A Deterministic Polynomial Public Key Algorithm over a Prime Galois Field GF(p)

Abstract

The ancient Vieta's formulas reveal the relationships between coefficients of an nth-degree polynomial and its roots. It is surprisingly found that there exists a hidden secret for a potential public key exchange: decoupling the product of all roots or constant term from summations of root products or coefficients of a polynomial to establish a keypair. The proposed deterministic polynomial public key algorithm or DPPK is built on the fact that a polynomial cannot be factorized without its constant term. DPPK allows the keypair generator to combine a base polynomial, eliminable during the decryption, with two solvable polynomials and creates two entangled polynomials. Two coefficient vectors of the entangled polynomials form a public key, and their constant terms, together with the two solvable polynomials, form the private key. By only publishing coefficients of polynomials without their constant terms, we greatly restrict polynomial factoring techniques for the private key extraction. We demonstrate that the time complexity, in terms of field operations, of the private key extraction from the known public key is a super-exponential difficulty O(p2) for classical attacks and an exponential difficulty O(p) for quantum attacks, respectively, in comparison with the low sub-exponential complexity for the PQC algorithms. The best-known deterministic complexity of the polynomial factoring problem for the secret key extraction from intercepted ciphertexts is O(npl/2) for classical attacks and O(pl/2) for quantum attacks, respectively, at the same complexity level as Grover's search algorithm. Performance comparisons with the PQC finalists for keypair generations, encryptions, and decryptions are presented.