It is well known that Stickelberger-Swan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on. We discuss this problem for type II pentanomials n...

It is well known that a characterization for the irreducibility of selfreciprocal binary pentanomials does not exist [1, 4]. In this work we divide the self-reciprocal binary pentanomials into four big families, in such a way that all members of one of these families are clearly reducible. Using the Berlekamp Algorithm for the factorization of binary polynomials [2], we prove that all members o...

The state-of-the-art Galois field GF ð2Þ multipliers offer advantageous space and time complexities when the field is generated by some special irreducible polynomial. To date, the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial (ESP). Unfortunately, there exist only a few irreducible ESPs in the range of interest ...

Efficient hardware implementations of arithmetic operations in the Galois field are highly desirable for several applications, such as coding theory, computer algebra and cryptography. Among these operations, multiplication is of special interest because it is considered the most important building block. Therefore, high-speed algorithms and hardware architectures for computing multiplication a...

Elements of a finite field, GF ð2Þ, are represented as elements in a ring in which multiplication is more time efficient. This leads to faster multipliers with a modest increase in the number of XOR and AND gates needed to construct the multiplier. Such multipliers are used in error control coding and cryptography. We consider rings modulo trinomials and 4-term polynomials. In each case, we sho...

Recently, the multipliers with subquadratic space complexity for trinomials and some specific pentanomials have been proposed. For such kind of multipliers, alternatively, we use double basis multiplication which combines the polynomial basis and the modified polynomial basis to develop a new efficient digit-serial systolic multiplier. The proposed multiplier depends on trinomials and almost eq...

Besides Karatsuba algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF (2) subquadratic multipliers. However, when GF (2) elements are represented using a shifted polynomial basis, this approach is currently appliable only to GF (2)s generated by all irreducible trinomials and a special type of irreducible pentanomials, not all general irreducible p...

Recently, finite field multipliers having high throughput rate and low-latency have gained great attention in emerging cryptographic systems, but such multipliers over GF(2) for National Institute Standard Technology (NIST) pentanomials are not so abundant. In this paper, we present two pairs of low-latency and highthroughput bit-parallel and digit-serial systolic multipliers based on NIST pent...

All primitive trinomials over GF (2) with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are X859433 +X288477 + 1 and its reciprocal. Also two examples of primitive pentanomials over GF (2) with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.