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

Symbolic Methods for Solving Algebraic Systems of Equations and Applications for Testing the Structural Stability

International audienceIn this work, we provide an overview of the classical symbolic techniques for solving algebraic systems of equations and show the interest of such techniques in the study of some problems in dynamical system theory, namely testing the structural stability of multidimensional systems

A General Polynomial Selection Method and New Asymptotic Complexities for the Tower Number Field Sieve Algorithm

In a recent work, Kim and Barbulescu had extended the tower number field sieve algorithm to obtain improved asymptotic complexities in the medium prime case for the discrete logarithm problem on $\mathbb{F}_{p^n}$ where $n$ is not a prime power. Their method does not work when $n$ is a composite prime power. For this case, we obtain new asymptotic complexities, e.g., $L_{p^n}(1/3,(64/9)^{1/3})$ (resp. $L_{p^n}(1/3,1.88)$ for the multiple number field variation) when $n$ is composite and a power of 2; the previously best known complexity for this case is $L_{p^n}(1/3,(96/9)^{1/3})$ (resp. $L_{p^n}(1/3,2.12)$). These complexities may have consequences to the selection of key sizes for pairing based cryptography. The new complexities are achieved through a general polynomial selection method. This method, which we call Algorithm-$\mathcal{C}$, extends a previous polynomial selection method proposed at Eurocrypt 2016 to the tower number field case. As special cases, it is possible to obtain the generalised Joux-Lercier and the Conjugation method of polynomial selection proposed at Eurocrypt 2015 and the extension of these methods to the tower number field scenario by Kim and Barbulescu. A thorough analysis of the new algorithm is carried out in both concrete and asymptotic terms

Challenges with Assessing the Impact of NFS Advances on the Security of Pairing-based Cryptography

In the past two years there have been several advances in Number Field Sieve (NFS) algorithms for computing discrete logarithms in finite fields $\mathbb{F}_{p^n}$ where $p$ is prime and $n > 1$ is a small integer. This article presents a concise overview of these algorithms and discusses some of the challenges with assessing their impact on keylengths for pairing-based cryptosystems

Asymptotic complexities of discrete logarithm algorithms in pairing-relevant finite fields

International audienceWe study the discrete logarithm problem at the boundary case between small and medium characteristic finite fields, which is precisely the area where finite fields used in pairing-based cryptosystems live. In order to evaluate the security of pairing-based protocols, we thoroughly analyze the complexity of all the algorithms that coexist at this boundary case: the Quasi-Polynomial algorithms, the Number Field Sieve and its many variants, and the Function Field Sieve. We adapt the latter to the particular case where the extension degree is composite, and show how to lower the complexity by working in a shifted function field. All this study finally allows us to give precise values for the characteristic asymptotically achieving the highest security level for pairings. Surprisingly enough, there exist special characteristics that are as secure as general ones