1. Fast Truncated Multiplication & Crypto Applications Laszlo Hars (Seagate Research)
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 operations than in the multiplication!

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

3. In the Paper Carry estimate, exact rounding for truncated products
Equivalence of LS and MS half products, 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