Fast Inversions in Small Finite Fields by Using Binary Trees

Yi, Haibo; Tang, Shaohua; Vemuri, Ranga

doi:10.1093/comjnl/bxw009

Cited by 9 publications

(4 citation statements)

References 26 publications

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“…We adopt a pipelined architecture in (2 4 ), which is the extension of the work in [30]. We use two binary trees for computing squares and inversions in (2 4 ), which are illustrated as follows:…”

Section: A Fast Inversion Based On Binary Trees We Suppose Thatmentioning

confidence: 99%

“…We propose a general architecture for efficient hardware implementations of Rainbow and enhance our design in three directions. First, we present a fast inversion in ((2 4 ) 2 ) based on binary trees, which is the extension of the work in [30]. Second, we present an efficient multiplication in ((2 4 ) 2 ) based on compact construction, which is the extension of the work in [27].…”

Section: Introductionmentioning

confidence: 99%

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Under Quantum Computer Attack: Is Rainbow a Replacement of RSA and Elliptic Curves on Hardware?

2018

Security and Communication Networks

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Among cryptographic systems, multivariate signature is one of the most popular candidates since it has the potential to resist quantum computer attacks. Rainbow belongs to the multivariate signature, which can be viewed as a multilayer unbalanced OilVinegar system. In this paper, we present techniques to exploit Rainbow signature on hardware meeting the requirements of efficient high-performance applications. We propose a general architecture for efficient hardware implementations of Rainbow and enhance our design in three directions. First, we present a fast inversion based on binary trees. Second, we present an efficient multiplication based on compact construction in composite fields. Third, we present a parallel solving system of linear equations based on GaussJordan elimination. Via further other minor optimizations and by integrating the major improvement above, we implement our design in composite fields on standard cell CMOS Application Specific Integrated Circuits (ASICs). The experimental results show that our implementation takes 4.9 us and 242 clock cycles to generate a Rainbow signature with the frequency of 50 MHz. Comparison results show that our design is more efficient than the RSA and ECC implementations.

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Section: A Fast Inversion Based On Binary Trees We Suppose Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Under Quantum Computer Attack: Is Rainbow a Replacement of RSA and Elliptic Curves on Hardware?

2018

Security and Communication Networks

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“…We improve the finite field multiplications via using composite field expression and design a finite field inversion based on binary trees described in [41]. Besides, we improve solving LSEs in a finite field based on the algorithm of Gauss-Jordan elimination and use the XOR gates to compute additions.…”

Section: Introductionmentioning

confidence: 99%

High-speed hardware architecture for implementations of multivariate signature generations on FPGAs

Nie

2018

J Wireless Com Network

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Multivariate signature belongs to Multivariate-Quadratic-Equations Public Key Cryptography (MPKC), which is secure to quantum computer attacks. Compared with RSA and ECC, it is required to speed up multivariate signature implementations. A high-speed hardware architecture for signature generations of a multivariate scheme is proposed in this paper. The main computations of signature generations of multivariate schemes are additions, multiplications, inversions, and solving systems of linear equations (LSEs) in a finite field. Thus, we improve the finite field multiplications via using composite field expression and design a finite field inversion via using binary trees. Besides, we improve solving LSEs in a finite field based on a variant algorithm of Gauss-Jordan elimination and use the XOR gates to compute additions. We implement the high-speed hardware architecture based on the above improvements on an Altera Stratix Field-Programmable Gate Array (FPGA), which shows that it takes only 90 clock cycles and 0.9 μs to generate a multivariate signature. The comparison shows that the hardware architecture is much faster than other implementations.

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“…Introduction. Finite field operations have gained increasing importance due to the fact that they are fundamental operations frequently encountered in areas of mathematic [1,5,23,26,27,29,[36][37][38] and engineering [4,7,16,28]. Generally, operations are computed via using a specific basis [2,3,6,9,10,13,[19][20][21][22]25,32,34,35], i.e.…”

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confidence: 99%

Efficient systolic multiplications in composite fields for cryptographic systems

Yi¹

2019

Discrete &Amp; Continuous Dynamical Systems - S

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Multiplications in finite fields are playing a key role in areas of cryptography and mathematic. We present approaches to exploit systolic architecture for multiplications in composite fields, which are expected to reduce the time-area product substantially. We design a pipelined architecture for multiplications in composite fields GF ((2 n) 2), where n is a positive integer. Besides, we design systolic architectures for multiplications and additions in finite fields GF (2 n). By integrating main improvements and other minor optimizations for multiplications in GF ((2 n) 2), the non-pipelined versions of our design takes 8n + 4 AND gates and 8n XOR gates to compute multiplications with the executing time of nT AN D + 4nT XOR , where T AN D and T XOR are delays of AND and XOR gates respectively; with the aid of pipelining, the pipelined version of our design has a throughput rate of one result per 2nT XOR. Other words, the time complexity and area complexity of our design are O(n). Thus, the complexity of time-area product of our design is O(n 2). Experimental results and comparisons show that our design provides significant reductions in executing time and area of multiplications. 2010 Mathematics Subject Classification. 12Y05.

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Fast Inversions in Small Finite Fields by Using Binary Trees

Cited by 9 publications

References 26 publications

Under Quantum Computer Attack: Is Rainbow a Replacement of RSA and Elliptic Curves on Hardware?

Under Quantum Computer Attack: Is Rainbow a Replacement of RSA and Elliptic Curves on Hardware?

High-speed hardware architecture for implementations of multivariate signature generations on FPGAs

Efficient systolic multiplications in composite fields for cryptographic systems

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