Preimage Selective Trapdoor Function: How to Repair an Easy Problem

Algorithm 2

Generating 𝐮 and 𝐯.

(1) For 𝑖=1,…,𝑛−1, randomly choose 𝛼_𝑖,𝛽_𝑖∈

   AIP_{⟨𝑖+1⟩12}∪AIP^𝑇_{⟨⟩𝑖+112} with repetition permitted.

(2) Randomly choose 2(𝑛−1) numbers 𝜁₂,…,𝜁_𝑛 and

   𝜂₂,…,𝜂_𝑛 and set 𝜁₁=𝜂₁=𝛼_𝑛=𝛽_𝑛=1 such that

   gcd(𝜁_𝑖,𝛼_𝑗)=gcd(𝜂_𝑖,𝛽_𝑗)=1 for 1≤𝑖,𝑗≤𝑛, and

   gcd(𝜁_𝑖,𝜁_𝑖+1)=gcd(𝜂_𝑖,𝜂_𝑖+1)=1 for 𝑖=1,…,𝑛−1.

(3) for  𝑖=1,…,𝑛  do

(4)     𝑢_𝑖=𝜁∏_𝑖^𝑛_𝑗=𝑖𝛼_𝑗,  𝑣_𝑖=𝜂∏_𝑖^𝑛_𝑗=𝑖𝛽_𝑗.   (★)

(5) end for

(6) return  𝐮=(𝑢₁,…,𝑢_𝑛), 𝐯=(𝑣₁,…,𝑣_𝑛).