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Efficient Montgomery multiplier construction method based on special penta-nomial expression
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A construction method and multiplier technology, applied in the direction of instruments, calculations, electrical digital data processing, etc., can solve problems that are not very simple, and achieve the effect of reducing the number of classifications
Inactive Publication Date: 2018-11-13
XINYANG NORMAL UNIVERSITY
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However, the squaring of the pentnomial is not very simple
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example 21
[0144] Example 2.1: Consider that in GF(2 5 ) using field multiplication represented by PB and the irreducible trinomial x 5 +x 4 +x 2 +x+1. Because its degree is an odd number, the present invention selects γ=x 3 +x 2 +1 for the Montgomery parameter. suppose is GF(2 5 ) on any two elements, the present invention splits A, B into A=A 1 +xA 2 2 ,B=x -1 B 1 +B 2 2 ,
[0145] in
[0146] A 1 =a 4 x 2 +a 2 x+a 0 ,A 2 =a 3 x+a 1 ,
[0147] B 1 =b 3 x 2 +b 1 x,B 2 =b 4 x 2 +b 2 x+b 0 .
[0148] According to equations (1) and (3), the present invention has
[0154] A 1 B 1 =(a 4 b 3 )x 4 +(a 2 b 3 +a 4 b 1 )x 3 +(a 0 b 3 +a 2 b 1 )x 2 +a 0 b 1 x,
[0155] A 2 B 3 =(a 3 b 4 )x 3 +(a 1 b 4 +a 3 b ...
example 3
[0205] Example 3.2: Consider S in Example 3.1 1 ,S 2 ,S 3 The calculation of , on the basis of the formulas in Appendix A and B, it is easy to get S 1 ,S 2 ,S 3 The coefficients are as follows:
[0206]
[0207]
[0208]
[0209] Obviously, S 1 +S 2 Each coefficient of is composed of at most 7 terms, so it can be found in 3T X Completion of the calculation; S 3 Each coefficient of contains at most 4 terms, which can be in 2T X complete the calculation. Therefore, S 1 ,S 2 ,S 3 The calculation of can be given by (17). 3. Reciprocal properties
[0210] So far, the present invention has only analyzed the C.1 type pentomial f(x)=x m +x m-1 +x k +x+1 in (or ) when the Montgomery multiplier. According to the description in Section 2.3, it is easy to know is a reciprocal polynomial of f(x), and in Can't make an appointment. Obviously, (or ) Pentanomials of this type also belong to those of type C.1. From Lemma 2, we can see that by choosing ap...
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Abstract
The invention discloses an efficient Montgomery multiplier construction method based on a special penta-nomial expression, and designs a low-complexity Montgomery bit parallel multiplier on a GF (2<m>) (Galois Field). The multiplier calculates GF multiplication generated by a type of special irreducible penta-nomial expression x<m>+x<m-1>+x<k>+x+1. According to the invention, a GPB (Generalized Polynomial Basis) squarer and a newly proposed divide-and-conquer PCHS ((ParkChang-Hong-Seo) algorithm are used, multiplication on the GF (2<m>) can be segmented into a combination of m / 2 degree polynomial multiplication and Montgomery / GPB Square operation, and the constructed multiplication has the characteristics of simple structure and the like, and thus, the efficient Montgomery multiplier construction method can be efficiently implemented. Compared to a current similar multiplier with the highest speed, the multiplier constructed by the method disclosed by the invention approximately can save one fourth of logic gate, and required time complexity can be matched with that of a previously proposed parallel multiplier which does not use the divide-and-conquer algorithm.
Description
technical field [0001] The invention belongs to the field of computer and information technology, and in particular relates to a method for constructing a high-efficiency Montgomery multiplier based on a special pentomial. Background technique [0002] Finite field GF(2 m ) (Galois Field) has important applications in many fields such as combinatorial design, coding theory, computer algebra and cryptography. More and more people began to study GF(2 m ) efficient implementation of multiplication. The main reason is that GF(2 m ) complex arithmetic operations including inversion and exponentiation can be implemented using multiplication. Today, bit-parallel multiplier architectures are common as more and more gates are integrated into a single chip. In recent years, many bit-parallel GFs (2 m ) multipliers are proposed in order to obtain lower space and time complexity. These schemes cover a wide range of cases including different basis representations and generator pol...
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