Complexity of structured totient preimage reconstruction

Determine the reconstruction complexity of the Structured Totient Preimage problem, which asks, given a shifted product x, a prime bit length λ, and a cardinality k, for any set of k distinct λ-bit primes whose shifted factors multiply to x.

Background

The paper defines the Structured Totient Preimage (STP) problem as the reconstruction of a set of distinct, equal-bit-length primes from the product of their shifted values p_i−1. Although the resulting relation is efficiently verifiable, the computational complexity of finding a valid preimage is not established.

The paper derives exact candidate-allocation counts and proves polynomial-time reconstruction for a restricted factored-input regime with fixed k and Ω(x)=O(log λ). It does not resolve the complexity of STP in general or in growing parameter regimes, leaving the broader reconstruction problem unresolved.