204 research outputs found

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

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    International audienceThe aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(pn)(p^n) where nn 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(p2)(p^2) whose cardinality is 180 digits (595 bits) long

    Solving discrete logarithms on a 170-bit MNT curve by pairing reduction

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    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

    Security Analysis of Pairing-based Cryptography

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    Recent progress in number field sieve (NFS) has shaken the security of Pairing-based Cryptography. For the discrete logarithm problem (DLP) in finite field, we present the first systematic review of the NFS algorithms from three perspectives: the degree α\alpha, constant cc, and hidden constant o(1)o(1) in the asymptotic complexity LQ(α,c)L_Q\left(\alpha,c\right) and indicate that further research is required to optimize the hidden constant. Using the special extended tower NFS algorithm, we conduct a thorough security evaluation for all the existing standardized PF curves as well as several commonly utilized curves, which reveals that the BN256 curves recommended by the SM9 and the previous ISO/IEC standard exhibit only 99.92 bits of security, significantly lower than the intended 128-bit level. In addition, we comprehensively analyze the security and efficiency of BN, BLS, and KSS curves for different security levels. Our analysis suggests that the BN curve exhibits superior efficiency for security strength below approximately 105 bit. For a 128-bit security level, BLS12 and BLS24 curves are the optimal choices, while the BLS24 curve offers the best efficiency for security levels of 160bit, 192bit, and 256bit.Comment: 8 figures, 8 tables, 5121 word

    Faster individual discrete logarithms in finite fields of composite extension degree

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    International audienceComputing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms in large and medium characteristic fields (e.g., {GF}(p2)(p^2), {GF}(p12)(p^{12})) are the Number Field Sieve and its variants (special, high-degree, tower). The best algorithms in small characteristic finite fields (e.g., {GF}(36509)(3^{6 \cdot 509})) are the Function Field Sieve, Joux's algorithm, and the quasipolynomial-time algorithm. The last step of this family of algorithms is the individual logarithm computation. It computes a smooth decomposition of a given target in two phases: an initial splitting, then a descent tree. While new improvements have been made to reduce the complexity of the dominating relation collection and linear algebra steps, resulting in a smaller factor basis (database of known logarithms of small elements), the last step remains at the same level of difficulty. Indeed, we have to find a smooth decomposition of a typically large element in the finite field. This work improves the initial splitting phase and applies to any nonprime finite field. It is very efficient when the extension degree is composite. It exploits the proper subfields, resulting in a much more smooth decomposition of the target. This leads to a new trade-off between the initial splitting step and the descent step in small characteristic. Moreover it reduces the width and the height of the subsequent descent tree

    Discrete logarithm computations over finite fields using Reed-Solomon codes

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    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

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

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    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

    Pairings in Cryptology: efficiency, security and applications

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    Abstract The study of pairings can be considered in so many di�erent ways that it may not be useless to state in a few words the plan which has been adopted, and the chief objects at which it has aimed. This is not an attempt to write the whole history of the pairings in cryptology, or to detail every discovery, but rather a general presentation motivated by the two main requirements in cryptology; e�ciency and security. Starting from the basic underlying mathematics, pairing maps are con- structed and a major security issue related to the question of the minimal embedding �eld [12]1 is resolved. This is followed by an exposition on how to compute e�ciently the �nal exponentiation occurring in the calculation of a pairing [124]2 and a thorough survey on the security of the discrete log- arithm problem from both theoretical and implementational perspectives. These two crucial cryptologic requirements being ful�lled an identity based encryption scheme taking advantage of pairings [24]3 is introduced. Then, perceiving the need to hash identities to points on a pairing-friendly elliptic curve in the more general context of identity based cryptography, a new technique to efficiently solve this practical issue is exhibited. Unveiling pairings in cryptology involves a good understanding of both mathematical and cryptologic principles. Therefore, although �rst pre- sented from an abstract mathematical viewpoint, pairings are then studied from a more practical perspective, slowly drifting away toward cryptologic applications

    Challenges with Assessing the Impact of NFS Advances on the Security of Pairing-based Cryptography

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    In the past two years there have been several advances in Number Field Sieve (NFS) algorithms for computing discrete logarithms in finite fields Fpn\mathbb{F}_{p^n} where pp is prime and n>1n > 1 is a small integer. This article presents a concise overview of these algorithms and discusses some of the challenges with assessing their impact on keylengths for pairing-based cryptosystems
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