Shifted Polynomial Basis Research Articles

Fast Hybrid Karatsuba Multiplier for Type II Pentanomials

We continue the study of Mastrovito form of Karatsuba (MK) multipliers under the shifted polynomial basis (SPB). An MK multiplier utilizes Karatsuba algorithm and Mastrovito approach to optimize polynomial multiplication and modular reduction, which lead to a better space and time tradeoff for all trinomials. Based on this work, we make two types of contributions: 1) We derive a new modular reduction formulation for constructing Mastrovito matrix associated with special types of pentanomials, i.e., Types I, II, and Type C.1 pentanomial. Through related formulations, we demonstrate that Type I pentanomial is less efficient than Type II because of a more complicated modular reduction under the same SPB; conversely, Type C.1 pentanomial is as good as Type II under generalized polynomial basis (GPB). 2) We introduce a new MK multiplier for Type II pentanomial. It is shown that our proposal is only one $T_{X}$ slower than the fastest quadratic multipliers for Type II pentanomial, but its space complexity is roughly 3/4 of those schemes. To the best of our knowledge, it is the first time for hybrid multiplier to achieve such a time delay bound.

On the Complexity of Hybrid $n$ -Term Karatsuba Multiplier for Trinomials

The ${n}$ -term Karatsuba algorithm (KA) is an extension of 2-term KA, which can obtain even fewer multiplications than the original one. The existing ${n}$ -term Karatsuba hybrid $\textit {GF}(2^{m})$ multipliers rely on factorization of ${m}$ or ${m}-1$ , so that put a confinement to these schemes. In this contribution, we use a new decomposition ${m}={n}\ell +{r}$ , such that $0\leq {r} and $0\leq {r} , and propose a novel ${n}$ -term Karatsuba hybrid $\textit {GF}(2^{m})$ multiplier for an arbitrary irreducible trinomial ${x}^{m}+{x}^{k}+1, {m}\geq 2{k}$ . Combined with shifted polynomial basis, a new approach (other than Mastrovito approach) is introduced to exploit the spatial correlations among different subexpressions. We investigate the explicit space and time complexities, and discuss related upper and lower bounds. As a main contribution, the flexibilities of ${n}, \ell $ and ${r}$ make our proposal approaching optimal for any given ${m}$ . The space complexity can achieve to ${m^{2}}/{2}+{O}({\sqrt {11} {m}^{3/2}}/{2})$ , which roughly matches the best result of current hybrid multipliers for special trinomials. Meanwhile, its time complexity is slightly higher than the counterparts, but can be improved for a new class of trinomials. In particular, we demonstrate that the hybrid multiplier for ${x}^{m}+{x}^{k}+1$ , where ${k}$ is approaching $\frac {m}{2}$ , can achieve a better space-time trade-off than any other trinomials.

Efficient Nonrecursive Bit-Parallel Karatsuba Multiplier for a Special Class of Trinomials

Recently, we present a novel Mastrovito form of nonrecursive Karatsuba multiplier for all trinomials. Specifically, we found that related Mastrovito matrix is very simple for equally spaced trinomial (EST) combined with classic Karatsuba algorithm (KA), which leads to a highly efficient Karatsuba multiplier. In this paper, we consider a new special class of irreducible trinomial, namely, xm+xm/3+1. Based on a three-term KA and shifted polynomial basis (SPB), a novel bit-parallel multiplier is derived with better space and time complexity. As a main contribution, the proposed multiplier costs about 2/3 circuit gates of the fastest multipliers, while its time delay matches our former result. To the best of our knowledge, this is the first time that the space complexity bound is reached without increasing the gate delay.

Mastrovito Form of Non-Recursive Karatsuba Multiplier for All Trinomials

We present a new type of bit-parallel non-recursive Karatsuba multiplier over $GF(2^m)$ generated by an arbitrary irreducible trinomial. This design effectively exploits Mastrovito approach and shifted polynomial basis (SPB) to reduce the time complexity and Karatsuba algorithm to reduce its space complexity. We show that this type of multiplier is only one $T_X$ slower than the fastest bit-parallel multiplier for all trinomials, where $T_X$ is the delay of one 2-input XOR gate. Meanwhile, its space complexity is roughly 3/4 of those multipliers. To the best of our knowledge, it is the first time that our scheme has reached such a time delay bound. This result outperforms previously proposed non-recursive Karatsuba multipliers.

