Scott Aaronson give interesting interpretation of Shor’s quantum
algorithm for factoring integer for the layman like me. What I
do not find is the reason why period finding is difficult
classically. The only implicit reason is that many smart people
have tried it, and did not find an algorithm. Here I want to
check the ideas in the Shor’s algorithm for the classical
approach. is the number we want to factor to be consistent with
Scott notation. The main purpose for the quantum algorithm is to
find the period of the function for some integer x. We need two
ingredients. First, can be represented as a binary number .
Second the natural representation of is . Two together give nice
representation of the period as a solution of the system of
polynomial equations. Following Scott presentation, we can write
, where To encode properties of we need following equations To
encode we need additionally and the final equations encoding
periodicity We can write many equations of the form (3) as many
as we need using differe