Faster Residue Multiplication Modulo 521-bit Mersenne Prime and an Application to ECC

We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz matrix-vector product. The total arithmetic cost of our proposed algorithms is less than that of existing algorithms, with algorithms for 64- and 32-bit residue mul...

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Bibliographic Details
Published in:IEEE transactions on circuits and systems. I, Regular papers Vol. 65; no. 8; pp. 2477 - 2490
Main Authors: Ali, Shoukat, Cenk, Murat
Format: Journal Article
Language:English
Published: New York IEEE 01.08.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
32- and 64-bit platforms
Algorithm design and analysis
Algorithms
Complexity theory
Elliptic curve cryptography
Limbs
Mathematical analysis
Matrix algebra
Matrix decomposition
Matrix methods
Mersenne prime
Multiplication
Multiplication & division
Residue multiplication
Timing
Toeplitz matrix-vector product
variable-and fixed-base scalar multiplication
ISSN:1549-8328, 1558-0806
Online Access:Get full text
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Summary:We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz matrix-vector product. The total arithmetic cost of our proposed algorithms is less than that of existing algorithms, with algorithms for 64- and 32-bit residue multiplication giving the best timing results on our test machine. The transition from 64- to 32-bit implementation is full of challenges because the number of limbs doubles and the limbs' bitlengths are cut in half. Without using any intrinsics or SIMD/assembly instructions in our implementation on an Intel(R) Core i5 - 6402P CPU @ 2.80GHz, we find 136 and 550 cycles for our 64-and 32-bit residue multiplications, respectively. In addition, we implement constant-time variable- and fixed-base scalar multiplication for the standard NIST curve P-521 and Edwards curve E-521.
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ISSN:1549-8328
1558-0806
DOI:10.1109/TCSI.2018.2791285