Area-Delay Efficient Digit-Serial Multiplier Based on <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math> </inline-formula>-Partitioning Scheme Combined With TMVP Block Recombination Approach

Shifted polynomial basis (SPB) and generalized polynomial basis (GPB) are two efficient bases of representation in binary extension fields, and are widely studied. In this paper, we use the GPB formulation to derive a new modified SPB (MSPB) representation for arbitrary irreducible trinomials and pentanomials. It is shown that the basis conversion from the MSPB to the SPB for trinomials is free of hardware cost. We have shown that multiplication based on SPB and MSPB representations can make use of Toeplitz matrix–vector product (TMVP) formulation. The existing TMVP block recombination (TMVPBR) approach is used here to derive an efficient $k$ -partitioning TMVPBR decomposition for digit-serial double basis multiplication that can achieve subquadratic space complexity. From synthesis results, we have shown that the proposed multiplier has less area and less area-delay product compared with the existing digit-serial multipliers. We also show that the proposed multiplier using $k$ -partitioning TMVPBR decomposition can provide a better tradeoff between time and space complexities.

Explicit formulae for Mastrovito matrix and its corresponding Toeplitz matrix for all irreducible pentanomials using shifted polynomial basis

We propose explicit formulae of the Mastrovito matrix M and its corresponding Toeplitz matrix T for an arbitrary irreducible pentanomial using shifted polynomial basis. We also give the complexity of the Toeplitz matrix for a pentanomial. This yields the complexity of a multiplier based on Toeplitz matrix–vector product (TMVP) for an arbitrary irreducible pentanomial for the first time. Moreover, we introduce a new type of pentanomials for which a multiplier based on TMVP is efficiently implemented. We show that the complexity of a subquadratic space complexity multiplier for such a special type of pentanomials is comparable with that for trinomials.

Area-Efficient Subquadratic Space-Complexity Digit-Serial Multiplier for Type-II Optimal Normal Basis of <formula formulatype="inline"><tex Notation="TeX">$GF(2^{m})$</tex> </formula> Using Symmetric TMVP and Block Recombination Techniques

The type-II optimal normal basis (ONB) is popularly used to represent $GF(2^{m})$ for elliptic curve cryptosystems. It is shown in the literature that multiplication in binary fields, including those represented by type-II ONB, shifted polynomial basis, and dual basis, can be transformed into non-symmetric Toeplitz matrix-vector product (TMVP) formulation. In this paper, we show that type-II ONB multiplication can be realized by two symmetric TMVPs (STMVP). Moreover, we have proposed a novel folded TMVP block recombination (TMVPBR) for the computation of STMVP. Based on the proposed folded TMVPBR approach, we have proposed a new digit-serial structure for type-II ONB multiplication, while traditional parallel ONB multipliers are based on non-symmetric TMVPBR approach to achieve subquadratic space complexity architecture. The proposed digit-serial structure also involves subquadratic space complexity. By the theoretical analysis as well as from synthesis result, however, we find that the proposed architecture has significantly less area and less area-delay product compared to the existing digit-serial type-II ONB multipliers.

$\schmi{GF(2^n)}$ Shifted Polynomial Basis Multipliers Based on Subquadratic Toeplitz Matrix-Vector Product Approach for All Irreducible Pentanomials

Besides Karatsuba’s algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$GF(2^n)$</tex> <mathgraphic fileref="han-ieq2-2297106.gif" graphicformat="GIF"/></formula> subquadratic multipliers. However, when <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$GF(2^n)$</tex><mathgraphic fileref="han-ieq3-2297106.gif" graphicformat="GIF"/> </formula> elements are represented using a polynomial basis or its generalization—shifted polynomial basis—this approach is currently appliable only to fields <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$GF(2^n)$</tex> <mathgraphic fileref="han-ieq4-2297106.gif" graphicformat="GIF"/></formula> generated by an irreducible trinomial or a special type of irreducible pentanomials, but not to a general irreducible pentanomial. 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.

