Fast Truncated Multiplication & Crypto Applications Laszlo Hars (Seagate Research)

1. Fast Truncated Multiplication & Crypto ApplicationsLaszlo Hars (Seagate Research) • www.hars.us/Papers/Truncated Products.pdf • Karatsuba, Toom-Cook not used much • Complex / Slow for short numbers • Mod Mult = 3 Mult time  ≈ 2 • Sub-quadratic multiplications:many non-multiplicative operations • Complexity paradox:faster multiplication from pieces • Do NO more auxiliary operationsthan in the multiplication!

2. Reciprocal recursion r2k = rk || rk(rk-x(2k)) • Barrett’s multiplication: ab mod m =ab−[ab/m]m = LS(ab)−(MS(ab)µ)m • β = MS2n(b/m), ab mod m = (aβ)m • Montgomery multiplication abd−n mod m = MS(ab)−(LS(ab)(−m−1))m • β= b(−m−1), abd−nmodm = ab−(aβ)m

3. In the Paper • Carry estimate, exact rounding for truncated products • Equivalence of LS and MS halfproducts, within a linear term • Truncated polynomial product algorithms adapted to long integers(half products, middle third products, third quarter products) • Lengths of MS and LS truncated products, faster than the full product • Fast initialization of the Newton reciprocal algorithm • Newton's reciprocal algorithm with only truncated product arithmetic (no external additions or subtractions) • Fast long integer division algorithms • Accelerated Barrett multiplication with Karatsuba complexity and faster • Barrett multiplication with constant • Sub-quadratic Montgomery multiplication • Montgomery multiplication with constant • Fast quad-length mod-multiplications on short arithmetic co-processors