The Experts below are selected from a list of 11856 Experts worldwide ranked by ideXlab platform
Johann Grosschadl - One of the best experts on this subject based on the ideXlab platform.
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a bit serial unified multiplier architecture for finite fields gf p and gf 2m
Cryptographic Hardware and Embedded Systems, 2001Co-Authors: Johann GrosschadlAbstract:The performance of elliptic curve cryptosystems is primarily determined by an efficient implementation of the arithmetic operations in the underlying finite field. This paper presents a hardware architecture for a unified multiplier which operates in two types of finite fields: GF(p) and GF(m). In both cases, the multiplication of field elements is performed by accumulation of partial-products to an intermediate result according to an MSB-first shift-and-add method. The reduction modulo the prime p (or the Irreducible Polynomial p(t), respectively) is interleaved with the addition steps by repeated subtractions of 2p and/or p (or p(t), respectively). A bit-serial multiplier executes a multiplication in GF(p) in approximately 1.5ċ⌈log2(p)⌉ clock cycles, and the multiplication in GF(m) takes exactly m clock cycles. The unified multiplier requires only slightly more area than that of the multiplier for prime fields GF(p). Moreover, it is shown that the proposed architecture is highly regular and simple to design.
Manuel Leone - One of the best experts on this subject based on the ideXlab platform.
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a new low complexity parallel multiplier for a class of finite fields
Cryptographic Hardware and Embedded Systems, 2001Co-Authors: Manuel LeoneAbstract:In this paper a new low complexity parallel multiplier for characteristic two finite fields GF(2m) is proposed. In particular our multiplier works with field elements represented through both Canonical Basis and Type I Optimal Normal Basis (ONB), provided that the Irreducible Polynomial generating the field is an All One Polynomial (AOP). The main advantage of the scheme is the resulting space complexity, significantly lower than the one provided by the other fast parallel multipliers currently available in the open literature and belonging to the same class.
Ferruh Ozbudak - One of the best experts on this subject based on the ideXlab platform.
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Polynomial multiplication over binary fields using charlier Polynomial representation with low space complexity
International Conference on Cryptology in India, 2010Co-Authors: Sedat Akleylek, Murat Cenk, Ferruh OzbudakAbstract:In this paper, we give a new way to represent certain finite fields GF(2 n ). This representation is based on Charlier Polynomials. We show that multiplication in Charlier Polynomial representation can be performed with subquadratic space complexity. One can obtain binomial or trinomial Irreducible Polynomials in Charlier Polynomial representation which allows us faster modular reduction over binary fields when there is no desirable such low weight Irreducible Polynomial in other representations. This representation is very interesting for NIST recommended binary field GF(2283) since there is no ONB for the corresponding extension. We also note that recommended NIST and SEC binary fields can be constructed with low weight Charlier Polynomials.
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improved Polynomial multiplication formulas over if using chinese remainder theorem
IEEE Transactions on Computers, 2009Co-Authors: Murat Cenk, Ferruh OzbudakAbstract:Let n and lscr be positive integers and f(x) be an Irreducible Polynomial over IF2 such that lscrdeg(f(x)) < 2n -1. We obtain an effective upper bound for the multiplication complexity of n-term Polynomials modulo f(x)lscr. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for Polynomial multiplication over IF2. We give improved formulae to multiply Polynomials of small degree over IF2. In particular we improve the best known multiplication complexities over IF2 in the literature in some cases.
Dahan Xavier - One of the best experts on this subject based on the ideXlab platform.
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Lexicographic Groebner bases of bivariate Polynomials modulo a univariate one
2021Co-Authors: Dahan XavierAbstract:Let T(x) in k[x] be a monic non-constant Polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate Polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an Irreducible Polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a Polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T. In particular no factorization or Groebner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Groebner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Groebner bases approach.Comment: Accepted at JSC. 50 pages, 6 tables, 8 figure
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Lexicographic Groebner bases of bivariate Polynomials modulo a univariate one
2020Co-Authors: Dahan XavierAbstract:Let T(x) in k[x] be a monic non-constant Polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate Polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an Irreducible Polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a Polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T. In particular no factorization or Groebner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Groebner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Groebner bases approach.Comment: 41 pages, 4 table
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Computation of gcd chain over the power of an Irreducible Polynomial
2018Co-Authors: Dahan XavierAbstract:A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic Polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to compute such a gcd chain remains challenging. This work treats completely the case of a triangular set T = (T_1 (x)) in one variable, namely a power of an Irreducible Polynomial. This seemingly "easy" case reveals the main steps necessary for treating the general case, and it allows to isolate the particular one step that does not directly extend and requires more care.Comment: (Added a full running example) 16 pages, 3 figures. Full version of extended abstract presented at ADG 201
Murat Cenk - One of the best experts on this subject based on the ideXlab platform.
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Polynomial multiplication over binary fields using charlier Polynomial representation with low space complexity
International Conference on Cryptology in India, 2010Co-Authors: Sedat Akleylek, Murat Cenk, Ferruh OzbudakAbstract:In this paper, we give a new way to represent certain finite fields GF(2 n ). This representation is based on Charlier Polynomials. We show that multiplication in Charlier Polynomial representation can be performed with subquadratic space complexity. One can obtain binomial or trinomial Irreducible Polynomials in Charlier Polynomial representation which allows us faster modular reduction over binary fields when there is no desirable such low weight Irreducible Polynomial in other representations. This representation is very interesting for NIST recommended binary field GF(2283) since there is no ONB for the corresponding extension. We also note that recommended NIST and SEC binary fields can be constructed with low weight Charlier Polynomials.
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improved Polynomial multiplication formulas over if using chinese remainder theorem
IEEE Transactions on Computers, 2009Co-Authors: Murat Cenk, Ferruh OzbudakAbstract:Let n and lscr be positive integers and f(x) be an Irreducible Polynomial over IF2 such that lscrdeg(f(x)) < 2n -1. We obtain an effective upper bound for the multiplication complexity of n-term Polynomials modulo f(x)lscr. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for Polynomial multiplication over IF2. We give improved formulae to multiply Polynomials of small degree over IF2. In particular we improve the best known multiplication complexities over IF2 in the literature in some cases.
