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Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography
, 2007
"... The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m log m log log m), for multiplication of mbit integers or (m − 1) st degree polynomials. Unfortunately, this method be ..."
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The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m log m log log m), for multiplication of mbit integers or (m − 1) st degree polynomials. Unfortunately, this method bears significant overhead due to the conversions between the time and frequency domains. This makes the original DFT based method impractical for small operands, e.g. less than 1000 bits in length as used in many applications. In this work, we investigate the application of the number theoretic transform (NTT), which found many applications in digital signal processing, to finite field multiplication with an emphasis on elliptic curve cryptography (ECC). Furthermore, we introduce an efficient algorithm for computing Montgomery products of polynomials in the frequency domain. Our algorithm performs the entire modular multiplication (including the reduction step) in the frequency domain, and thus eliminates costly back and forth conversions improving upon the straightforward NTT approach. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to elliptic curve cryptography (ECC). This paper is an expanded version of the earlier paper [22] on the same topic which, for the first time, proposes the use of frequency domain arithmetic for ECC and shows that it can be efficient.
A Stateoftheart Elliptic Curve Cryptographic Processor Operating in the Frequency Domain
"... Abstract. We propose a novel area/time efficient ECC processor architecture which performs all finite field arithmetic operations in the discrete Fourier domain. The proposed architecture utilizes a class of Optimal Extension Fields (OEF) GF (q m) where the field characteristic is a Mersenne prime q ..."
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Abstract. We propose a novel area/time efficient ECC processor architecture which performs all finite field arithmetic operations in the discrete Fourier domain. The proposed architecture utilizes a class of Optimal Extension Fields (OEF) GF (q m) where the field characteristic is a Mersenne prime q = 2 n − 1 and m = n. The main advantage of our architecture is that it achieves extension field modular multiplication in the discrete Fourier domain with only a linear number of base field GF (q) multiplications in addition to a quadratic number of simpler operations such as addition and bitwise rotation. We achieve an area between 25k and 50k equivalent gates for the implementation of OEFs of size 169, 289 and 361 bits. With its low area and high speed, the proposed architecture is well suited for elliptic curve cryptography in small device environments such as sensor networks. The work at hand presents the first hardware implementation of a frequency domain multiplier suitable for elliptic curve cryptography and the first hardware implementation of ECC in the frequency domain.
Optimal Extension Field Inversion in the Frequency Domain
"... Abstract. In this paper, we propose an adaptation of the ItohTsujii algorithm to the frequency domain for efficient inversion in a class of Optimal Extension Fields. To the best of our knowledge, this is the first time a frequency domain finite field inversion algorithm is proposed for elliptic cur ..."
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Abstract. In this paper, we propose an adaptation of the ItohTsujii algorithm to the frequency domain for efficient inversion in a class of Optimal Extension Fields. To the best of our knowledge, this is the first time a frequency domain finite field inversion algorithm is proposed for elliptic curve cryptography. We believe the proposed algorithm would be well suited especially for efficient lowpower hardware implementation of elliptic curve cryptography using affinecoordinates in constrained small devices such as smart cards and wireless sensor network nodes.
An Efficient Hardware Architecture for Spectral Hash Algorithm
"... The Spectral Hash algorithm is one of the Round 1 candidates for the SHA3 family, and is based on spectral arithmetic over a finite field, involving multidimensional discrete Fourier transformations over a finite field, data dependent permutations, Rubictype rotations, and affine and nonlinear fun ..."
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The Spectral Hash algorithm is one of the Round 1 candidates for the SHA3 family, and is based on spectral arithmetic over a finite field, involving multidimensional discrete Fourier transformations over a finite field, data dependent permutations, Rubictype rotations, and affine and nonlinear functions. The underlying mathematical structures and operations pose interesting and challenging tasks for computer architects and hardware designers to create fast, efficient, and compact ASIC and FPGA realizations. In this paper, we present an efficient hardware architecture for the full 512bit hash computation using the spectral hash algorithm. We have created a pipelined implementation on a Xilinx Virtex4 XC4VLX20011 FPGA which yields 100 MHz and occupies 38,328 slices, generating a throughput of 51.2 Gbps. Our fully parallel implementation shows that the spectral hash algorithm is about 100 times faster than the fastest SHA1 implementation, while requiring only about 13 times as many logic slices. 1.