Partial fraction decomposition is a fundamental technique in mathematics that expresses a rational function as a sum of simpler fractions. Although rational functions have appeared in several cryptographic constructions, their rich algebraic structure has not been systematically explored as a direct foundation for building cryptographic primitives. In this work, we identify and exploit two key properties of partial fraction decomposition: (1) the decomposition property itself, which enables efficient set membership testing, and (2) a novel linear independence property derived from the non-singularity of Cauchy matrices, which enables threshold cryptography. We present two main applications. First, we construct a key-value commitment scheme in which the dictionary is represented as a linear combination of partial fractions. Our scheme achieves constant-size commitments (a single group element) and proofs, supports homomorphic updates enabling stateless operation, and provides efficient membership and non-membership proofs through simple pairing equations. We also introduce credential-based key-value commitments, in which keys are registered using Boneh-Boyen signatures, enabling applications in permissioned settings. Second, we construct a dynamic threshold encryption scheme leveraging the linear independence of partial fraction products. Our scheme achieves constant-size ciphertexts, supports public preprocessing of public keys into a succinct encryption key, enables dynamic selection of the decryption threshold, and provides robustness through share verification without random oracles. In particular, we obtain the shortest known CPA-secure ciphertext, consisting of three group elements, together with a logarithmic-size preprocessed encryption key. We prove security in the standard model under new q-type assumptions and establish their generic hardness in the generic bilinear group model. Our work demonstrates that working directly with the algebraic structure of rational fractions, rather than converting them into polynomial representations, yields elegant and efficient constructions with concrete advantages over prior work.