Fast Rounding in Multiprecision . . .

to Section III, the factorized algorithm computes (14) and (15) with 36 multiplications and 90 additions. Other symmetries of (14) and (15) can be exploited: the first terms of the sums in (14) and also the first of d, and the second of di', the first of d; and the second of dd, the first of di' and the second of d; are pairwise equal, therefore, only 18 multiplications and 48 additions in GF(2) are needed. Without taking into account the time for input/output operations, a completely narallel realization of the Massey-Omura multiplier (with elementary nrocessors canable of the binarv GF(2) onerations between two boerand) muliinlies in five clock pulses~ the factorized algorithm multiplies in six (but with a greater communication complex%y). 1) An important part of the algorithmic principle presented in Section II is the factorization along the chain of fields (10). This principle can be applied to algorithms different from the one considered in Section III. For example, the algorithm presented in [20] is intrinsically sequential and factoring it along the chain (10) gives a much more parallel procedure. 2) Consider the basic algorithm between F,-;+ 1 and Fs-i+z in (12). The coefficients uk,((j+ 1)) in (7) are fixed elements of F,-;+ 2 so they induce a linear transformation into F,-i+2 which can be computed in no more than (m;-i *. . m1)2 operations in F. With this modification in the case m = 2 " (example l), the factorized algorithm requires no more than m2(l + s/4) multiplications in F. 3) Algorithms based on the bilinear renresentation of Section II allow a highly parallel implementation which computes a product in a finite field GF(2 ") in time log2 n. This is true also for the factorized algorithm of Section III with the modification of remark 2. It is worthwhile to notice that multiplication algorithms derived from efficient multiplication and division algorithms for polynomials (as the FFT and the Schonhage-Strassen algorithms [ 11, [lo], [ 141, [ 151) do not allow a parallel implementation running in time linear in log2 n. 4) The basis of E over F resulting from the algorithm factorization exhibits a " nested structure " as in (16). It is worthwhile to notice that, in general, it is not a normal basis for Fs-i+ 1 over F [13]. 5) The matrix Ac5) of the bilinear renresentation (2) associated with the basis (16) in example …