R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone. Computational Method and Apparatus for Finite Field Multiplication. U.S. Patent 4,745,568, May 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....elements, so these are useful for computing large exponentiations and multiplicative inverses [13, 11, 1] Also, the multiplication table of a normal basis is symmetric, so suitable for hardware implementation. This is the basis for the multiplier of Massey Omura [16] and that of Onyszchuk et al. [18]. Recently, Gao et al. 7, 8] have proposed a novel method to perform fast multiplication with a normal basis generated by Gau periods. The main idea is to embed a field in a larger ring, perform multiplication (using the Fast Fourier Transform) there and then convert the result back to the ....

I.M. Onyszchuk, R.C. Mullin, and S.A. Vanstone. Computational method and apparatus for finite field multiplication. U.S. Patent No.4,745,568, 1988.

....on the choice of basis, the mathematical formula for a GF (2 m ) multiplication can be quite di erent, thus making major di erences in practical implementation. Currently, it seems that normal basis representation (especially optimal normal basis) offers the best performance in hardware [9 11], while in software polynomial basis representation is more ecient [2, 3, 8] For interoperability, it is desirable to support both types of basis in software, which can be done either by implementing arithmetic in both bases or by implementing one basis together with basis conversion algorithms. ....

....m 1 M i=1 a k i b t1[i] k b t2[i] k : 3) 3 Related Work 3.1 Hardware In formula (1) when a new bit c k needs to be computed, the coecients of both a and b are rotated to the left by one bit. This fact is useful for ecient hardware implementation of normal basis multiplication [9 11], since the same circuit that represents M can be repeatedly used and each coecient can be computed in one clock cycle. Even though the sequence of operations for each coecient is easily parallelized in hardware, it is quite dicult to mimic the same technique in a software implementation since ....

R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone. Computational Method and Apparatus for Finite Field Multiplication. U.S. Patent 4,745,568, May 1988.

....151] including single chip exponentiators for the fields F 2 127 [152] F 2 155 [3] and F 2 332 [56] and an encryption processor for F 2 593 [114] for public key cryptography. These products are based on multiplication schemes due to Massey and Omura [95] and Mullin, Onyszchuk and Vanstone [105] by using normal bases to represent finite fields and choosing appropriate algorithms for the arithmetic. Interestingly, the advantage of using a normal basis representation was noticed by Hensel [61] in 1888, long before finite field theory found its practical applications. The complexity of the ....

I.M. Onyszchuk, R.C. Mullin and S.A. Vanstone, "Computational method and apparatus for finite field multiplication", U.S. patent #4,745,568, May 1988.

....can be computed with O(n=log q n) multiplications in F q n for q small (compared to n) with a storage for O(n= log 2 q n) elements of F q n . The question is how to implement multiplication efficiently under normal bases. Hardware implementations of large finite fields (Massey and Omura 1986, Onyszchuk et al. 1988, Calmos 1988, Rosati 1989, Agnew et al. 1991 and Agnew et al. 1993) exploit the symmetry in the multiplication table of a normal basis. In an attempt to minimize hardware cost, Mullin et al. 1989) introduce optimal normal bases. For a normal basis N = ff; ff q ; Delta Delta Delta ; ff q ....

....q i expressed in the basis N itself is at least 2n Gamma 1. If it is equal to 2n Gamma 1, then N is called optimal. Under an optimal normal basis, hardware cost (i.e. the number of cell connections) is minimized to 2n Gamma 1 (for q = 2) in the designs used by Massey and Omura (1986) and Onyszchuk et al. 1988). But the total number of operations in F q required for one multiplication in F q n is still about n 2 , and exponentiation in F q n needs about n 3 = log n operations in F q . Can we reconcile fast multiplication and division with normal bases We answer this question affirmatively in ....

Onyszchuk, I., Mullin, R., Vanstone, S. (1988). Computational method and apparatus for finite field multiplication. United States Patent, 4,745,568.

....permutability in more detail below. It is known that the complexity of a normal basis B for F 2 N =F 2 satisfies C(B) 2N Gamma 1, and for certain fields it is possible to find a normal basis satisfying C(B) 2N Gamma 1. Such bases are called optimal normal bases , or ONB for short. See [1, 3, 4, 10, 11, 12] for further information about normal bases. In [2, 6, 14, 16] an idea was introduced to perform computations in F 2 N with complexity smaller than 2N Gamma 1, while preserving many of the nice properties of (optimal) normal bases. Briefly, the idea is to write F 2 N as the quotient field of a ....

I. Onyszchuk, R. Mullin, S. Vanstone, Computational method and apparatus for finite field multiplication, United States Patent 4745568, May 17, 1988.

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R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone. Computational Method and Apparatus for Finite Field Multiplication. U.S. Patent 4,745,568, May 1988.

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