Subquadratic Space Complexity Research Articles

Novel Systolization of Subquadratic Space Complexity Multipliers Based on Toeplitz Matrix–Vector Product Approach

Systolic finite field multiplier over $GF(2^{m})$ , because of its superior features such as high throughput and regularity, is highly desirable for many demanding cryptosystems. On the other side, however, obtaining high-performance systolic multiplier with relatively low hardware cost is still a challenging task due to the fact that the systolic structure usually involves large area complexity. Based on this consideration, in this paper, we propose to carry out two novel coherent interdependent efforts. First, a new digit-serial multiplication algorithm based on polynomial basis over binary field $(GF(2^{m}))$ is proposed. Novel Toeplitz matrix–vector product (TMVP)-based decomposition strategy is employed to derive an efficient subquadratic space complexity. Second, The proposed algorithm is then innovatively mapped into a low-complexity systolic multiplier, which involves less area-time complexities than the existing ones. A series of resource optimization techniques also has been applied on the multiplier which optimizes further the proposed design (it is the first report on digit-serial systolic multiplier based on TMVP approach covering all irreducible polynomials, to the best of our knowledge). The following complexity analysis and comparison confirm the efficiency of the proposed multiplier, that is, it has lower area-delay product (ADP) than the existing ones. The extension of the proposed multiplier for bit-parallel implementation is also considered in this paper.

Digit-Serial Versatile Multiplier Based on a Novel Block Recombination of the Modified Overlap-Free Karatsuba Algorithm

Overlap-Free Karatsuba Algorithm (OFKA) combined with block recombination approach (OFKABR) can improve the complexity of the original OFKA to obtain efficient implementation of polynomial basis multiplication over finite field $GF(2^{m})$ . In this paper, we have further proposed a modified OFKABR (MOFKABR) strategy to reduce the space and time complexities of the best-known method. The proposed strategy is also extended to obtain a low-complexity versatile multiplier, which is designed to support the generalized multiplication on a wider range of field size for $m_{1}\leq m\leq m_{\lambda }$ . Using the proposed MOFKABR approach, the proposed digit-serial versatile multiplier can achieve subquadratic space complexity when compared with the existing digit-serial versatile multipliers.

Subquadratic Space Complexity Multiplier Using Even Type GNB Based on Efficient Toeplitz Matrix-Vector Product

Multiplication schemes based on Toeplitz matrix-vector product (TMVP) have been proposed by many researchers. TMVP can be computed using the recursive two-way and three-way split methods, which are composed of four blocks. Among them, we improve the space complexity of the component matrix formation (CMF) block. This result derives the improvements of multiplication schemes based on TMVP. Also, we present a subquadratic space complexity $GF(2^m)$ multiplier with even type Gaussian normal basis (GNB). In order to design the multiplier, we formulate field multiplication as a sum of two TMVPs and efficiently compute the sum. As a result, for type 2 and 4 GNBs, the proposed multipliers outperform other similar schemes. The proposed type 6 GNB is the first subquadrtic space complexity multiplier with its explicit complexity formula.

Efficient multiplier based on hybrid approach for Toeplitz matrix–vector product

We propose a hybrid approach for a Toeplitz matrix–vector product (TMVP) of size k⋅2i3j, where k≥1 and i,j≥0. It is possible to make trade-offs between time and space complexities for a TMVP by choosing values k, i, and j properly. We show that the multiplier based on the proposed hybrid TMVP approach has lower space as well as time complexities than other subquadratic space complexity multipliers for five fields recommended by NIST. Moreover, for those five fields, the space complexities of the proposed multiplier are reduced by a minimum 59% and a maximum 77% compared with quadratic space complexity multiplier.

New Block Recombination for Subquadratic Space Complexity Polynomial Multiplication Based on Overlap-Free Approach

In this paper, we present new parallel polynomial multiplication formulas which result in subquadratic space complexity. The schemes are based on a recently proposed block recombination of polynomial multiplication formula. The proposed two-way, three-way, and four-way split polynomial multiplication formulas achieve the smallest space complexities. Moreover, by providing area-time tradeoff method, the proposed formulas enable one to choose a parallel formula for polynomial multiplication which is suited for a design environment.

