In 1978, Yao studied the subset product problem and proved its NP-completeness. Later, Galbraith, Zobernig, and Li considered a prime-based modular variant and studied its average-case hardness. More recently, Li introduced the general problem of unknown-coefficient multivariate exponential system solving and studied its hardness systematically in an abstract setting. Li's framework implies a noisy modular variant of subset product as a special case. In this paper, we study this noisy subset product problem in a concrete setting, prove equivalence relations among several related variants, and use these results to construct a post-quantum non-alternating key exchange scheme, with a natural one-sided interpretation as a key encapsulation mechanism. Unlike other noise-based schemes, such as lattice-based schemes, our construction does not rely on an additional reconciliation plaintext. Instead, the receiver is able to recover the sender's ephemeral secret key itself. Moreover, we reduce indistinguishability security directly to the search version of the noisy subset product problem, without relying on a separate decisional assumption.