A Chinese Remainder Theorem Approach to Bit-Parallel GF(2^n) Polynomial Basis Multipliers for Irreducible Trinomials

We show that the step “modulo the degree-n field generating irreducible polynomial” in the classical definition of the GF(2^n) multiplication operation can be avoided. This leads to an alternative representation of the finite field multiplication operation. Combining this representation and the Chinese Remainder Theorem, we design bit-parallel GF(2^n) multipliers for irreducible trinomials u^n + u^k + 1 on GF(2) where 1 < k &#8804; n=2. For some values of n, our architectures have the same time complexity as the fastest bit-parallel multipliers – the quadratic multipliers, but their space complexities are reduced. Take the special irreducible trinomial u^(2k) + u^k + 1 for example, the space complexity of the proposed design is reduced by about 1=8, while the time complexity matches the best result. Our experimental results show that among the 539 values of n such that 4 < n < 1000 and x^n+x^k+1 is irreducible over GF(2) for some k in the range 1 < k &#8804; n=2, the proposed multipliers beat the current fastest parallel multipliers for 290 values of n when (n &#8722; 1)=3 &#8804; k &#8804; n=2: they have the same time complexity, but the space complexities are reduced by 8.4% on average

A Trace Based GF(2n)GF(2^n) Inversion Algorithm

By associating Fermat\u27s Little Theorem based $GF(2^n)$ inversion algorithms with the multiplicative Norm function, we present an additive Trace based $GF(2^n)$ inversion algorithm. For elements with Trace value 0, it needs 1 less multiplication operation than Fermat\u27s Little Theorem based algorithms in some $GF(2^n)$s

Toeplitz matrix-vector product based GF(2^n) shifted polynomial basis multipliers for all irreducible pentanomials

Besides Karatsuba algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF(2^n) subquadratic multipliers. However, when GF(2^n) elements are represented using a shifted polynomial basis, this approach is currently appliable only to GF(2^n)s generated by all irreducible trinomials and a special type of irreducible pentanomials, not all general irreducible pentanomials. The reason is that no transformation matrix, which transforms the Mastrovito matrix into a Toeplitz matrix, has been found. In this article, we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial. Because there is no known value of n for which either an irreducible trinomial or an irreducible pentanomial does not exist, this transformation matrix makes the TMVP approach a universal tool, i.e., it is applicable to all practical GF(2^n)s

Bit-Parallel GF(2n)GF(2^{n}) Squarer Using Shifted Polynomial Basis

We present explicit formulae and complexities of bit-parallel shifted polynomial basis (SPB) squarers in finite field $GF(2^{n})$s generated by general irreducible trinomials $x^{n}+x^{k}+1$ ($0< k <n$) and type-II irreducible pentanomials $x^{n}+x^{k+1}+x^{k}+x^{k-1}+1$ ($3<k<(n-3)/2$). The complexities of the proposed squarers match or slightly outperform the previous best results. These formulae can also be used to design polynomial basis Montgomery squarers without any change. Furthermore, we show by examples that XOR gate numbers of SPB squarers are different when different shift factors in the SPB definition, i.e., parameter $v$ in ${\{}x^{i-v}|0\leq i\leq n-1 {\}}$, are used. This corrects previous misinterpretation

New GF(2n) Parallel Multiplier Using Redundant Representation

A new GF(2n) redundant representation is presented. Squaring in the representation is almost cost-free. Based on the representation, two multipliers are proposed. The XOR gate complexity of the first multiplier is lower than a recently proposed normal basis multiplier when CN (the complexity of the basis) is larger than 3n-1

Low Complexity MDS Matrices Using GF(2n)GF(2^n) SPB or GPB

While $GF(2^n)$ polynomial bases are widely used in symmetric-key components, e.g. MDS matrices, we show that even low time/space complexities can be achieved by using $GF(2^n)$ shifted polynomial bases (SPB) or generalized polynomial bases (GPB)

GF(2n)GF(2^n) Bit-Parallel Squarer Using Generalized Polynomial Basis For a New Class of Irreducible Pentanomials

We present explicit formulae and complexities of bit-parallel $GF(2^{n})$ squarers for a new class of irreducible pentanomials $x^{n}+x^{n-1}+x^{k}+x+1$, where $n$ is odd and $1<k<(n-1)/2$. The squarer is based on the generalized polynomial basis of $GF(2^{n})$. Its gate delay matches the best results, while its XOR gate complexity is $n+1$, which is only about 2/3 of the current best results