The Structured Totient Preimage Problem: Reconstruction, Collisions, and Cryptographic Implications

Abstract

We define and study the Structured Totient Preimage (STP) problem as a restricted reconstruction relation with a direct cryptographic motivation. Let $p_1,\ldots,p_k$ be distinct primes of the same bit length and reveal only $x=\prod_{i=1}^k(p_i-1)$. Given $(x,\lambda,k)$, STP asks for any set of $k$ distinct $\lambda$-bit primes satisfying this product. The relation is efficiently verifiable, but its reconstruction complexity is not known. We establish three concrete results. First, for factored $x$ we derive the exact number of ordered exponent allocations and a bound showing that direct reconstruction is polynomial for fixed $k$ when $\Omega(x)=O(\log\lambda)$; this rules out that regime as a basis for a strong hardness claim. Second, we give exhaustive algorithms for reconstruction and collision analysis. Third, we exhaustively evaluate 28 parameter pairs, with $2\leq k\leq5$, up to $\lambda=16$ for pairs and 4,588,935 prime sets in the largest census. The data quantify non-injectivity through collision participation, maximum multiplicity, and conditional ambiguity in bits. These results isolate STP from general inverse-totient computation and motivate a Structured Totient Preimage Assumption for explicitly growing parameter families. Under such an assumption, STP becomes a candidate preimage-resistant relation whose implications for commitments, proofs of knowledge of multiplicative witnesses, and authentication can be stated precisely. The paper establishes the computational foundation and parameter constraints for those constructions; it does not claim a security reduction or post-quantum hardness.