A Comparison of VLSI Architecture of Finite Field Multipliers Using Dual, Normal, or Standard Bases
IEEE Transactions on Computers
A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases
Information and Computation
Designs, Codes and Cryptography
Finite field inversion over the dual basis
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Montgomery Multiplication in GF(2^k
Designs, Codes and Cryptography
Double-Basis Multiplicative Inversion Over GF(2m)
IEEE Transactions on Computers
Fast Galois field arithmetic for elliptic curve cryptography and error control codes
Fast Galois field arithmetic for elliptic curve cryptography and error control codes
Elliptic curve cryptography on smart cards without coprocessors
Proceedings of the fourth working conference on smart card research and advanced applications on Smart card research and advanced applications
Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography and Codes
Designs, Codes and Cryptography
Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Fast Key Exchange with Elliptic Curve Systems
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
A comparison of different finite fields for use in elliptic curve cryptosystems
A comparison of different finite fields for use in elliptic curve cryptosystems
Constructing tower extensions of finite fields for implementation of pairing-based cryptography
WAIFI'10 Proceedings of the Third international conference on Arithmetic of finite fields
An analysis of affine coordinates for pairing computation
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Efficient hardware implementation of elliptic curve cryptography over GF(pm)
WISA'05 Proceedings of the 6th international conference on Information Security Applications
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We introduce a new tower field representation, optimal tower fields (OTFs), that facilitates efficient finite field operations. The recursive direct inversion method we present has significantly lower complexity than the known best method for inversion in optimal extension fields (OEFs), i.e., Itoh-Tsujii's inversion technique. The complexity of our inversion algorithm is shown to be O(m^2), significantly better than that of the Itoh-Tsujii algorithm, i.e., O(m^2(\log_2m)). This complexity is further improved to O(m^{\log_23}) by utilizing the Karatsuba-Ofman algorithm. In addition, we show that OTFs may be converted to OEF representation via a simple permutation of the coefficients and, hence, OTF operations may be utilized to achieve the OEF arithmetic operations whenever a corresponding OTF representation exists. While the original OTF multiplication and squaring operations require slightly more additions than their OEF counterparts, due to the free conversion, both OTF operations may be achieved with the complexity of OEF operations.