We define and study the Structured Totient Preimage (STP) problem as a restricted reconstruction relation with a direct cryptographic motivation. Let p1,…,pk be distinct primes of the same bit length and reveal only x=∏i=1k(pi−1)
Nearby in the stack
. Given
(x,λ,k)
, STP asks for any set of
k
distinct
λ
-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
Ω(x)=O(logλ)
; 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≤k≤5
, up to
λ=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.