A construction of 3-dimensional lattice sieve for number field sieve over F_{p^n}

The security of pairing-based cryptography is based on the hardness of solving the discrete logarithm problem (DLP) over extension field F_{p^n} of characteristic p and degree n. Joux et al. proposed an asymptotically fastest algorithm for solving DLP over F_{p^n} (JLSV06-NFS) as the extension of the number field sieve over prime field F _p (JL03-NFS). The lattice sieve is often used for a large-scaled experiment of solving DLP over F_p by the number field sieve. Franke and Kleinjung proposed a 2-dimensional lattice sieve which efficiently enumerates all the points in a given sieve region of the lattice. However, we have to consider a sieve region of more than 2 dimensions in the lattice sieve of JLSV06-NFS. In this paper, we extend the Franke-Kleinjung method to 3-dimensional sieve region. We construct an appropriate basis using the Hermite normal form, which can enumerate the points in a given sieve region of the 3-dimensional lattice. From our experiment on F_{p^{12}} of 303 bits, we are able to enumerate more than 90\% of the points in a sieve region in the lattice generated by special-q. Moreover, we implement the number field sieve using the proposed 3-dimensional lattice sieve. Our implementation of the JLSV06 over F_{p^6} of 240 bits is about as efficient as that of the current record over F_{p^6} using 3-dimensional line sieve by Zajac

Discrete logarithm computations over finite fields using Reed-Solomon codes

Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2) operations over GF(q), operating on a q x q matrix with (h+2) q non-zero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of GF(q^h); this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations

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

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 GF(p6)GF(p^6)

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

The Tower Number Field Sieve

The security of pairing-based crypto-systems relies on the difficulty to compute discrete logarithms in finite fields GF(p^n) where n is a small integer larger than 1. The state-of-art algorithm is the number field sieve (NFS) together with its many variants. When p has a special form (SNFS), as in many pairings constructions, NFS has a faster variant due to Joux and Pierrot. We present a new NFS variant for SNFS computations, which is better for some cryptographically relevant cases, according to a precise comparison of norm sizes. The new algorithm is an adaptation of Schirokauer\u27s variant of NFS based on tower extensions, for which we give a middlebrow presentation

On Improving Integer Factorization and Discrete Logarithm Computation using Partial Triangulation

The number field sieve is the best-known algorithm for factoring integers and solving the discrete logarithm problem in prime fields. In this paper, we present some new improvements to various steps of the number field sieve. We apply these improvements on the current 768-bit discrete logarithm record and show that we are able to perform the overall computing time in about 1260 core$\cdot$years using these improvements instead of 2350 core$\cdot$years using the best known parameters for this problem. Moreover, we show that the pre-computation phase for a 768-bit discrete logarithm problem, that allows for example to build a massive decryption tool of IPsec traffic protected by the Oakley group~1, was feasible in reasonable time using technologies available before the year 2000

Breaking pairing-based cryptosystems using ηT\eta_T pairing over GF(397)GF(3^{97})

There are many useful cryptographic schemes, such as ID-based encryption, short signature, keyword searchable encryption, attribute-based encryption, functional encryption, that use a bilinear pairing. It is important to estimate the security of such pairing-based cryptosystems in cryptography. The most essential number-theoretic problem in pairing-based cryptosystems is the discrete logarithm problem (DLP) because pairing-based cryptosystems are no longer secure once the underlining DLP is broken. One efficient bilinear pairing is the $\eta_T$ pairing defined over a supersingular elliptic curve $E$ on the finite field $GF(3^n)$ for a positive integer $n$. The embedding degree of the $\eta_T$ pairing is $6$; thus, we can reduce the DLP over $E$ on $GF(3^n)$ to that over the finite field $GF(3^{6n})$. In this paper, for breaking the $\eta_T$ pairing over $GF(3^n)$, we discuss solving the DLP over $GF(3^{6n})$ by using the function field sieve (FFS), which is the asymptotically fastest algorithm for solving a DLP over finite fields of small characteristics. We chose the extension degree $n=97$ because it has been intensively used in benchmarking tests for the implementation of the $\eta_T$ pairing, and the order (923-bit) of $GF(3^{6\cdot 97})$ is substantially larger than the previous world record (676-bit) of solving the DLP by using the FFS. We implemented the FFS for the medium prime case (JL06-FFS), and propose several improvements of the FFS, for example, the lattice sieve for JL06-FFS and the filtering adjusted to the Galois action. Finally, we succeeded in solving the DLP over $GF(3^{6\cdot 97})$. The entire computational time of our improved FFS requires about 148.2 days using 252 CPU cores. Our computational results contribute to the secure use of pairing-based cryptosystems with the $\eta_T$ pairing

Discrete Logarithm in GF(2809) with FFS

International audienceThe year 2013 has seen several major complexity advances for the discrete logarithm problem in multiplicative groups of small- characteristic finite fields. These outmatch, asymptotically, the Function Field Sieve (FFS) approach, which was so far the most efficient algorithm known for this task. Yet, on the practical side, it is not clear whether the new algorithms are uniformly better than FFS. This article presents the state of the art with regard to the FFS algorithm, and reports data from a record-sized discrete logarithm computation in a prime-degree extension field