A Hardware-Accelerated ECDLP with High-Performance Modular Multiplication

Algorithm 3

Vector inversion using Montgomery's trick [23].

Input: 𝑧₁,𝑧₂,…𝑧_𝑀∈𝔽_𝑞

Output: 𝑧₁⁻¹,𝑧₂⁻¹,…𝑧_𝑀⁻¹∈𝔽_𝑞

 // Preprocessing

 𝑥₁←𝑧₁

 for 𝑖=2→𝑀 do

  𝑥_𝑖←𝑥_𝑖−1⋅𝑧_𝑖 mod 𝑞

 // Inversion

 𝑦_𝑀←𝑥_𝑀⁻¹

 // Postprocessing

 for 𝑖=𝑀−1→1 do

  𝑦_𝑖←𝑦_𝑖+1⋅𝑧_𝑖+1 mod 𝑞

  𝑧_𝑖+1←𝑦_𝑖+1⋅𝑥_𝑖 mod 𝑞

 𝑧₁←𝑦₁

 return 𝑧₁⁻¹,𝑧₂⁻¹,…𝑧_𝑀⁻¹