Low-Complexity Digit-Serial and Scalable SPB/GPB Multipliers Over Large Binary Extension Fields Using (b,2)-Way Karatsuba Decomposition

Shifted polynomial basis (SPB) and generalized polynomial basis (GPB) are two variations of polynomial basis representation. SPB/GPB have potential for efficient bit-level and digit-level implementations of multiplication over binary extension fields. This paper presents a (b,2)-way KA decomposition for digit-serial multiplication with low-space complexity. Based on the proposed parallel (b,2)-way KA scheme, we derive a novel scalable SPB/GPB multiplier. Analytical results show that the proposed multiplier could achieve the desired trade-off between space and time complexities. Our proposed multiplier is modular, regular, and suitable for very-large-scale integration (VLSI) implementations. It involves significantly less area complexity, less computation time and less energy consumption compared to the existing digit-serial and scalable multipliers.

Formulas for cube roots in [formula omitted] using shifted polynomial basis

Evaluation of cube roots in characteristic three finite fields is required for Tate (or modified Tate) pairing computation. The Hamming weight of x1/3 means that the number of nonzero coefficients in the polynomial representation of x1/3 in F3m=F3[x]/(f), where f∈F3[x] is an irreducible polynomial. The Hamming weight of x1/3 determines the efficiency of cube roots computation for characteristic three finite fields. Ahmadi et al. found the Hamming weight of x1/3 using polynomial basis [4]. In this paper, we observe that shifted polynomial basis (SPB), a variation of polynomial basis, can reduce Hamming weights of x1/3 and x2/3. Moreover, we provide the suitable SPB that eliminates modular reduction process in cube roots computation.

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg—de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg—de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.

The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation

The Kuramoto-Sivashinsky equation is a kind of high-order nonlinear evolution equation which can describe complicated chaotic nature. Due to the existence of high-order derivatives in the equation, the shape functions violate the consistency conditions when using traditional element-free Galerkin method which adopts high-order polynomial basis functions to construct the shape functions. In order to solve the problems encountered in the traditional element-free Galerkin method, a kind of element-free Galerkin method adopting the shifted polynomial basis functions is presented in this paper. Compared with the traditional element-free Galerkin method, the Galerkin principle is still used to discrete the equation in this method, but the shape functions are constructed by moving least squares based on the shifted polynomial basis functions. Numerical results for the Kuramoto-Sivashinsky equation having traveling wave solution and chaotic nature prove the validity of the presented method.

Fast Bit-Parallel Shifted Polynomial Basis Multiplier Using Weakly Dual Basis Over $GF(2^{m})$

In this paper, we present a new method to compute the Mastrovito matrix for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF</i> (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) generated by an arbitrary irreducible polynomial using weakly dual basis of shifted polynomial basis. In particular, we derive the explicit formulas of the proposed multiplier for special type of irreducible pentanomial <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k3</sup> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k2</sup> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k1</sup> +1 with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ≤ ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> )/2 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> <; min(2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> /2). As a result, the time complexity of the proposed multiplier matches or outperforms the previously known results. On the other hand, the number of XOR gates of the proposed multiplier is slightly greater than the best known results.

Digit-Level Semi-Systolic and Systolic Structures for the Shifted Polynomial Basis Multiplication Over Binary Extension Fields

Finite field multiplication is one of the most important operations in the finite field arithmetic. In this paper, we study semi-systolic and systolic implementations of the shifted polynomial basis multiplication and propose low time complexity semi-systolic and systolic array structures. We show that our proposed semi-systolic multiplier is faster than its existing counterparts available in the literature. Our application-specified integrated circuit (ASIC) implementation of the proposed semi-systolic multiplier demonstrates that reduction in time complexity is achieved without imposing hardware overhead. Furthermore, our proposed systolic array shifted polynomial basis (SPB) multiplier has a low time complexity for general irreducible polynomials.

