198 research outputs found
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
Solving a 676-Bit Discrete Logarithm Problem in GF(36n )
Pairings on elliptic curves over finite fields are crucial for constructing various cryptographic schemes. The \eta_T pairing on supersingular curves over GF(3^n) is particularly popular since it is efficiently implementable. Taking into account the Menezes-Okamoto-Vanstone (MOV) attack, the discrete logarithm problem (DLP) in GF(3^{6n}) becomes a concern for the security of cryptosystems using \eta_T pairings in this case. In 2006, Joux and Lercier proposed a new variant of the function field sieve in the medium prime case, named JL06-FFS. We have, however, not yet found any practical implementations on JL06-FFS over GF(3^{6n}). Therefore, we first fulfilled such an implementation and we successfully set a new record for solving the DLP in GF(3^{6n}), the DLP in GF(3^{6 \cdot 71}) of 676-bit size. In addition, we also compared JL06-FFS and an earlier version, named JL02-FFS, with practical experiments. Our results confirm that the former is several times faster than the latter under certain conditions
Collecting relations for the number field sieve in
International audienceIn order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in GF(p^6) with the Number Field Sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-q strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-q and various enumeration strategies
Improvements on the Individual Logarithm Step in Extended Tower Number Field Sieve
The hardness of discrete logarithm problem over finite fields is the foundation of many cryptographic protocols. When the characteristic of the finite field is medium or large, the state-of-art algorithms for solving the corresponding problem are the number field sieve and its variants. There are mainly three steps in such algorithms: polynomial selection, factor base logarithms computation, and individual logarithm computation. Note that the former two steps can be precomputed for fixed finite field,
and the database containing factor base logarithms can be used by the last step for many times. In certain application circumstances, such as Logjam attack, speeding up the individual logarithm step is vital.
In this paper, we devise a method to improve the individual logarithm step by exploring subfield structures. Our method is based on the extended tower number field sieve algorithm,
and achieves more significant improvement when the extension degree has a large proper factor. We also perform some experiments to illustrate our algorithm and confirm the result
Higher dimensional sieving for the number field sieve algorithms
International audienceSince 2016 and the introduction of the exTNFS (extended Tower Number Field Sieve) algorithm, the security of cryptosystems based on non-prime finite fields, mainly the paring and torus-based one, is being reassessed. The feasibility of the relation collection, a crucial step of the NFS variants, is especially investigated. It usually involves polynomials of degree one, i.e., a search space of dimension two. However, exTNFS uses bivariate polynomials of at least four coefficients. If sieving in dimension two is well described in the literature, sieving in higher dimension received significantly less attention. We describe and analyze three different generic algorithms to sieve in any dimension for the NFS algorithms. Our implementation shows the practicability of dimension four sieving, but the hardness of dimension six sieving
Efficient subgroup exponentiation in quadratic and sixth degree extensions
This paper describes several speedups for computation in the order p + 1 subgroup of F*(p2) and the order p2 - p + 1 subgroup of F*(p6). These results are in a way complementary to LUC and XTR, where computations in these groups are sped up using trace maps. As a side result, we present an efficient method for XTR with p ≡ 3 mod
A Generalisation of the Conjugation Method for Polynomial Selection for the Extended Tower Number Field Sieve Algorithm
In a recent work, Kim and Barbulescu showed how to combine previous polynomial selection methods with the extended tower
number field sieve algorithm to obtain improved complexity for the discrete logarithm problem on finite fields
for the medium prime case and where is composite and not a prime-power. A follow up work by Sarkar and Singh presented a
general polynomial selection method and showed how to lower the complexity in the medium prime case even when is composite
and a prime-power. This complexity, though, was higher than what was reported for the case of composite and not a prime-power.
By suitably combining the Conjugation method of polynomial selection proposed earlier by Barbulescu et al. with the extended tower
number field sieve algorithm, Jeong and Kim showed that the same asymptotic complexity is achieved for any composite .
The present work generalises the polynomial selection method of Jeong and Kim for all composite . Though the best complexity that can
be achieved is not lowered, there is a significant range of finite fields for which the new algorithm achieves complexity which
is lower than all previously proposed methods
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