Pseudorandom Function from Learning Burnside Problem

We present three progressively refined pseudorandom function (PRF) constructions based on the learning Burnside homomorphisms with noise (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>n</mi></msub></semantics></math></inline-formula>-LHN) assumption. A key challenge in this approach is error management, which we address by extracting errors from the secret key. Our first design, a direct pseudorandom generator (PRG), leverages the lower entropy of the error set (<i>E</i>) compared to the Burnside group (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>r</mi></msub></semantics></math></inline-formula>). The second, a parameterized PRG, derives its function description from public parameters and the secret key, aligning with the relaxed PRG requirements in the Goldreich–Goldwasser–Micali (GGM) PRF construction. The final indexed PRG introduces public parameters and an index to refine efficiency. To optimize computations in Burnside groups, we enhance concatenation operations and homomorphisms from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>n</mi></msub></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>r</mi></msub></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≫</mo><mi>r</mi></mrow></semantics></math></inline-formula>. Additionally, we explore algorithmic improvements and parallel computation strategies to improve efficiency.