On Class Group Computations Using the Number Field Sieve

The best practical algorithm for class group computations in imaginary quadratic number fields (such as group structure, class number, discrete logarithm computations) is a variant of the quadratic sieve factoring algorithm. Paradoxical as it sounds, the principles of the number field sieve, in a strict sense, could not be applied to number field computations, yet. In this article we give an indication of the obstructions. In particular, we first present fundamental core elements of a number field sieve for number field computations of which it is absolutely unknown how to design them in a useful way. Finally, we show that the existence of a number field sieve for number field computations with a running time asymptotics similar to that of the genuine number field sieve likely implies the existence of an algorithm for elliptic curve related computational problems with subexponential running time

Improving NFS for the Discrete Logarithm Problem in Non-prime Finite Fields

International audienceThe aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF$(p^n)$ where $n$ is a small integer greater than 1. Though less studied than the small characteristic case or the prime field case, the difficulty of this problem is at the heart of security evaluations for torus-based and pairing-based cryptography. The best known method for solving this problem is the Number Field Sieve (NFS). A key ingredient in this algorithm is the ability to find good polynomials that define the extension fields used in NFS. We design two new methods for this task, modifying the asymptotic complexity and paving the way for record-breaking computations. We exemplify these results with the computation of discrete logarithms over a field GF$(p^2)$ whose cardinality is 180 digits (595 bits) long

New Complexity Trade-Offs for the (Multiple) Number Field Sieve Algorithm in Non-Prime Fields

The selection of polynomials to represent number fields crucially determines the efficiency of the Number Field Sieve (NFS) algorithm for solving the discrete logarithm in a finite field. An important recent work due to Barbulescu et al. builds upon existing works to propose two new methods for polynomial selection when the target field is a non-prime field. These methods are called the generalised Joux-Lercier (GJL) and the Conjugation methods. In this work, we propose a new method (which we denote as $\mathcal{A}$) for polynomial selection for the NFS algorithm in fields $\mathbb{F}_{Q}$, with $Q=p^n$ and $n>1$. The new method both subsumes and generalises the GJL and the Conjugation methods and provides new trade-offs for both $n$ composite and $n$ prime. Let us denote the variant of the (multiple) NFS algorithm using the polynomial selection method ``{X} by (M)NFS-{X}. Asymptotic analysis is performed for both the NFS-$\mathcal{A}$ and the MNFS-$\mathcal{A}$ algorithms. In particular, when $p=L_Q(2/3,c_p)$, for $c_p\in [3.39,20.91]$, the complexity of NFS-$\mathcal{A}$ is better than the complexities of all previous algorithms whether classical or MNFS. The MNFS-$\mathcal{A}$ algorithm provides lower complexity compared to NFS-$\mathcal{A}$ algorithm; for $c_p\in (0, 1.12] \cup [1.45,3.15]$, the complexity of MNFS-$\mathcal{A}$ is the same as that of the MNFS-Conjugation and for $c_p\notin (0, 1.12] \cup [1.45,3.15]$, the complexity of MNFS-$\mathcal{A}$ is lower than that of all previous methods

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

Extended Tower Number Field Sieve with Application to Finite Fields of Arbitrary Composite Extension Degree

We propose a generalization of exTNFS algorithm recently introduced by Kim and Barbulescu (CRYPTO 2016). The algorithm, exTNFS, is a state-of-the-art algorithm for discrete logarithm in $\mathbb{F}_{p^n}$ in the medium prime case, but it only applies when $n=\eta\kappa$ is a composite with nontrivial factors $\eta$ and $\kappa$ such that $\gcd(\eta,\kappa)=1$. Our generalization, however, shows that exTNFS algorithm can be also adapted to the setting with an arbitrary composite $n$ maintaining its best asymptotic complexity. We show that one can solve discrete logarithm in medium case in the running time of $L_{p^n}(1/3, \sqrt[3]{48/9})$ (resp. $L_{p^n}(1/3, 1.71)$ if multiple number fields are used), where $n$ is an \textit{arbitrary composite}. This should be compared with a recent variant by Sarkar and Singh (Asiacrypt 2016) that has the fastest running time of $L_{p^n}(1/3, \sqrt[3]{64/9})$ (resp. $L_{p^n}(1/3, 1.88)$) when $n$ is a power of prime 2. When $p$ is of special form, the complexity is further reduced to $L_{p^n}(1/3, \sqrt[3]{32/9})$. On the practical side, we emphasize that the keysize of pairing-based cryptosystems should be updated following to our algorithm if the embedding degree $n$ remains composite

Hard Instances of the Constrained Discrete Logarithm Problem

The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds

Breaking the encryption scheme of the Moscow Internet voting system

This work is a merger of arXiv:1908.09170 and arXiv:1908.05127.International audienceIn September 2019, voters for the election at the Parliament of the city of Moscow were allowed to use an Internet voting system. The source code of it had been made available for public testing. In this paper we show two successful attacks on the encryption scheme implemented in the voting system. Both attacks were sent to the developers of the system, and both issues had been fixed after that.The encryption used in this system is a variant of ElGamal over finite fields. In the first attack we show that the used key sizes are too small. We explain how to retrieve the private keys from the public keys in a matter of minutes with easily available resources.When this issue had been fixed and the new system had become available for testing, we discovered that the new implementation was not semantically secure. We demonstrate how this newly found security vulnerability can be used for counting the number of votes cast for a candidate

Asymptotic complexities of discrete logarithm algorithms in pairing-relevant finite fields

International audienceWe study the discrete logarithm problem at the boundary case between small and medium characteristic finite fields, which is precisely the area where finite fields used in pairing-based cryptosystems live. In order to evaluate the security of pairing-based protocols, we thoroughly analyze the complexity of all the algorithms that coexist at this boundary case: the Quasi-Polynomial algorithms, the Number Field Sieve and its many variants, and the Function Field Sieve. We adapt the latter to the particular case where the extension degree is composite, and show how to lower the complexity by working in a shifted function field. All this study finally allows us to give precise values for the characteristic asymptotically achieving the highest security level for pairings. Surprisingly enough, there exist special characteristics that are as secure as general ones