Abstract: Disclosed are methods and systems to provide distributed computation within a Fully Homomorphic Encryption (FHE) system by using g-adic properties to separate a ciphertext into multiple ciphertexts for each Hensel digit level. A number t of computation units may individually perform addition and/or multiplication of each Hensel digit level on each of the computation units and then reconstruct the resulting value from the result ciphertext of each computation unit using p-adic and g-adic operations. Accordingly, computation burdens may be distributed to several computation units.

Abstract: Disclosed are methods and systems to use p-adic numbers to permit a RSA cryptosystem to send rational numbers or to add randomness to the RSA cryptosystem. An embodiment may convert at the source device a rational number to an integer as p-adic based Hensel code representation of the rational number at the source device and then recover the rational number at the destination device by reversing the Hensel code back to the original rational number. Another embodiment may use a g-adic inverse of a message value together with a random number to obtain a different rational number to encrypt for each different random number resulting in different ciphertexts representing the same message value while still recovering the original message value despite having a different ciphertexts for the same message value. The various embodiments further retain the multiplicative homomorphism of the RSA cryptosystem since the p-adic Hensel codes are also multiplicative homomorphic.

Abstract: Disclosed are methods and systems to provide public and private-key leveled Fully Homomorphic Encryption (FHE) systems using Hensel Codes and p-adic and g-adic properties for encryption and decryption that also provide for homomorphic arithmetic operations on encrypted ciphertexts. A source device may encrypt the ciphertext of a message using Hensel Codes, then deliver the ciphertext to either a destination device or an intermediary device. When the intermediary device receives the ciphertext from the source device, the intermediary device may homomorphically perform Hensel Code arithmetic computations with the ciphertext and at least one additional ciphertext and send the result ciphertext to the destination device. The destination device decrypts the ciphertext, giving the original message when no computations have been performed by the intermediary device, or the unencrypted result equivalent to the unencrypted computations performed on the ciphertexts by the intermediary device.

Abstract: Disclosed are methods and systems to provide distributed computation within a Fully Homomorphic Encryption (FHE) system by using g-adic properties to separate a ciphertext into multiple ciphertexts for each Hensel digit level. A number t of computation units may individually perform addition and/or multiplication of each Hensel digit level on each of the computation units and then reconstruct the resulting value from the result ciphertext of each computation unit using p-adic and g-adic operations. Accordingly, computation burdens may be distributed to several computation units.