Methods and systems for encryption and homomorphic encryption systems using geometric algebra and hensel codes

Abstract

Disclosed are methods and systems to encrypt/decrypt a data message using Geometric Algebra and Hensel encoding (i.e., finite p-adic arithmetic). The security key(s), message data, and ciphertext are all represented as Geometric Algebra multivectors where a sum of the coefficients of an individual multivector is equal to the numeric value of the corresponding message or security key. Various Geometric Algebra operations with the message and security key multivectors act to encrypt/decrypt the message data. Each coefficient of the security key and message multivectors is further Hensel encoded to provide additional confusion/diffusion for the encrypted values. The Geometric Algebra operations permit homomorphic operations for adding, subtracting, multiplication and division of ciphertext multivectors such that the resulting ciphertext, when decrypted, is equal to corresponding mathematical operations using the unencrypted values. The additional Hensel encoding of the coefficients of the multivectors does not impede the homomorphic aspects of the Geometric Algebra encryption operations. Operations for security key updates and exchanges are also provided.