A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases
Information and Computation
Optimal normal bases in GF(pn)
Discrete Applied Mathematics
Structure of parallel multipliers for a class of fields GF(2m)
Information and Computation
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
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Computing special powers in finite fields: extended abstract
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Algorithms for exponentiation in finite fields
Journal of Symbolic Computation
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LATIN '95 Proceedings of the Second Latin American Symposium on Theoretical Informatics
IEEE Transactions on Computers
Efficient Multiplication Using Type 2 Optimal Normal Bases
WAIFI '07 Proceedings of the 1st international workshop on Arithmetic of Finite Fields
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
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In the 1990s and early 2000s several papers investigated the relative merits of polynomial-basis and normal-basis computations for F2n. Even for particularly squaring-friendly applications, such as implementations of Koblitz curves, normal bases fell behind in performance unless a type-I normal basis existed for F2n. In 2007 Shokrollahi proposed a new method of multiplying in a type-II normal basis. Shokrollahi's method efficiently transforms the normal-basis multiplication into a single multiplication of two size-(n+1) polynomials. This paper speeds up Shokrollahi's method in several ways. It first presents a simpler algorithm that uses only size-n polynomials. It then explains how to reduce the transformation cost by dynamically switching to a 'type-II optimal polynomial basis' and by using a new reduction strategy for multiplications that produce output in type-II polynomial basis. As an illustration of its improvements, this paper explains in detail how the multiplication overhead in Shokrollahi's original method has been reduced by a factor of 1.4 in a major cryptanalytic computation, the ongoing attack on the ECC2K-130 Certicom challenge. The resulting overhead is also considerably smaller than the overhead in a traditional low-weight-polynomial-basis approach. This is the first state-of-the-art binary-elliptic-curve computation in which type-II bases have been shown to outperform traditional low-weight polynomial bases.