Fast hybrid Karatsuba multiplier for Type II pentanomials

We continue the study of Mastrovito form of Karatsuba multipliers under the shifted polynomial basis (SPB), recently introduced by Li et al. (IEEE TC (2017)). A Mastrovito-Karatsuba (MK) multiplier utilizes the Karatsuba algorithm (KA) to optimize polynomial multiplication and the Mastrovito approach to combine it with the modular reduction. The authors developed a MK multiplier for all trinomials, which obtain a better space and time trade-off compared with previous non-recursive Karatsuba counterparts. Based on this work, we make two types of contributions in our paper. FORMULATION. We derive a new modular reduction formulation for constructing Mastrovito matrix associated with Type II pentanomial. This formula can also be applied to other special type of pentanomials, e.g. Type I pentanomial and Type C.1 pentanomial. Through related formulations, we demonstrate that Type I pentanomial is less efficient than Type II one because of a more complicated modular reduction under the same SPB; conversely, Type C.1 pentanomial is as good as Type II pentanomial under an alternative generalized polynomial basis (GPB). EXTENSION. 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 bit-parallel multipliers for Type II pentanomial, but its space complexity is roughly 3/4 of those schemes, where $T_X$ is the delay of one 2-input XOR gate. To the best of our knowledge, it is the first time for hybrid multiplier to achieve such a time delay bound

A new class of irreducible pentanomials for polynomial-based multipliers in binary fields

We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ of the form $f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1$. Let $m=2b+c$ and use $f$ to define the finite field extension of degree $m$. We give the exact number of operations required for computing the reduction modulo $f$. We also provide a multiplier based on Karatsuba algorithm in $\mathbb{F}_2[x]$ combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm

Low Complexity Finite Field Multiplier for a New Class of Fields

Finite fields is considered as backbone of many branches in number theory, coding theory, cryptography, combinatorial designs, sequences, error-control codes, and algebraic geometry. Recently, there has been considerable attention over finite field arithmetic operations, specifically on more efficient algorithms in multiplications. Multiplication is extensively utilized in almost all branches of finite fields mentioned above. Utilizing finite field provides an advantage in designing hardware implementation since the ground field operations could be readily converted to VLSI design architecture. Moreover, due to importance and extensive usage of finite field arithmetic in cryptography, there is an obvious need for better and more efficient approach in implementation of software and/or hardware using different architectures in finite fields. This project is intended to utilize a newly found class of finite fields in conjunction with the Mastrovito algorithm to compute the polynomial multiplication more efficiently

Efficient Square-based Montgomery Multiplier for All Type C.1 Pentanomials

In this paper, we present a low complexity bit-parallel Montgomery multiplier for $GF(2^m)$ generated with a special class of irreducible pentanomials $x^m+x^{m-1}+x^k+x+1$. Based on a combination of generalized polynomial basis (GPB) squarer and a newly proposed square-based divide and conquer approach, we can partition field multiplications into a composition of sub-polynomial multiplications and Montgomery/GPB squarings, which have simpler architecture and thus can be implemented efficiently. Consequently, the proposed multiplier roughly saves 1/4 logic gates compared with the fastest multipliers, while the time complexity matches previous multipliers using divide and conquer algorithms

Reconfigurable implementation of GF(2^m) bit-parallel multipliers

Hardware implementations of arithmetic operations over binary finite fields GF(2^m) are widely used in several important applications, such as cryptography, digital signal processing and error-control codes. In this paper, efficient. reconfigurable implementations of bit-parallel canonical basis multipliers over binary fields generated by type II irreducible pentanomials f_(y) = y^m + y^(n+2) + y^(n+1) + y^n + 1 are presented. These pentanomials are important because all five binary fields recommended by NIST for ECDSA can be constructed using such polynomials. In this work, a new approach for CF(2^m) multiplication based on type II pentanomials is given and several post-place and route implementation results in Xilinx Artix-7 FPGA are reported. Experimental results show that the proposed multiplier implementations improve the area x time parameter when compared with similar multipliers found in the literature

Novel Single and Hybrid Finite Field Multipliers over GF(2m) for Emerging Cryptographic Systems