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.

Subquadratic space complexity Gaussian normal basis multipliers overGF(2m) based on Dickson–Karatsuba decomposition

Gaussian normal basis (GNB) of the even-type is popularly used in elliptic curve cryptosystems. Efficient GNB multipliers could be realised by Toeplitz matrix-vector decomposition to realise subquadratic space complexity architectures. In this study, Dickson polynomial representation is proposed as an alternative way to represent an GNB of characteristic two. The authors have derived a novel recursive Dickson–Karatsuba decomposition to achieve a subquadratic space-complexity parallel GNB multiplier. By theoretical analysis, it is shown that the proposed subquadratic multiplier saves about 50% bit-multiplications compared with the corresponding subquadratic GNB multiplication using Toeplitz matrix-vector product approach.

Efficient Digit-Serial KA-Based Multiplier Over Binary Extension Fields Using Block Recombination Approach

It is well-known that the $(a,b)$ -way Karatsuba algorithm (KA) with $a\neq b$ is used for efficient digit-serial multiplication with subquadratic space complexity architecture. In this paper, based on $(a,b)$ -way KA decomposition, we have derived a novel $k$ -way block recombination KA (BRKA) decomposition for digit-serial multiplication. The proposed $k$ -way BRKA is formed by a power of 2 polynomial decomposition. By theoretical analysis, it is shown that $k$ -way BRKA can provide the necessary tradeoff between space and time complexity. Using (4,2)-way KA to construct the proposed $k$ -way BRKA architecture in $GF(2^{409})$ , it is shown that the proposed 2-way BRKA approach requires less area, and the proposed 8-way BRKA approach requires less computation time and less area-time product compared to compared the existing $(a,b)$ -way KA decomposition.

Efficient Subquadratic Space Complexity Architectures for Parallel MPB Single- and Double-Multiplications for All Trinomials Using Toeplitz Matrix-Vector Product Decomposition

Subquadratic multiplication algorithm has received significant attention of cryptographic hardware researchers for efficient implementation public-key cryptosystems. In this paper, we derive a new shifted MPB (SMPB) representation based on modified polynomial basis (MPB). We have shown that by using MPB and SMPB, the proposed double basis multiplication can be transformed into Toeplitz matrix-vector product (TMVP) structure. Furthermore, by employing this formulation of double basis multiplication, we show that three-operand multiplication over GF(2 m ) for all trinomials can be realized efficiently by the recursive TMVP (RTMVP) formulation. To perform the three-operand multiplication with the RTMVP formulation, we have derived a new RTMVP decomposition scheme. The proposed single- and double-multiplications can, respectively, use TMVP and RTMVP decompositions to achieve subquadratic space complexity architectures. By theoretical analysis, it is shown that the proposed subquadratic multipliers involve significantly less space complexity and less computation time compared to the existing subquadratic multipliers using TMVP and Karatsuba algorithms. Moreover, our proposed double-multiplication design can be used in several applications involving successive multiplications, such as exponentiation, inversion, and elliptic curve point multiplication.

Low-Latency Digit-Serial Systolic Double Basis Multiplier over <formula formulatype="inline"> <tex Notation="TeX">$\mbi GF{(2^m})$</tex> </formula> Using Subquadratic Toeplitz Matrix-Vector Product Approach

Recently in cryptography and security, 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 equally space pentanomials (AESPs), and utilizes the subquadratic Toeplitz matrix-vector product scheme to derive a low-latency digit-serial systolic architecture. If the selected digit-size is $d$ bits, the proposed digit-serial multiplier for both polynomials, i.e., trinomials and AESPs, requires the latency of $2\left\lceil \!{\sqrt {{m \over d}}} \right\rceil\! $ , while traditional ones take at least $O\left({\left\lceil {{m \over d}} \right\rceil} \right)$ clock cycles. Analytical and application-specific integrated circuit (ASIC) synthesis results indicate that both the area and the time $ \times $ area complexities of our proposed architecture are significantly lower than the existing digit-serial systolic multipliers.

