Indiscreet discrete logarithms

In 2013 and 2014 a revolution took place in the understanding of the discrete logarithm problem (DLP) in finite fields of small characteristic. Consequently, many cryptosystems based on cryptographic pairings were rendered completely insecure, which serves as a valuable reminder that long-studied so-called hard problems may turn out to be far easier than initially believed. In this article, Robert Granger gives an overview of the surprisingly simple ideas behind some of the breakthroughs and the many computational records that have so far resulted from them.

Breaking `128-bit Secure' Supersingular Binary Curves

Robert Granger

The discrete logarithm problem (DLP) in finite fields of small characteristic recently enjoyed a dramatic series of breakthrough results and computational records, with its (heuristic) complexity dropping from subexponential to quasi-polynomial. While these results asymptotically render any cryptosystem relying on the hardness of such DLPs unusable, a question remained over whether the new techniques can weaken or indeed break any of the parameters proposed in the literature for pairing-based cryptographic protocols at the industry-standard 128-bit security level. In this talk I will first describe the ideas underlying the recent developments and then introduce some techniques which allow one to answer this question affirmatively. This is joint work with Thorsten Kleinjung and Jens Zumbragel.

2014

Arithmetic and geometric structures in cryptography

Benjamin Pierre Charles Wesolowski

We explore a few algebraic and geometric structures, through certain questions posed by modern cryptography. We focus on the cases of discrete logarithms in finite fields of small characteristic, the structure of isogeny graphs of ordinary abelian varieties, and the geometry of ideals in cyclotomic rings. The presumed difficulty of computing discrete logarithms in certain groups is essential for the security of a number of communication protocols deployed today. One of the most classic choices for the underlying group is the multiplicative group of a finite field. Yet this choice is showing its age, and particularly when the characteristic of the field is small: recent algorithms allow to compute logarithms efficiently in these groups. However, these methods are only heuristic: they seem to always work, yet we do not know how to prove it. In the first part, we propose to study these methods in the hope to get a better understanding, notably by revealing the geometric structures at play. A more modern choice is the group of rational points of an elliptic curve defined over a finite field. There, the difficulty of the discrete logarithm problem seems at its peak. More generally, the group of rational points of an abelian variety (notably the Jacobian of a curve of small genus) could be appropriate. One of the main tools for studying discrete logarithms on such objects is the notion of isogeny: a morphism from a variety to another one, which allows, among other things, to transfer the computation of a logarithm. Whereas the theory for elliptic curves is already mature, little is known about the structures formed by these isogenies (the isogeny graphs) for varieties of higher dimension. In the second part, we study the structure of isogeny graphs of absolutely simple, ordinary abelian varieties, with a few consequences regarding discrete logarithms on Jacobians of hyperelliptic curves of genus 2, the main object of concern of so-called hyperelliptic cryptography. The security of quite a few protocols, notably those relying on discrete logarithms, would collapse in front of an adversary equipped with a large-scale quantum computer. This perspective motivates cryptographers to study problems that would resist this technological feat. One of the major directions is cryptography based on Euclidean lattices, relying on the difficulty to find short vectors in a given lattice. For efficiency, one benefits from considering lattices endowed with more structure, such as the ideals of a cyclotomic field. In the third part, we study the geometry of these ideals, and show that a quantum computer allows to efficiently find much shorter vectors in these ideals than is currently possible in generic lattices.

EPFL2018

On the Cryptanalysis of Public-Key Cryptography

Joppe Willem Bos

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient "sloppy reduction." The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units.