The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications

Abstract Sequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards functions (PRF) recently, based on the power residue symbols, due to their efficiency multi-party setting. The security these PRFs is not known be reducible standard assumptions. In this work, we show that key-recovery attacks against PRF are equivalent solving a specific family multivariate quadratic (MQ) equation system over finite prime field. This new perspective sheds some light complexity PRF. We conduct algebraic cryptanalysis resulting MQ instance. currently techniques fall short sparse systems. Furthermore, build novel applications PRF, e.g., verifiable random function (verifiable) oblivious (programmable) PRFs.