Faster Multiplication in GF(2)[x]

Brent, Richard P.; Gaudry, Pierrick; Thomé, Emmanuel; Zimmermann, Paul

doi:10.1007/978-3-540-79456-1_10

Cited by 43 publications

(42 citation statements)

References 16 publications

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“…Other asymptotically faster algorithms exist for GF (2); the most interesting ones are due to Cantor and Schönhage [21,22]. Another approach is the use of segmentation, also known as Kronecker-Schönhage's trick, but the threshold between Toom-Cook and all these methods is far above 100 000 bits [23].…”

Section: Fast Polynomial Productmentioning

confidence: 99%

“…The given algorithm overwrites its input, so it should be slightly modified for general use; however, all the algorithms presented in [24], and proposed here, works with in-place operations. Other operations used in Toom-Cook are bit-shifts, but they can be avoided with word-alignment [23] in software implementation, and they have practically no cost in hardware implementations.…”

Section: Cost Of Exact Divisionsmentioning

confidence: 99%

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A New Analysis of the McEliece Cryptosystem Based on QC-LDPC Codes

Baldi

Bodrato

Chiaraluce

2008

Lecture Notes in Computer Science

108

120

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Abstract. We improve our proposal of a new variant of the McEliece cryptosystem based on QC-LDPC codes. The original McEliece cryptosystem, based on Goppa codes, is still unbroken up to now, but has two major drawbacks: long key and low transmission rate. Our variant is based on QC-LDPC codes and is able to overcome such drawbacks, while avoiding the known attacks. Recently, however, a new attack has been discovered that can recover the private key with limited complexity. We show that such attack can be avoided by changing the form of some constituent matrices, without altering the remaining system parameters. We also propose another variant that exhibits an overall increased security level. We analyze the complexity of the encryption and decryption stages by adopting efficient algorithms for processing large circulant matrices. The Toom-Cook algorithm and the short Winograd convolution are considered, that give a significant speed-up in the cryptosystem operations.

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Section: Fast Polynomial Productmentioning

confidence: 99%

Section: Cost Of Exact Divisionsmentioning

confidence: 99%

A New Analysis of the McEliece Cryptosystem Based on QC-LDPC Codes

Baldi

Bodrato

Chiaraluce

2008

Lecture Notes in Computer Science

108

120

View full text Add to dashboard Cite

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“…We use here the univariate basis C i introduced previously, which makes multiplication straightforward. However, several push-down and lift-up operations are now required tothe gf2x package [4], which provide the basic univariate polynomial arithmetic needed here. Our implementation handles three NTL classes of finite fields: GF2 for p = 2, zz_p for word-size p and ZZ_p for arbitrary p.…”

Section: Frobenius and Pseudotracementioning

confidence: 99%

Fast arithmetics in artin-schreier towers over finite fields

Feo

Schost

2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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An Artin-Schreier tower over the finite field F p is a tower of field extensions generated by polynomials of the form X p − X − α. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.

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“…Our search used a new algorithm [5,6] relying on fast arithmetic in GF(2) [x], whose details are given in [2]. Another significant improvement over previous work is that certificates were produced; this enables one to check easily that the claimed nonprimitive trinomials are indeed reducible.…”

Section: Remarks the Integers R Listed Inmentioning

confidence: 99%

Ten new primitive binary trinomials

Brent

Zimmermann

2008

Math. Comp.

Self Cite

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Abstract. We exhibit ten new primitive trinomials over GF(2) of record degrees 24 036 583, 25 964 951, 30 402 457, and 32 582 657. This completes the search for the currently known Mersenne prime exponents.Primitive trinomials of degree up to 6 972 593 were previously known [4]. We have completed a search for all known Mersenne prime exponents [7]. Ten new primitive trinomials were found. Our results are summarized in the following theorem: Theorem 1. For the integers r listed in Table 1, the primitive trinomials x r +x s +1 of degree r over GF(2) are exactly those given in Table 1, and the corresponding reciprocal trinomials x r + x r−s + 1. Proof. From the GIMPS Project [7], the integers r listed in Table 1 are exponents of Mersenne primes 2 r − 1. Thus, irreducible trinomials of degree r are necessarily primitive. Irreducibility of the trinomials listed in Table 1 follows from the authors' computations, using the new algorithm described in [5,6] (verified using the algorithm of [3] and independently verified by Allan Steel using Magma). Finally, the fact that no irreducible trinomials were missed during the search, for those degrees r, follows from the certificates given on the authors' web pages [1]. Table 1 are the known Mersenne exponents of the form r = ±1 mod 8 in the interval [100 000, 32 582 657]. For smaller exponents, omitted to save space, see [10] or our web site [1]. According to the GIMPS Project [7], the list is complete for r ≤ 16 300 000. Known Mersenne exponents of the form r = ±3 mod 8 for r > 5 cannot be the degrees of irreducible trinomials due to Swan's theorem [12]; the possibility x r +x 2 +1 permitted by Swan's theorem is easily ruled out in all known cases with r > 5; see the authors' web site [1]. Remarks. The integers r listed inOur search used a new algorithm [5,6] relying on fast arithmetic in GF(2) [x], whose details are given in [2]. Another significant improvement over previous work is that certificates were produced; this enables one to check easily that the claimed nonprimitive trinomials are indeed reducible. A certificate is simply an encoding of a nontrivial factor of smallest degree. A 2.4Ghz Intel Core 2 takes only 15 minutes to check the certificates of all 16 291 325 reducible trinomials (s ≤ r/2) of degree r = 32 582 657 with our check-ntl program based on NTL [11].

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Faster Multiplication in GF(2)[x]

Cited by 43 publications

References 16 publications

A New Analysis of the McEliece Cryptosystem Based on QC-LDPC Codes

A New Analysis of the McEliece Cryptosystem Based on QC-LDPC Codes

Fast arithmetics in artin-schreier towers over finite fields

Ten new primitive binary trinomials

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