Candidate weak pseudorandom functions in AC0 · MOD2

Pseudorandom functions (PRFs) play a fundamental role in symmetric-key cryptography. However, they are inherently complex and cannot be implemented in the class AC⁰(MOD₂). Weak pseudorandom functions (weak PRFs) do not suffer from this complexity limitation, yet they suffice for many cryptographic applications. We study the minimal complexity requirements for constructing weak PRFs. To this end • We conjecture that the function family F_A (x) = g (Ax), where A is a random square GF(2) matrix and g is a carefully chosen function of constant depth, is a weak PRF. In support of our conjecture, we show that functions in this family are inapproximable by GF(2) polynomials of low degree and do not correlate with any fixed Boolean function family of subexponential size. • We study the class AC⁰ · MOD₂ that captures the complexity of our construction. We conjecture that all functions in this class have a Fourier coefficient of magnitude exp(-poly log n) and prove this conjecture in the case when the MOD₂ function is typical. • We investigate the relation between the hardness of learning noisy parities and the existence of weak PRFs in AC⁰ · MOD₂. We argue that such a complexity-driven approach can play a role in bridging the gap between the theory and practice of cryptography.