Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Structure computation and discrete logarithms in finite abelian p-groups

We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G.Comment: 23 pages, minor edit

Security Estimates for Quadratic Field Based Cryptosystems

We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field. The algorithms used incorporate improvements over previously-used algorithms, and extensive numerical results are presented demonstrating their efficiency. This data is used as the basis for extrapolations, used to provide recommendations for parameter sizes providing approximately the same level of security as block ciphers with $80,$ $112,$ $128,$ $192,$ and $256$-bit symmetric keys

Computing discrete logarithms in subfields of residue class rings

Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete logarithms in small characteristic finite fields share an interesting feature in common with the earlier medium prime function field sieve method \cite{jl}. To solve discrete logarithms in a finite extension of a finite field \F, a polynomial h(x) \in \F[x] of a special form is constructed with an irreducible factor g(x) \in \F[x] of the desired degree. The special form of $h(x)$ is then exploited in generating multiplicative relations that hold in the residue class ring \F[x]/h(x)\F[x] hence also in the target residue class field \F[x]/g(x)\F[x]. An interesting question in this context and addressed in this paper is: when and how does a set of relations on the residue class ring determine the discrete logarithms in the finite fields contained in it? We give necessary and sufficient conditions for a set of relations on the residue class ring to determine discrete logarithms in the finite fields contained in it. We also present efficient algorithms to derive discrete logarithms from the relations when the conditions are met. The derived necessary conditions allow us to clearly identify structural obstructions intrinsic to the special polynomial $h(x)$ in each of the aforementioned methods, and propose modifications to the selection of $h(x)$ so as to avoid obstructions.Comment: arXiv admin note: substantial text overlap with arXiv:1312.167

Quantum algorithms for problems in number theory, algebraic geometry, and group theory

Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in Quantum Computation/Information at Kinki Universit

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

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes