Montgomery Modular Multiplication on Reconfigurable Hardware: Systolic versus Multiplexed Implementation

Algorithm 1

Montgomery modular multiplication.

Require:   𝑁=∑^𝑚−1_𝑖=0(2^𝑘)^𝑖𝑛_𝑖, 𝑛_𝑖∈{0,1,…,2^𝑘−1}

 𝐵=∑^𝑚−1_𝑖=0(2^𝑘)^𝑖𝑏_𝑖,𝑏_𝑖∈{0,1,…,2^𝑘−1}

 𝐴=∑^𝑚−1_𝑖=0(2^𝑘)^𝑖𝑎_𝑖, 𝑎_𝑖 ∈ {0,1,…,2^𝑘−1}, 𝑅=(2^𝑘)^𝑚

 𝐴,𝐵<2𝑁; 𝑁<𝑅=2^𝑘𝑚; 𝑁′=−𝑁⁻¹mod(2^𝑘).

return 𝑆_𝑖+1=ABR⁻¹mod𝑁

𝑆₀=0

For 𝑖=0 to 𝑚−1  do

 𝑞_𝑖=((𝑆₀+𝑎_𝑖×𝑏₀)𝑁′)mod(2^𝑘)

 𝑆_𝑖+1=(𝑆_𝑖+𝑞_𝑖×𝑁+𝑎_𝑖×𝐵)/2^𝑘

End  for