With the rapid development of economic and technical progress, designers and users of various kinds of ICs and emerging embedded systems like body-embedded chips and wearable devices are increasingly facing security issues. All of these demands from customers push the cryptographic systems to be faster, more efficient, more reliable and safer. On the other hand, multiplier over GF(2m) as the most important part of these emerging cryptographic systems, is expected to be high-throughput, low-complexity, and low-latency. Fortunately, very large scale integration (VLSI) digital signal processing techniques offer great facilities to design efficient multipliers over GF(2m). This dissertation focuses on designing novel VLSI implementation of high-throughput low-latency and low-complexity single and hybrid finite field multipliers over GF(2m) for emerging cryptographic systems. Low-latency (latency can be chosen without any restriction) high-speed pentanomial basis multipliers are presented. For the first time, the dissertation also develops three high-throughput digit-serial multipliers based on pentanomials. Then a novel realization of digit-level implementation of multipliers based on redundant basis is introduced. Finally, single and hybrid reordered normal basis bit-level and digit-level high-throughput multipliers are presented. To the authors knowledge, this is the first time ever reported on multipliers with multiple throughput rate choices. All the proposed designs are simple and modular, therefore suitable for VLSI implementation for various emerging cryptographic systems

A new approach in building parallel finite field multipliers

A new method for building bit-parallel polynomial basis finite field multipliers is proposed in this thesis. Among the different approaches to build such multipliers, Mastrovito multipliers based on a trinomial, an all-one-polynomial, or an equally-spacedpolynomial have the lowest complexities. The next best in this category is a conventional multiplier based on a pentanomial. Any newly presented method should have complexity results which are at least better than those of a pentanomial based multiplier. By applying our method to certain classes of finite fields we have gained a space complexity as n2 + H - 4 and a time complexity as TA + ([ log2(n-l) ]+3)rx which are better than the lowest space and time complexities of a pentanomial based multiplier found in literature. Therefore this multiplier can serve as an alternative in those finite fields in which no trinomial, all-one-polynomial or equally-spaced-polynomial exists

An Efficient CRT-based Bit-parallel Multiplier for Special Pentanomials

The Chinese remainder theorem (CRT)-based multiplier is a new type of hybrid bit-parallel multiplier, which can achieve nearly the same time complexity compared with the fastest multiplier known to date with reduced space complexity. However, the current CRT-based multipliers are only applicable to trinomials. In this paper, we propose an efficient CRT-based bit-parallel multiplier for a special type of pentanomial $x^m+x^{m-k}+x^{m-2k}+x^{m-3k}+1, 5k<m\leq 7k$. Through transforming the non-constant part $x^m+x^{m-k}+x^{m-2k}+x^{m-3k}$ into a binomial, we can obtain relatively simpler quotient and remainder computations, which lead to faster implementation with reduced space complexity compared with classic quadratic multipliers. Moreover, for some $m$, our proposal can achieve the same time delay as the fastest multipliers for irreducible Type II and Type C.1 pentanomials of the same degree, but the space complexities are reduced

Fast bit-parallel binary multipliers based on type-I pentanomials

In this paper, a fast implementation of bit-parallel polynomial basis (PB) multipliers over the binary extension field GF(2^m) generated by type-I irreducible pentanomials is presented. Explicit expressions for the coordinates of the multipliers and a detailed example are given. Complexity analysis shows that the multipliers here presented have the lowest delay in comparison to similar bit-parallel PB multipliers found in the literature based on this class of irreducible pentanomials. In order to prove the theoretical complexities, hardware implementations over Xilinx FPGAs have also been performed. Experimental results show that the approach here presented exhibits the lowest delay with a balanced Area x Time complexity when it is compared with similar multipliers

Low-delay FPGA-based implementation of finite field multipliers

Arithmetic operations over binary extension fields GF(2^m) have many important applications in domains such as cryptography, code theory and digital signal processing. These applications must be fast, so low-delay implementations of arithmetic circuits are required. Among GF(2^m) arithmetic operations, field multiplication is considered the most important one. For hardware implementation of multiplication over binary finite fields, irreducible trinomials and pentanomials are normally used. In this brief, low-delay FPGA-based implementations of bit-parallel GF(2^m) polynomial basis multipliers are presented, where a new multiplier based on irreducible trinomials is given. Several post-place and route implementation results in Xilinx Artix-7 FPGA for different GF(2^m) finite fields are reported. Experimental results show that the proposed multiplier exhibits the best delay, with a delay improvement of up to 4.7%, and the second best Area x Time complexities when compared with similar multipliers found in the literature