On the discrete logarithm problem in finite fields of fixed characteristic

For $q$ a prime power, the discrete logarithm problem (DLP) in $\mathbb{F}_{q}$ consists in finding, for any $g \in \mathbb{F}_{q}^{\times}$ and $h \in \langle g \rangle$, an integer $x$ such that $g^x = h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime $p$ there exist infinitely many explicit extension fields $\mathbb{F}_{p^n}$ in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions $\mathbb{F}_{p^n}$ in expected quasi-polynomial time.Comment: 15 pages, 2 figures. To appear in Transactions of the AM

The Discrete Logarithm Problem in Finite Fields of Small Characteristic

Computing discrete logarithms is a long-standing algorithmic problem, whose hardness forms the basis for numerous current public-key cryptosystems. In the case of finite fields of small characteristic, however, there has been tremendous progress recently, by which the complexity of the discrete logarithm problem (DLP) is considerably reduced. This habilitation thesis on the DLP in such fields deals with two principal aspects. On one hand, we develop and investigate novel efficient algorithms for computing discrete logarithms, where the complexity analysis relies on heuristic assumptions. In particular, we show that logarithms of factor base elements can be computed in polynomial time, and we discuss practical impacts of the new methods on the security of pairing-based cryptosystems. While a heuristic running time analysis of algorithms is common practice for concrete security estimations, this approach is insufficient from a mathematical perspective. Therefore, on the other hand, we focus on provable complexity results, for which we modify the algorithms so that any heuristics are avoided and a rigorous analysis becomes possible. We prove that for any prime field there exist infinitely many extension fields in which the DLP can be solved in quasi-polynomial time. Despite the two aspects looking rather independent from each other, it turns out, as illustrated in this thesis, that progress regarding practical algorithms and record computations can lead to advances on the theoretical running time analysis -- and the other way around.Die Berechnung von diskreten Logarithmen ist ein eingehend untersuchtes algorithmisches Problem, dessen Schwierigkeit zahlreiche Anwendungen in der heutigen Public-Key-Kryptographie besitzt. Für endliche Körper kleiner Charakteristik sind jedoch kürzlich erhebliche Fortschritte erzielt worden, welche die Komplexität des diskreten Logarithmusproblems (DLP) in diesem Szenario drastisch reduzieren. Diese Habilitationsschrift erörtert zwei grundsätzliche Aspekte beim DLP in Körpern kleiner Charakteristik. Es werden einerseits neuartige, erheblich effizientere Algorithmen zur Berechnung von diskreten Logarithmen entwickelt und untersucht, wobei die Laufzeitanalyse auf heuristischen Annahmen beruht. Unter anderem wird gezeigt, dass Logarithmen von Elementen der Faktorbasis in polynomieller Zeit berechnet werden können, und welche praktischen Auswirkungen die neuen Verfahren auf die Sicherheit paarungsbasierter Kryptosysteme haben. Während heuristische Laufzeitabschätzungen von Algorithmen für die konkrete Sicherheitsanalyse üblich sind, so erscheint diese Vorgehensweise aus mathematischer Sicht unzulänglich. Der Aspekt der beweisbaren Komplexität für DLP-Algorithmen konzentriert sich deshalb darauf, modifizierte Algorithmen zu entwickeln, die jegliche heuristische Annahme vermeiden und dessen Laufzeit rigoros gezeigt werden kann. Es wird bewiesen, dass für jeden Primkörper unendlich viele Erweiterungskörper existieren, für die das DLP in quasi-polynomieller Zeit gelöst werden kann. Obwohl die beiden Aspekte weitgehend unabhängig voneinander erscheinen mögen, so zeigt sich, wie in dieser Schrift illustriert wird, dass Fortschritte bei praktischen Algorithmen und Rekordberechnungen auch zu Fortentwicklungen bei theoretischen Laufzeitabschätzungen führen -- und umgekehrt

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

Polynomial Time Bounded Distance Decoding near Minkowski's Bound in Discrete Logarithm Lattices

International audienceWe propose a concrete family of dense lattices of arbitrary dimension n in which the lattice Bounded Distance Decoding (BDD) problem can be solved in determin-istic polynomial time. This construction is directly adapted from the Chor-Rivest cryptosystem (IEEE-TIT 1988). The lattice construction needs discrete logarithm computations that can be made in deterministic polynomial time for well-chosen parameters. Each lattice comes with a deterministic polynomial time decoding algorithm able to decode up to large radius. Namely, we reach decoding radius within O(log n) Minkowski's bound, for both 1 and 2 norms

Границы сбалансированной степени вложения для криптографии на билинейных спариваниях

Вводится формула для расчёта границ сбалансированной степени вложения гиперэллиптической кривой. Вычислены текущие границы для кривых рода 1-3. Для кривых с известными алгоритмами генерации, наименьшими р-значениями и степенями вложения от 1 до 10 вычислен диапазон значений, которому принадлежит уровень безопасности кривой

A deterministic algorithm for the discrete logarithm problem in a semigroup

The discrete logarithm problem (DLP) in a finite group is the basis for many protocols in crypto-graphy. The best general algorithms which solve this problem have a time complexity of O(root N logN) and a space complexity of O(root N), where N is the order of the group. (If N is unknown, a simple modification would achieve a time complexity of (root N(logN)(2)).) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this article is a deterministic algorithm for solving the DLP in a semi-group. Specifically, let x be an element in a semigroup having finite order N-x. The article provides an algorithm, which, given any element y is an element of < x), provides all natural numbers m with x(m) = y, and has time complexity (root N-x(logN(x))(2)) steps. The article also gives an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely

Computing Discrete Logarithms

We describe some cryptographically relevant discrete logarithm problems (DLPs) and present some of the key ideas and constructions behind the most efficient algorithms known that solve them. Since the topic encompasses such a large volume of literature, for the finite field DLP we limit ourselves to a selection of results reflecting recent advances in fixed characteristic finite fields

Faster Privacy-Preserving Location Proximity Schemes

Peer reviewe

An Analysis of the Algebraic Group Model

The algebraic group model (AGM), formalized by Fuchsbauer, Kiltz, and Loss, has recently received significant attention. One of the appealing properties of the AGM is that it is viewed as being (strictly) weaker than the generic group model (GGM), in the sense that hardness results for algebraic algorithms imply hardness results for generic algorithms, and generic reductions in the AGM (namely, between the algebraic formulations of two problems) imply generic reductions in the GGM. We highlight that as the GGM and AGM are currently formalized, this is not true: hardness in the AGM may not imply hardness in the GGM, and a generic reduction in the AGM may not imply a similar reduction in the GGM

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented