B. Ito and S. Tsujii. Structure of a Parallel Multiplier for a Class of Fields GF(2 ). Information and Compuers, 83:21--40, 1989. 9  Home/Search   Document Not in Database   Summary   Related Articles   Check

....devised a practical implementation of the Weil descent to compute discrete logarithms in F 2 N for composite N s. This result seemed to preclude the use of composite binary fields for elliptic curve cryptography. On the other hand, Silverman [21] and independently in 1989 ItoTsujii [9]) proved that basic field operations can be implemented very quickly on certain composite binary extensions, namely extensions F 2 p 1 , with prime p such that 2 is a primitive root modulo p, which we will call Silverman fields. The goal of the present article is to resuscitate elliptic curve ....

B. Ito and S. Tsujii. Structure of a Parallel Multiplier for a Class of Fields GF(2 ). Information and Compuers, 83:21--40, 1989. 9

....The bit serial and hybrid architectures of this modified multiplier have lower complexity compared to the previously reported normal basis multipliers. A constant multiplier using the redundant representation is also considered. We should mention other related work here. Itoh and Tsujii [14] constructed a multiplier for a class of fields defined by irreducible all one polynomials (AOPs) and equally spaced polynomials (ESPs) Wolf [22] found a simple multiplication architecture for irreducible AOP s. Drolet [4] uses maximum subfields in cyclotomic rings. Silverman [19] considered a ....

T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of fields GF(# ). Inform. and Comput., 83:21--40, 1989.

....Table 5.1 shows that the introduction of composite fields GF ( 2 n ) m ) GF (2 k ) leads to significantly improved parallel multipliers with respect to the number of mod 2 adders and multipliers if compared to traditional architectures such as introduced in Section 3. 1 or [HWB92b] IT89] Moreover, the multiplier has also a lower gate count for all fields Multipliers over Composite Fields 57 u u u u k k ffifl fflfi k ffifl fflfi k k ffifl fflfi k u u ffifl fflfi k k k ffifl fflfi k k k k k k k k k k k k k ffifl fflfi ffifl fflfi ffifl fflfi ffifl fflfi ffifl fflfi ffifl fflfi ....

T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of fields GF (2 k ). Inform. and Comp., 83:21--40, 1989.

....point of view since most advanced arithmetic functions such as inversion and exponentiation are based on multiplication. There have been considerable research efforts in the development of VLSI suited multiplier architectures over the last decade which is reflected by numerous journal publications [31, 29, 11, 12, 20, 10, 9, 5] and recent dissertations [21, 8, 6, 24, 13] Multipliers can be classified into bit serial and bit parallel architectures. Since there is a space time trade off, bit parallel multipliers tend to be faster which makes them attractive for many applications. The penalty, however, are higher hardware ....

T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of fields GF (2 k ). Inform. and Comp., 83:21--40, 1989.

....RI 02912 USA jhs math.brown.edu Abstract The complexity of the multiplication operation in finite fields is of interest for both theoretical and practical reasons. For example, an optimal normal basis for F 2 N has complexity 2N Gamma1. For certain values of N , a construction described in [2, 6, 14, 16] allows multiplication of complexity N 1 to be performed in F 2 N by working in a larger ring R of dimension N 1 over F 2 . In this paper we give a complete classification of all such rings and show that the cited construction is the only one that also possesses an important permutability ....

....with 2 elements, and let R = F 2 N to be a finite field extension of k. Such fields are used extensively in cryptography, and there is a considerable literature devoted to the problem of efficiently implementing the multiplication operation in F 2 N in both hardware and software, see for example [1, 2, 4, 5, 6, 7, 8, 13, 15, 16, 17, 18]. A particularly nice sort of basis for F 2 N =F 2 is a normal basis. Normal bases allow extremely rapid squaring of elements, and they have a nice permutability property that allows all of the k ij multipliers to be easily derived from the multipliers with k = 1. We will discuss ....

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B. Ito, S. Tsujii, Structure of a parallel multipliers for a class of fields GF(2 m ). Information and Computers 83 (1989), 21--40.

....and inversion are very complex operations. Our contribution is concerned with space and area optimization of architectures for these two operations in GF (2 8 ) There is a wealth of literature on general arithmetic architectures for Galois fields [FBT96a, FBT96b, KRV93, HWB92, Mas89, IT89, Fen89, MK89] However, there appears to be only one publication which deals specifically with application of Galois field arithmetic architectures to reconfigurable platforms [Kli95] Our contribution provides a detailed and systematic study of two multiplication and two inversion architectures ....

T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of fields GF (2 k ). Inform. and Comp., 83:21--40, 1989.

....them seems promising and may yield a considerably reduced over all encoder complexity. Although there have been considerable research efforts The research was done while the author was with the Institute for Experimental Mathematics, University of Essen, Germany. WTS 85, STP86, HTDR88, IT89, Mas89, HWB92b, HWB92a, KRV93] in the area of general finite field multipliers, to our knowledge this is the first report of a systematic approach to optimizing constant multiplication over Galois fields GF (2 n ) In addition to RS encoders, constant multipliers with low complexity are also of ....

T. Itoh and S. Tsujii. Structure of parallel multipliers for a class of fields GF (2 k ). Inform. and Comp., 83:21--40, 1989.

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