Sieve Method for Polynomial Linear Equivalence

Algorithm 1

Multiplicative sieve: MulSieve 𝔽_𝑞,𝒮,𝒫,𝐱_𝑖,𝑆_𝑖.

(1) Set the set 𝑆^×_𝑖=Φ. //Φ  denotes an empty set.

(2) Choose an element 𝑎_𝑖≠0,1 uniformly and randomly from 𝔽_𝑞 and compute 𝒫(𝑎_𝑖𝐱_𝑖).

(3) for each vector 𝐲∈𝑆_𝑖  do

(4)  Compute 𝒮(𝑎_𝑖𝐲).

(5)   if  𝒮(𝑎_𝑖𝐲)=𝒫(𝑎_𝑖𝐱_𝑖)  then

(6)   Add y to the set 𝑆^×_𝑖.

(7)   end if

(8) end for

(9) return  𝑆^×_𝑖.