26 research outputs found
New Complexity Trade-Offs for the (Multiple) Number Field Sieve Algorithm in Non-Prime Fields
The selection of polynomials to represent number fields crucially determines the efficiency of the Number Field Sieve
(NFS) algorithm for solving the discrete logarithm in a finite field. An important recent work due to Barbulescu et al. builds upon
existing works to propose two new methods for polynomial selection when the target field is a non-prime field. These methods are
called the generalised Joux-Lercier (GJL) and the Conjugation methods. In this work, we propose a new method (which we denote
as ) for polynomial selection for the NFS algorithm in fields , with and .
The new method both subsumes and generalises the GJL and the Conjugation methods and provides new trade-offs for both composite
and prime. Let us denote the variant of the (multiple) NFS algorithm using the polynomial selection method ``{X} by (M)NFS-{X}.
Asymptotic analysis is performed for both the NFS- and the MNFS- algorithms.
In particular, when , for , the complexity of NFS- is better than the complexities
of all previous algorithms whether classical or MNFS. The MNFS- algorithm provides lower complexity compared to
NFS- algorithm; for , the complexity of MNFS-
is the same as that of the MNFS-Conjugation and for , the complexity of MNFS-
is lower than that of all previous methods
Relation collection using Pollard special-q sieving to solve integer factorization and discrete logarithm problem
A nonuniform algorithm for the hidden number problem in subgroups
Abstract. Boneh and Venkatesan have proposed a polynomial time algorithm in a non-uniform model for recovering a ”hidden ” element α ∈ IFp, where p is prime, from very short strings of the most significant bits of the residue of αt modulo p for several randomly chosen t ∈ IFp. Here we modify the scheme and amplify the uniformity of distribution of the ‘multipliers ’ t and thus extend this result to subgroups of IF ∗ p, which are more relevant to practical usage. As in the work of Boneh and Venkatesan, our result can be applied to the bit security of Diffie–Hellman related encryption schemes starting with subgroups of very small size, including all cryptographically interesting subgroups