A Hardware-Accelerated ECDLP with High-Performance Modular Multiplication

Algorithm 2

Sliding window modular exponentiation [24].

Input: 𝑧∈𝔽_𝑞, 𝑒[𝑡∶0], 𝑘≥1

Output: 𝑧^𝑒 mod 𝑞

 // Precomputation

 𝑝₀←𝑧, 𝑠←𝑧⋅𝑧 mod 𝑞

 for 𝑖=1→2^𝑘−1−1 do

  𝑝_𝑖←𝑝_𝑖−1⋅𝑠

 // Inversion

 𝑟←1

 𝑖←𝑡

 while 𝑖≥0 do

  if 𝑒_𝑖=0 then

   𝑟←𝑟⋅𝑟 mod 𝑞

   𝑖←𝑖−1

  else

   Find longest string 𝑒[𝑖∶𝑙] where 𝑒[𝑙]=1

   and 𝑖−𝑙+1≤𝑘

   𝑟←𝑟⋅𝑝_{𝑒[𝑖∶𝑙]} mod 𝑞

   for 𝑗=0→(𝑖−𝑙+1) do

    𝑟←𝑟⋅𝑟 mod 𝑞

   𝑖←𝑖−𝑙−1

 return 𝑟