Rate-1 Trapdoor Functions From The Diffie-Hellman Problem

Trapdoor functions (TDFs) are one of the fundamental building blocks in cryptography. Studying the underlying assumptions and the efficiency of the resulting instantiations is therefore of both theoretical and practical interest. In this work we improve the input-toimage rate of TDFs based on the Diffie-Hellman problem. Specifically, we present:(a) A rate-1 TDF from the computational Diffie-Hellman (CDH) assumption, improving the result of Garg, Gay, and Hajiabadi [EUROCRYPT 2019], which achieved linear-size outputs but with large constants. Our techniques combine non-binary alphabets and high-rate error-correcting codes over large fields.(b) A rate-1 deterministic public-key encryption satisfying blocksource security from the decisional Diffie-Hellman (DDH) assumption. While this question was recently settled by Dottling et al. [CRYPTO 2019], our scheme is conceptually simpler and concretely more efficient. We demonstrate this fact by implementing our construction.