A.Pincin, "A New Algorithm for Multiplication in Finite Fields", IEEE Trans on Computers, Vol 38, No 7, pp 1045 - 1049, July '89  Home/Search   Document Not in Database   Summary   Related Articles   Check

....29 3.2.3 Other Architectures In this subsection references to two more schemes are provided which use subfields of Galois fields. Since the schemes are less relevant for the architectures to be developed in the thesis, they will only briefly be mentioned. The first architecture is by Pincin [Pin89] It is a parallel normal base multiplier over GF (2 k ) which uses arithmetic in subfields. The architecture is suited for a decomposition in multiple subfields, which are named descending chain of fields. For fields GF (2 2 s ) the computational complexity of the architecture is of ....

A. Pincin. A new algorithm for multiplication in finite fields. IEEE Transactions on Computers, 38(7):1045--1049, July 1989.

...., because we can calculate all of operations over the subfield and memorize them in advance, and then we can reduce a field operation to subfield operations using word operations. Pincin proposed an algorithm for the finite field F p n using a normal basis over subfields and successive extension [P89] The algorithm combines several operations which are basically the same operation into one operation for enhancing the speed. It is effective if n has many divisors. Harper, Menezes, and Vanstone implemented arithmetic operations over F 2 104 using a standard basis over subfield F 2 8 [HMV93] ....

....we have ff 2 n = ff 1. That is; ff is a normal basis generator, and [ff ff 2 n ] ff ff 1] holds. 4 Arithmetic Operations in a Quadratic Extension Field with Characteristic 2 Using a Successive Extension Pincin pointed out that a successive extension using a normal basis is effective [P89] and Paar pointed out that a quadratic extension using a standard basis is effective [P96, Sect. 4.2] for multiplication. So, we expect that other operations which are a squaring and an inversion are also effective. This section discusses successive quadratic extension and compares with a ....

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A. Pincin. A New Algorithm for Multiplication in Finite Fields. IEEE Transactions on computers, Vol. 38, No. 7, pp. 1045--1049, 1989.

....The extension field GF (2) x] Id(p(x) is denoted by GF (2 n ) where Id(p(x) is an ideal of p(x) n is the extension degree. It is known that for every positive integer n there exists an irreducible polynomial with degree n. For finding irreducible polynomials over GF (2) see the references [20, 21, 23, 24]. In Appendix B we list the irreducible polynomials over GF (2 n ) 2 n 68) primitive elements, and normal generators used in our experiments. Definition 2 Let F : GF (2 n ) GF (2 n ) be a function with an n bit input x = x n01 ; x 0 ) and an n bit output y = y n01 ; ....

....) is done in the same way as the polynomial basis case. Normal basis allows for a very fast squaring: it can be done by one shift operation, but multiplication is more complex than in polynomial basis. The Massey Omura algorithm[10] and some improved algorithms have been proposed for multiplication[20], but the normal basis representation seems more appropriate for hardware, as some references reported[3, 4, 5, 15, 25] 2 k th power operation by using normal basis requires only k cyclic shift operations, and it can be computed very fast. Example 1 Let n = 3, p(x) x 3 x 1; A = a 2 ....

A. Pincin, "A New Algorithm for Multiplication in Finite Fields," IEEE Transactions on Computers, vol.38, No.7, pp.1045--1049, 1989.

....this reason it is attractive to provide architectures with low computational complexity for efficient hardware implementations. More recently there have also been proposals for multipliers with complexities below the k 2 bound. These architectures are either based on multiple field extensions [2, 28], on fast convolution algorithms [1] or on both [25] see [24, Section 3.2] for an overview. One algorithm which was found to be particularly suited in this context is the Karatsuba Ofman 1 algorithm (KOA) which will be looked at in more detail in the following section. In this contribution a ....

A. Pincin. A new algorithm for multiplication in finite fields. IEEE Transactions on Computers, 38(7):1045--1049, July 1989.

....Galois field with four elements. It has also presented an efficient method for realizing the circuits that operate over the fields of larger size, provided that the field size is a multiple of four. When implementing the operations over fields that are much larger than four, the approach used in [87] can lead to successful designs. It is possible that some applications in coding can use the GF4 field directly. As a consequence, our circuits can be successfully employed without creating the circuits that operate over the higher order fields. We note a recent paper [112] that exemplifies such a ....

Pincin, A., A New Algorithm for Multiplication in Finite Fields, IEEE Trans. on Computers, Vol. C-38 No. 7, pp 1049-1049, July 1989.

.... of O(k 2 ) This paper presents a new architecture of a bit parallel, i.e. fast, multiplier for extension fields of GF (2) with a significantly improved space complexity. The application of (multiple) field extensions to multipliers has been proposed before in [7] 8] by Afanasyev and in [9] by Pincin. The results there show also a low space complexity although the internal structure of the multiplier is different. We consider finite fields GF (2 n ) with n 1. The elements of an extension field GF ( 2 n ) m ) may be represented in the standard (or canonical) base as ....

A. Pincin, "A new algorithm for multiplication in finite fields," IEEE Trans. Comp., vol. 38, pp. 1045--1049, July 1989.

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A.Pincin, "A New Algorithm for Multiplication in Finite Fields", IEEE Trans on Computers, Vol 38, No 7, pp 1045 - 1049, July '89

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