A note about binary finite fields multiplication on FPGA F. Garcia Crespi, F. Vicedo, R. Guti´ errez , Katya G. Llamazares, P. Garrido, S. Alcaraz Departamento de F´ ısica y Arquitectura de Computadores Universidad Miguel Hermnandez Elche, 03202, Spain A. Grediaga Universidad de Alicante Dep. Tecnolog´ ıa Inform,´ atica y Computaci´ on Ap. 99,Alicante, Spain J.J. Climent Dep. Ciencia de la Computaci´ o e Inteligencia Artificial Ap. 99,Alicante, Spain ABSTRACT This paper present a notes about a hardware architecture over FPGAs for multiplication in binary fields GF (2 m ) us- ing a matrix representation of the elements of GF (2 m ). KEY WORDS Finite Field Arithmetic, cryptography, finite field multipli- cation 1 Introduction Finite fields are increasingly important for many applica- tions in cryptography and algebraic coding theory [5]. Cer- tain properties of the binary finite field GF (2 m ) like its “carry-free” arithmetic make it very attractive for hard- ware implementation. Another advantage of GF (2 m ) is the availability of different equivalent representations of the field elements, e.g. polynomial bases, normal bases, or dual bases. According to the different basis representations, a va- riety of algorithms and architectures for multiplication in GF (2 m ) have been proposed. Efficient implementation of the field arithmetic in GF (2 m ) depends enormously on the particular basis used for the finite field. From an architectural point of view, a polynomial basis multiplier can be realised in a bit-serial, digit- serial, or bit-parallel fashion. For area-restricted devices like smart cards, the bit-serial architecture offers a fair area/performance trade-off. In this paper we presents a method for multiplication in GF (2 m ), where the fields elements are represented as matrices. 2 Finite Fields Arithmetic 2.1 Representation of the Field Elements Abstractly, a finite field (or Galois field) consists of a finite set of elements together with the description of two opera- tions (addition and multiplication) that can be performed on pairs of field elements. These operations must possess cer- tain properties — associativity and commutativity of both addition and multiplication, distributivity, existence of an additive identity and a multiplicative identity, and existence of additive inverses as well as multiplicative inverses. The order of a finite field is the number of field elements it con- tains, and it is traditional to denote a finite field of order m as GF (m). GF (2) is the smallest possible finite field; it just contains the integers 0 and 1 as field elements. Addi- tion and multiplication are performed modulo 2, therefore the addition is equivalent to the logical XOR, and the mul- tiplication corresponds to the logical AND. GF (2 m ) is called a characteristic two field or a bi- nary finite field. It can be viewed as a vector space of di- mension m over the field GF(2). That is, there exist m elements x 0 ,x 1 ,x 2 ,...,x m-1 in GF (2 m ) such that each element x ∈ GF (2 m ) can be uniquely written in the form: x = a 0 x 0 + a 1 x 1 ... + a m-1 x m-1 where a i ∈ GF (2 ). The binary finite field GF (2 m ) contains 2 m elements, whereby m is a non-zero positive integer. Each of these 2 m elements can be uniquely represented with a polynomial of degree up to m-1 with coefficients from GF(2). For exam- ple, if a(x) is an element in GF (2 m ), then one can have a(x) = m-1 ∑ i=0 a i x i 2.2 Addition and Multiplication Such a set { x 0 ,x 1 ,x 2 ...x m-1 } is called a basis of GF (2 m ) over GF(2). Given such a basis, a field element x can be represented as the bit string (a 0 a 1 ...a m-1 ). Ad- dition of field elements is performed by bit-wise XOR-ing the vector representations. The multiplication rule depends on the basis selected. There are many different bases of GF (2 m ) over GF(2). Some bases lead to more efficient software or hardware implementations of the arithmetic in GF (2 m ) than other bases. The most popular two kinds of bases used are the polynomial bases and the normal bases. In the polynomial bases, the field arithmetic is im- plemented as polynomial arithmetic modulo f (x). In