thesis, 113 pages, 1 tabl
A New Ranking Function for Polynomial Selection in the Number Field Sieve
International audienceThis article explains why the classical Murphy-E ranking function might fail to correctly rank polynomial pairs in the Number Field Sieve, and proposes a new ranking function
Montgomery's method of polynomial selection for the number field sieve
The number field sieve is the most efficient known algorithm for factoring
large integers that are free of small prime factors. For the polynomial
selection stage of the algorithm, Montgomery proposed a method of generating
polynomials which relies on the construction of small modular geometric
progressions. Montgomery's method is analysed in this paper and the existence
of suitable geometric progressions is considered
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