Normal bases via general Gauss periods

Feisel, Sandra; Gathen, Joachim von zur; Shokrollahi, Mohammad Amin

doi:10.1090/s0025-5718-99-00988-6

Cited by 40 publications

(38 citation statements)

References 8 publications

(12 reference statements)

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“…For odd values of m, it is well known [15] that a Gaussian normal basis of type k or (m, k) always exists for some k ≥ 1. Since mk + 1 is a prime with m = odd, it follows that k is even.…”

Section: We Havementioning

confidence: 99%

Efficient Linear Array for Multiplication in GF(2 m ) Using a Normal Basis for Elliptic Curve Cryptography

Kwon

Gaj

Kim

et al. 2004

Lecture Notes in Computer Science

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Abstract. We present a new sequential normal basis multiplier over GF (2 m ). The gate complexity of our multiplier is significantly reduced from that of Agnew et al. and is comparable to that of Reyhani-Masoleh and Hasan, which is the lowest complexity normal basis multiplier of the same kinds. On the other hand, the critical path delay of our multiplier is same to that of Agnew et al. Therefore it is supposed to have a shorter or the same critical path delay to that of Reyhani-Masoleh and Hasan. Moreover our method of using a Gaussian normal basis makes it easy to find a basic multiplication table of normal elements. So one can easily construct a circuit array for large finite fields, GF (2 m ) where m = 163, 233, 283, 409, 571, i.e. the five recommended fields by NIST for elliptic curve cryptography.

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Section: We Havementioning

confidence: 99%

Efficient Linear Array for Multiplication in GF(2 m ) Using a Normal Basis for Elliptic Curve Cryptography

Kwon

Gaj

Kim

et al. 2004

Lecture Notes in Computer Science

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“…We use the standard notation 1S, K2 for the subgroup generated by the elements in S and K together, and G/K, or % ) , for the quotient group of G by K. Next we apply this theorem to some problems in "nite "elds that arise in the work of Feisel et al [7] on constructing normal bases from Gauss periods.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In recent years, special Gauss periods have been successfully used to construct normal bases of low complexity [4,7,11,17] and for implementation of "nite "elds [2,3,19]. While Gauss periods can be de"ned in any "nite Galois extension of an arbitrary "eld (see Pohst and Zassenhaus [20, pp.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Abelian Groups, Gauss Periods, and Normal Bases

Gao

2001

Finite Fields and Their Applications

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“…However one may repeat the same arguments as in Lemma 2 and 3 based on the more general Gauss periods [20] …”

Section: Lemma 3 If a Gaussian Normal Basis Of Type (2 K) Is Used Fmentioning

confidence: 93%

Fast Irreducibility Testing for XTR Using a Gaussian Normal Basis of Low Complexity

Kwon

Kim

Hong

2004

Selected Areas in Cryptography

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Abstract. XTR appeared in 2000 is a very promising alternative to elliptic curve cryptosystem. Though the basic idea behind XTR is very elegant and universal, one needs to restrict the primes p such as p ≡ 2 (mod 3) for optimal normal bases since it involves many multiplications in GF (p 2 ). Moreover the restriction p ≡ 2 (mod 3) is consistently used to improve the time complexity for irreducibility testing for XTR polynomials. In this paper, we propose that a Gaussian normal basis of type (2, k) for small k can also be used for efficient field arithmetic for XTR when p ≡ 2 (mod 3). Furthermore we give a new algorithm for fast irreducibility testing and finding a generator of XTR group when p ≡ 1 (mod 3). Also we present an explicit generator of XTR group which does not need any irreducibility testing when there is a Gaussian normal basis of type (2, 3) in GF (p 2 ). We show that our algorithms are simple to implement and the time complexity of our methods are comparable to the best ones proposed so far.

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Normal bases via general Gauss periods

Cited by 40 publications

References 8 publications

Efficient Linear Array for Multiplication in GF(2 m ) Using a Normal Basis for Elliptic Curve Cryptography

Efficient Linear Array for Multiplication in GF(2 m ) Using a Normal Basis for Elliptic Curve Cryptography

Abelian Groups, Gauss Periods, and Normal Bases

Fast Irreducibility Testing for XTR Using a Gaussian Normal Basis of Low Complexity

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