Fast Truncated Multiplication

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 multiplicationsmany
  non-multiplicative operations
• Complexity paradoxfaster multiplication from
  pieces
• Do NO more auxiliary operationsthan in the
  multiplication!

2

• Reciprocal recursion r2k  rk  rk ? (rk ?
  -x(2k))
• Barretts multiplication a b mod m a b -
  a b / m m LS(a b) - (MS(a b)?µ) ? m
• ß   MS2n(b / m), a b mod m  (a ? ß ) ? m
• Montgomery multiplication a b d -n mod m
   MS(a b) - (LS(a b) ? (-m-1)) ? m
• ß b ? (-m-1), a b d -n mod m
  a ? b - (a ? ß) ? 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