By Richard A. DeMillo
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Extra info for Foundations of Secure Computation
The cost of the algorithm is mainly that of the four convolutions AB mod n (2 ±1), cµ mod (2n −1) and γN mod (2n +1), which cost 24 M (n) altogether. However in cµ mod (2n − 1) and γN mod (2n + 1), the operands µ and N are invariant, therefore their Fourier transform can be precomputed, which saves 31 M (n). A further saving of 61 M (n) is obtained by keeping d and δ in Fourier space in steps 6 and 8, performing the subtraction d − δ in Fourier space, and performing only one inverse transform for step 9.
2) also translates to the LSB context. More precisely, in Algorithm REDC, one wants to modify the divisor N so that the “quotient selection” q ← µci mod β at step 5 becomes trivial. A natural choice is to have µ = 1, which corresponds to N ≡ −1 mod β. The multiplier k used in Svoboda division is thus here simply the parameter µ in REDC. The Montgomery-Svoboda algorithm obtained works as follows: 1. first compute N ′ = µN , with N ′ < β n+1 ; 2. then perform the n − 1 first loops of REDC, replacing µ by 1, and N by N ′; 3.
Thus several algorithms in this Chapter are just LSB-variants of algorithms discussed in Chapter 1 (see Fig. 1). 5 Link with Polynomials As in Chapter 1, a strong link exists between modular arithmetic and arithmetic on polynomials. One way of implementing finite fields Fq with q = pn is to work with polynomials in Fp [x], which are reduced modulo a monic irreducible polynomial f (x) ∈ Fp [x] of degree n. , modulo f (x). Some algorithms work in the ring (Z/nZ)[x], where n is a composite integer.
Foundations of Secure Computation by Richard A. DeMillo