Discrete logarithms in curves over finite fields

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

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

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

Efficient dot product over word-size finite fields

We want to achieve efficiency for the exact computation of the dot product of two vectors over word-size finite fields. We therefore compare the practical behaviors of a wide range of implementation techniques using different representations. The techniques used include oating point representations, discrete logarithms, tabulations, Montgomery reduction, delayed modulus

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

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

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

Quantum resource estimates for computing elliptic curve discrete logarithms

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added. ASIACRYPT 201

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