VLSI Architectures for Computing Multiplications and Inverses in GF(2m)
IEEE Transactions on Computers
Introduction to finite fields and their applications
Introduction to finite fields and their applications
Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases
Information and Computation
Self-complementary normal bases in finite fields
SIAM Journal on Discrete Mathematics
Optimal normal bases in GF(pn)
Discrete Applied Mathematics
Discrete Applied Mathematics
Finite field inversion over the dual basis
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
GF(2m) Multiplication and Division Over the Dual Basis
IEEE Transactions on Computers
Efficient Algorithms for Elliptic Curve Cryptosystems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
A Fast Software Implementation for Arithmetic Operations in GF(2n)
ASIACRYPT '96 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Fast Key Exchange with Elliptic Curve Systems
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Efficient Bit Serial Multiplication Using Optimal Normal Bases of Type II in GF (2m)
ISC '02 Proceedings of the 5th International Conference on Information Security
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In this paper we consider a special type of dual basis for finite fields, GF(2m), where the variants of m are presented in section 2. We introduce our field representing method for efficient field arithmetic(such as field multiplication and field inversion). It reveals a very effective role for both software and hardware(VLSI) implementations, but the aspect of hardware design of its structure is out of this manuscript and so, here, we deal only the case of its software implementation(the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of several advantageous characteristics of our method is that (1) the field multiplication in GF(2m) can be constructed only by m + 1 vector rotations and the same amount of vector XOR operations, (2) there is required no additional work load such as basis changing(from standard to dual basis or from dual basis to standard basis as the conventional dual based arithmetic does), (3) the field squaring is only bit-by-bit permutation and it has a good regularity for its implementation, and (4) the field inversion process is available to both cases of its implementation using Fermat's Theorem and using almost inverse algorithm[14], especially the case of using the almost inverse algorithm has an additional advantage in finding(computing) its complete inverse element(i.e., there is required no pre-computed table of the values, x-k, k = 1, 2, . . .).