Subquadratic Space Complexity Multiplier for <i>GF</i>(2<sup>n</sup>) Using Type 4 Gaussian Normal Bases

Subquadratic space complexity multipliers for optimal normal bases (ONBs) have been proposed for practical applications. However, for the Gaussian normal basis (GNB) of type t > 2 as well as the normal basis (NB), there is no known subquadratic space complexity multiplier. In this paper, we propose the first subquadratic space complexity multipliers for the type 4 GNB. The idea is based on the fact that the finite field GF(2 n ) with the type 4 GNB can be embedded into fields with an ONB.

Subquadratic Space Complexity Multiplier for <i>GF</i>(2<sup>n</sup>) Using Type 4 Gaussian Normal Bases

Subquadratic space complexity multipliers for optimal normal bases (ONBs) have been proposed for practical applications. However, for the Gaussian...

A New Representation of Elements of Binary Fields with Subquadratic Space Complexity Multiplication of Polynomials

A New Representation of Elements of Binary Fields with Subquadratic Space Complexity Multiplication of Polynomials

Toeplitz Matrix Approach for Binary Field Multiplication Using Quadrinomials

In the recent past, subquadratic space complexity multipliers have been proposed for binary fields defined by irreducible trinomials and some specific pentanomials. For such multipliers, alternative irreducible polynomials can also be used, in particular, nearly all one polynomials (NAOPs) seem to be better than pentanomials. For improved efficiency, multiplication modulo an NAOP is performed via modulo a quadrinomial whose degree is one more than that of the original NAOP. In this paper, we present a Toeplitz matrix-vector product based approach for multiplication modulo a quadrinomial. We obtain a fully parallel multiplier with a subquadratic space complexity. The Toeplitz matrix-vector product-based approach is also interesting in the design of sequential multipliers. We present two such multipliers that process a two-bit digit every clock cycle. Field-programmable gate-array implementations of the proposed sequential as well as fully parallel multipliers for the field size of 163 are also presented.

Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication Based on Toeplitz Matrix-Vector Product

In this paper, we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity. The method is based on decomposing the building blocks of the Fan-Hasan subquadratic Toeplitz matrix-vector multiplier. We reduce the space complexity of their architecture by recombining the building blocks. In comparison to other similar schemes available in the literature, our proposal presents a better space complexity while having the same time complexity. We also show that block recombination can be used for efficient implementation of the GHASH function of Galois Counter Mode (GCM).

Subquadratic Space Complexity Binary Field Multiplier Using Double Polynomial Representation

This paper deals with binary field multiplication. We use the bivariate representation of binary field called Double Polynomial System (DPS) presented in . This concept generalizes the composite field representation to every finite field. As shown in , the main interest of DPS representation is that it enables to use Lagrange approach for multiplication, and in the best case, Fast Fourier Transform approach, which optimizes Lagrange approach. We use here a different strategy from to perform reduction, and we also propose in this paper, some new approaches for constructing DPS. We focus on DPS, which provides a simpler and more efficient method for coefficient reduction. This enables us to avoid a multiplication required in the Montgomery reduction approach of , and thus to improve the complexity of the DPS multiplier. The resulting algorithm proposed in the present paper is subquadratic in space O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.31</sup> ) and logarithmic in time. The space complexity is 33 percent better than in and 18 percent faster. It is asymptotically more efficient than the best known method (specifiably more efficient than when n ≥ 3,000). Furthermore, our proposal is available for every n and not only for n a power of two or three.

Alternative to the Karatsuba algorithm for software implementations of GF(2n) multiplications

A new approach to subquadratic space complexity GF(2n) multipliers has been proposed recently. The corresponding algorithm for software implementations is developed. While its recursive implementation is as simple as that of the Karatsuba algorithm, it requires much less memory to store the look-up table. Therefore it is quite suitable for memory-constrained applications, for example smart cards.

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