Solving a 676-Bit Discrete Logarithm Problem in GF(36n )

Hayashi, Takuya; Shinohara, Naoyuki; Wang, Lihua; Matsuo, Shin’ichiro; Shirase, Masaaki; Takagi, Tsuyoshi

doi:10.1007/978-3-642-13013-7_21

Cited by 11 publications

(14 citation statements)

References 26 publications

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“…This culminated with a record set by Joux and Lercier who computed discrete logarithms in F 2 613 and surpassed the previous record set by Thomé [8] in 2001 using Coppersmith's algorithm [9]. A long period with essentially no activity on the topic has followed; it ended recently with a new record [10], [11] set over F 3 6×97 , which took advantage of the fact that the extension was composite; this particular case is of practical interest in pairing-based cryptography.…”

Section: Introductionmentioning

confidence: 99%

Relation Collection for the Function Field Sieve

Detrey

Gaudry²,

Videau³

2013

2013 IEEE 21st Symposium on Computer Arithmetic

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Abstract-In this paper, we focus on the relation collection step of the Function Field Sieve (FFS), which is to date the best algorithm known for computing discrete logarithms in smallcharacteristic finite fields of cryptographic sizes. Denoting such a finite field by Fpn , where p is much smaller than n, the main idea behind this step is to find polynomials of the formwhich, when considered as principal ideals in carefully selected function fields, can be factored into products of low-degree prime ideals. Such polynomials are called "relations", and current record-sized discrete-logarithm computations need billions of those.Collecting relations is therefore a crucial and extremely expensive step in FFS, and a practical implementation thereof requires heavy use of cache-aware sieving algorithms, along with efficient polynomial arithmetic over Fp[t]. This paper presents the algorithmic and arithmetic techniques which were put together as part of a new public implementation of FFS, aimed at medium-to record-sized computations.

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Section: Introductionmentioning

confidence: 99%

Relation Collection for the Function Field Sieve

Detrey

Gaudry²,

Videau³

2013

2013 IEEE 21st Symposium on Computer Arithmetic

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“…The next important class of problem contains the sieving-based index calculus algorithm for factoring [1,5,15] and discrete logarithms [11][12][13]. In this class, the largest phase of the computation (the sieving phase) is embarrassingly parallel, however, it produces a large amount of data which needs to be collected in a centralized place.…”

Section: Typical Cryptanalytic Applicationsmentioning

confidence: 99%

A Tutorial on High Performance Computing Applied to Cryptanalysis

Joux

2012

Advances in Cryptology – EUROCRYPT 2012

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“…Moreover, free relation [19] provides additional relations without computation with a sieving algorithm.…”

Section: Polynomial Selection Phasementioning

confidence: 99%

“…The reduced system of linear equations is solved using the parallel Lanczos method [4,19] or other methods, and the discrete logarithms of elements in the factor base are obtained:…”

Section: Linear Algebra Phasementioning

confidence: 99%

“…The cardinality of the subgroup of the supersingular elliptic curve is 151 bits, and that of GF (3 6·97 ) is 923 bits. The size of our target DLP is 247 bits larger than the previous world record of solving the DLP over GF (3 6·71 ), whose cardinality is 676 bits [19]. The current world record for solving an ECDLP is the 112-bit ECDLP [14].…”

Section: Introductionmentioning

confidence: 99%

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Breaking Pairing-Based Cryptosystems Using η T Pairing over GF(397)

Hayashi

Shimoyama

Shinohara

et al. 2012

Advances in Cryptology – ASIACRYPT 2012

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Abstract. 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 ηT 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 ηT 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 ηT 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 ηT pairing, and the order (923-bit) of GF (3 6·97 ) 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 6·97 ). 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 ηT pairing.

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Solving a 676-Bit Discrete Logarithm Problem in GF(36n )

Cited by 11 publications

References 26 publications

Relation Collection for the Function Field Sieve

Relation Collection for the Function Field Sieve

A Tutorial on High Performance Computing Applied to Cryptanalysis

Breaking Pairing-Based Cryptosystems Using η T Pairing over GF(397)

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