Low complexity bit parallel multiplier for [formula omitted] generated by equally-spaced trinomials

Based on the shifted polynomial basis (SPB), a high efficient bit-parallel multiplier for the field GF ( 2 m ) defined by an equally-spaced trinomial (EST) is proposed. The use of SPB significantly reduces time delay of the proposed multiplier and at the same time Karatsuba method is combined with SPB to decrease space complexity. As a result, with the same time complexity, approximately 3/4 gates of previous multipliers are used in the proposed multiplier.

A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields

Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2n) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers. For example, with n being a power of 2, the space complexity is about 8 percent better, while the asymptotic gate delay is about 33 percent better, respectively. Another advantage of the proposed matrix-vector product approach is that it can also be used to design subquadratic space complexity polynomial, dual, weakly dual, and triangular basis parallel multipliers. To the best of our knowledge, this is the first time that subquadratic space complexity parallel multipliers are proposed for dual, weakly dual, and triangular bases. A recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers. This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

Efficient parallel multiplier in shifted polynomial basis

In this paper we study the multiplication in fields F2^n using the Shifted Polynomial Basis (SPB) representation of Fan and Dai [H. Fan, Y. Dai, Fast bit-parallel GF(2^n) multiplier for all trinomials, IEEE Transactions on Computers 54 (4) (2005) 485-490]. We give a simpler construction than in Fan and Dai (2005) of the matrix associated to the SPB used to perform the field multiplication. We present also a novel parallel architecture to multiply in SPB. This multiplier have a smaller time complexity (for good field it is equal to T[email protected]?log2(n)@?TX) than all previously presented architecture. For practical field F2^n, i.e., for [email protected]?163, this roughly improves the delay by 10%. On the other hand the space complexity is increased by 25%: the space complexity is a little greater than the time gain.

Low Complexity Bit-Parallel Squarer for GF(2n) Defined by Irreducible Trinomials

We present a bit-parallel squarer for GF(2n) defined by an irreducible trinomial xn + xk + 1 using a shifted polynomial basis. The proposed squarer requires TX delay and at most ⌈n/2⌉ XOR gates, where TX is the delay of one XOR gate. As a result, the squarer using the shifted polynomial basis is more efficient than one using the polynomial basis except for k = 1 or n/2.

Relationship between GF(2^m) Montgomery and Shifted Polynomial Basis Multiplication Algorithms

Applying the matrix-vector product idea of the Mastrovito multiplier to the GF(2^{m}) Montgomery multiplication algorithm, we present a new parallel multiplier for irreducible trinomials. This multiplier and the corresponding shifted polynomial basis (SPB) multiplier have the same circuit structure for the same set of parameters. Furthermore, by establishing isomorphisms between the Montgomery and the SPB constructions of GF(2^{m}), we show that the Montgomery algorithm can be used to perform the SPB multiplication without any changes and vice versa.

Efficient Bit-Parallel Multiplier for Irreducible Pentanomials Using a Shifted Polynomial Basis

In this paper, we present a bit-parallel multiplier for GF(2^m) defined by an irreducible pentanomial x^m+x^{k_3}+x^{k_2}+x^{k_1}+1, where 1\leq k_1 < k_2 <k_3\leq m/2. In order to design an efficient bit-parallel multiplier, we introduce a shifted polynomial basis and modify a reduction matrix presented by Reyhani-Masoleh and Hasan. As a result, the time complexity of the proposed multiplier is T_A + (3+ \lceil \log _2(m-1)\rceil)T_X, where T_A and T_X are the delay of one AND and one XOR gate, respectively. This result matches or outperforms the previously known results. On the other hand, the proposed multiplier has the same space complexity as the previously known multipliers except for special types of irreducible pentanomials. Note that its hardware architecture is similar to that presented by Reyhani-Masoleh and Hasan.