Bit-parallel Polynomial Basis Multipliers Research Articles

High-Speed Polynomial Basis Multipliers Over GF(2m) for Special Pentanomials

Efficient hardware implementations of arithmetic operations in the Galois field $GF(2^{m})$ 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 are highly required. In this paper, bit-parallel polynomial basis multipliers over the binary field $GF(2^{m})$ generated using type II irreducible pentanomials are considered. The multiplier here presented has the lowest time complexity known to date for similar multipliers based on this type of irreducible pentanomials.

New efficient bit-parallel polynomial basis multiplier for special pentanomials

We present a bit-parallel polynomial basis multiplier based on a new divide-and-conquer approach using squaring. In particular, we apply the proposed approach to special types of irreducible pentanomials called as types I and II pentanomials, and induce explicit formulae and complexities of the proposed multiplier for these types of pentanomials. As a result, the proposed multiplier for type I pentanomials has almost the same time complexity, but about 25% reduced space complexity compared with the best known results in the literature. For type II pentanomials, we obtain the multiplier which has the lowest time complexity and about 25% reduced space complexity than the best known polynomial basis multipliers.

Efficient Polynomial Basis Multipliers for Type-II Irreducible Pentanomials

Bit-parallel polynomial basis multipliers over the finite field GF(2m) generated using type-II irreducible pentanomials are considered in this brief. The bit-parallel multipliers presented here have the lowest delay known to date for similar multipliers based on this type of irreducible pentanomials, with a very small increase of XOR gates.

Low complexity bit-parallel polynomial basis multipliers over binary fields for special irreducible pentanomials

Finite field GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF(2m) multiplication is of special interest because it is considered the most important building block. GF(2m) multipliers present reduced space and time complexities when the field is generated by some special irreducible polynomials. Among these, irreducible pentanomials of degree m are specially important because they are abundant and there are several eligible candidates for a given m. In this paper, we consider bit-parallel polynomial basis multipliers over the finite field GF(2m) generated using type 2 irreducible pentanomials, for which explicit formulas and algorithms for the computation of the products are given. In this contribution, two new subclasses of type 2 irreducible pentanomials are also introduced. The theoretical complexity analysis proves that the bit-parallel multipliers here presented have the lowest number of XOR gates known to date for similar polynomial basis multipliers based on this type of irreducible pentanomials, while the number of AND gates and the time complexity match the best known results found in the literature.

Non-XOR approach for low-cost bit-parallel polynomial basis multiplier over GF(2m)

Finite field arithmetic has been widely used in many cryptosystems, particularly in the elliptic curve cryptosystem (ECC) and the advanced encryption standard (AES) as a method for speeding up their encryption/decryption processes. Low-cost design for finite field arithmetic is more attractive for various mobile applications. It is a factor that a large number of Exclusive OR (XOR) gates have been used in the arithmetic operations under the traditional finite field arithmetic implementation. Thus, the cost of the traditional finite field arithmetic cannot be effectively lowered, because a typical XOR gate design consists of 12 transistors. To address this, a novel non-XOR approach consisting of eight transistors, for realising low-cost polynomial basis (PB) multiplier over GF(2 m ) was developed in this study. The authors proposed that non-XOR architecture for bit-parallel PB multiplier uses the multiplexer function instead of the traditional XOR function in its design. Based on the proposed non-XOR methodology, three popular low-cost irreducible polynomials - trinomial, pentanomial and all-one-polynomial - are proposed and designed in this study. The results indicate that the proposed non-XOR architecture can reduce space complexity by 22-, compared with that of the traditional design.

Bit-Parallel Polynomial Basis Multiplier for New Classes of Finite Fields

In this paper, three small classes of finite fields GF$(2^m)$ are found for which low complexity bit-parallel multipliers are proposed. The proposed multipliers have lower complexities compared to those based on the irreducible pentanomials. It is also shown that there does not always exist an irreducible all-one polynomial, equally-spaced polynomial, or trinomial for the new classes of fields.