Abstract: Methods and apparatus for providing secure user identification or digital signatures based on evaluation of constrained polynomials. In an exemplary user identification technique, a prover sends a verifier a commitment signal representative of a first polynomial satisfying a first set of constraints. The verifier sends the prover a challenge signal representative of a second polynomial satisfying a second set of constraints. The prover generates a response signal as a function of (i) information used to generate the commitment signal, (ii) a challenge signal, and (iii) a private key polynomial of the prover, such that the response signal is representative of a third polynomial satisfying a third set of constraints. The verifier receives the response signal from the prover, and authenticates the identity of the prover by evaluating a function of information contained in at least a subset of (i) the commitment signal, (ii) the challenge signal, (iii) the response signal and (iv) a public key of the prover.