Applicability of randomized masked period-finding analysis to deterministic measurement sets

Prove that the randomized analysis of period finding on superpositions of simple periodic signals—used to model superposition masking of width S—applies to the deterministic remainder set R that arises when factoring RSA moduli using masked approximate modular exponentiation and frequency-basis measurement. Specifically, demonstrate that the quantum Fourier transform measurement probabilities and success-rate suppression predicted under random R accurately characterize the actual deterministic R encountered in Ekerå–Håstad-style period finding.

Background

To avoid measuring approximation error in the low bits of the modular exponentiation output, the paper uses superposition masking of a proportion S of the output space before measurement. The resulting post-measurement input state is a superposition of periodic signals determined by a remainder set R, and the author analyzes the success rate using a randomized model where R is chosen uniformly at random among sets of a given size.

Although numerical experiments suggest the real (deterministic) R behaves similarly to the randomized case, the analysis relies on the conjecture that the randomized behavior carries over. Establishing this would justify the cost model that estimates a mask of proportion S incurs an effective failure probability S and scales shots by 1/(1−S).