| 摘要 |
A system for producing a quotient B/A, where A and B are elements of GF(22M), 2M+1 is prime and 2 is a primitive element of GF(2M+1), first determines A-1 and then multiplies B by A-1. The system uses a (2M+1)-bit representation for A and produces, directly from A, an element C=A2M+1, where C also is an element of GF(22M) which is a subfield of GF(2M). The system produces M+1 bits to represent C by performing bit manipulations that are equivalent to permuting the (2M+1)-bits to produce A2M and multiplying the permuted bits by A. The bit manipulations are: c0=SIGMAaiai; c1=SIGMAaiai+1 . . . cM=SIGMAaiai+M where the aj's and cj's are the coefficients of A and C, respectively. The system retrieves C-1 from a (2M-1)-element lookup table and multiplies C-1=A-2M+1 by A2M to produce A-1. This multiplication is performed as the exclusive-OR'ing of M+1 sums of cyclically shifted copies of A2M , which is a permutation of the 2M+1 bits of A. The multiplication of B by A-1 is also performed as the exclusive-OR'ing of cyclically shifted copies of a (2M+1)-bit representation of B.
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