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Finite Field Multiplier Using Redundant Representation
 IEEE Transactions on Computers
, 2002
"... This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our ar ..."
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This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide areatime tradeoffs which enable us to implement the multipliers in a partialparallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that when there is an optimal normal basis, the proposed bitserial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have advantage in using the redundant representation. Index terms: Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Efficient multiplication using type 2 optimal normal bases
"... Abstract. In this paper we propose a new structure for multiplication using optimal normal bases of type 2. The multiplier uses an efficient linear transformation to convert the normal basis representations of elements of Fqn to suitable polynomials of degree at most n over Fq. These polynomials ar ..."
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Abstract. In this paper we propose a new structure for multiplication using optimal normal bases of type 2. The multiplier uses an efficient linear transformation to convert the normal basis representations of elements of Fqn to suitable polynomials of degree at most n over Fq. These polynomials are multiplied using any method which is suitable for the implementation platform, then the product is converted back to the normal basis using the inverse of the above transformation. The efficiency of the transformation arises from a special factorization of its matrix into sparse matrices. This factorization — which resembles the FFT factorization of the DFT matrix — allows to compute the transformation and its inverse using O(n log n) operations in Fq, rather than O(n 2) operations needed for a general change of basis. Using this technique we can reduce the asymptotic cost of multiplication in optimal normal bases of type 2 from 2M(n) + O(n) reported by Gao et al. (2000) to M(n) + O(n log n) operations in Fq, where M(n) is the number of Fqoperations to multiply two polynomials of degree n − 1 over Fq. We show that this cost is also smaller than other proposed multipliers for n> 160, values which are used in elliptic curve cryptography.
Gauß Periods in Finite Fields
"... In this survey, we review two recent applications of a venerable tool: Gauß periods. In Section 2, we describe Gauß' original construction, and how it can be used to generate normal bases in extensions of finite fields. Section 3 contains the first application: finding elements of exponentially larg ..."
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In this survey, we review two recent applications of a venerable tool: Gauß periods. In Section 2, we describe Gauß' original construction, and how it can be used to generate normal bases in extensions of finite fields. Section 3 contains the first application: finding elements of exponentially large order in certain finite fields. This can be viewed as a step towards solving the famous open problem of finding efficiently a primitive element in a given finite field. A pleasant feature is that the prime factorization of the order of the multiplicative group is not required. In Section 4 we give another example of the method, yielding a different kind of bound: among the q shifts + a of an element of an extension of F q , where a runs through F q , at most one has "small" order. The second application, in Section 5, deals with efficient exponentiation...
Finite Fields in AXIOM
 ATR/5) (NP2522), The Numerical Algorithm Group, Downer’s
, 1992
"... Finite fields play an important role for many applications (e.g. coding theory, cryptography). There are different ways to construct a finite field for a given prime power. The paper describes the different constructions implemented in AXIOM. These are polynomial basis representation, cyclic group r ..."
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Finite fields play an important role for many applications (e.g. coding theory, cryptography). There are different ways to construct a finite field for a given prime power. The paper describes the different constructions implemented in AXIOM. These are polynomial basis representation, cyclic group representation, and normal basis representation. Furthermore, the concept of the implementation, the used algorithms and the various datatype coercions between these representations are discussed. Address of authors: Vangerowstr. 18, Postfach 10 30 68, D6900 Heidelberg, Germany, email: grabm@dhdibm1.bitnet resp. adscheer@dhdibm1.bitnet Contents 1 Introduction 4 2 Basic theory and notations 5 3 Categories for finite field domains 7 4 General finite field functions 8 4.1 E as an algebra of rank n over F : : : : : : : : : : : : : : : : : : 8 4.2 The F [X]module structure of E : : : : : : : : : : : : : : : : : : 10 4.3 The cyclic group E : : : : : : : : : : : : : : : : : : : : : : : : ...
Exponentiation Using Addition Chains For Finite Fields
, 2000
"... We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and nor ..."
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We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and normal bases. We introduce weighted qaddition chains to derive efficient algorithms, and report on implementation results for our methods.