Sieve Method for Polynomial Linear Equivalence

Algorithm 3

Differential sieve: DifSieve 𝔽_𝑞,𝒮,𝒫,𝐱_𝑖,𝑆_𝑖.

(1)   Set the set 𝑆⁻_𝑖=Φ.

(2)   Randomly choose an element 𝑎_𝑖≠0,1 from 𝔽_𝑞 and a nonzero vector

   𝐳_𝑖∈𝔽^𝑛_𝑞.

(3)   Compute 𝒫𝐱_𝑖+𝐳_𝑖 and 𝒫𝐱_𝑖+𝑎_𝑖𝐳_𝑖.

(4)   Compute the pre-images sets 𝒮⁻¹𝒫𝐱_𝑖+𝐳_𝑖 and 𝒮⁻¹𝒫𝐱_𝑖+𝑎_𝑖𝐳_𝑖.

(5)   for each 𝐲∈𝑆_𝑖  do

(6)      if  𝑎_𝑖−1𝐲∈𝑎_𝑖𝒮⁻¹𝒫𝐱_𝑖+𝐳_𝑖−𝒮⁻¹𝒫𝐱_𝑖+𝑎_𝑖𝐳_𝑖  then

(7)       Add y to 𝑆⁻_𝑖.

(8)      end if

(9)   end for

(10) return  𝑆⁻_𝑖