Universal One-way Functions Research Articles

Semigroups and one-way functions

We study the complexity classes 𝖯 and 𝖭𝖯 through a semigroup 𝖿𝖯 ("polynomial-time functions"), consisting of all polynomially balanced polynomial-time computable partial functions. The semigroup 𝖿𝖯 is non-regular if and only if 𝖯 ≠ 𝖭𝖯. The one-way functions considered here are based on worst-case complexity (they are not cryptographic); they are exactly the non-regular elements of 𝖿𝖯. We prove various properties of 𝖿𝖯, e.g. that it is finitely generated. We define reductions with respect to which certain universal one-way functions are 𝖿𝖯-complete.

A Complete Public-Key Cryptosystem

We present a cryptosystem which is complete for the class of probabilistic public-key cryp- tosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal, this class contains probabilistic encryption of Goldwasser-Micali as well as Ajtai-Dwork and NTRU cryptosystems. The latter two are known to make errors with some small positive probability. To our best knowledge, no complete public-key cryptosystem has been known before, whether encryp- tion/decryption errors are allowed or not. At the same time, other complete primitives such as Levin's universal one-way function (1) have been known for a long time.