Abstract: Disclosed are methods and systems to provide homomorphic compatible, p-adic arithmetic based encoding and decoding of rational numbers to integers and back to rational numbers for use with existing Fully Homomorphic Encryption (FHE) systems. Embodiments support both Approximate Greatest Common Devisor (AGCD) systems such as those with an Integer—Dijk, Gentry, Halevi, and Vaikuntanathan (IDGHV) scheme, and Ring Learning With Error (RLWE) systems such as a Fan and Vercauteren (FV) scheme modified for encrypting integers (ModFV). Encoded integers are provided to an FHE system on a source device that may optionally deliver the encrypted ciphertext to an intermediary device for performance of homomorphic algebra operations, and, the resultant and/or original ciphertext is delivered to a destination device for decryption of the ciphertext, and decoding of the decrypted integer back to a rational number.

Abstract: Disclosed are methods and system for providing secure arithmetic equality and comparison using probabilistic rounding in the domain of Farey rationals with quadratic residues for a Multi-Party Computation (MPC) system. Embodiments securely provide for evaluation of encoded rational numbers for arithmetic equality equal to zero or not, probabilistic modulo of two rational numbers, probabilistic modulo of a rational number by a public value of two, a test for whether a rational number is greater-than-zero for quadratic residues and a general greater-than-zero test for a rational number which are quadratic non-residues.

Abstract: Disclosed are methods and systems to provide a rounding protocol to permit extended use of homomorphic multiplication and division for a Multi-Party Computation (MPC) system, such as Mercury. The various embodiments operate on fixed-point data using the rounding protocol to provide an approximation of the precision of the fixed-point encoded variable operands of the multiplication or division operation. The various embodiments further use a unique uniform random bit protocol to provide a random integer for the rounding protocol.

Abstract: Disclosed are methods and systems to provide homomorphic compatible encryption of rational numbers using a modified Brakerski, Fan, and Vercauteren (BFV) homomorphic encryption scheme where classical polynomial rings in BFV homomorphic encryption are replaced by Laurent polynomial rings. This allows for encryption/decryption of rational numbers without the need to encode and decode the numbers before encryption and after decryption, respectively. Encoded rational numbers are provided to the modified BFV system on a source device that may optionally deliver the encrypted ciphertext to an intermediary device for performance of homomorphic algebra operations, and, the resultant or original ciphertext is delivered to a destination device for decryption of the ciphertext into a rational number.

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 encoding and decoding using of p-adic arithmetic, and inverse p-adic arithmetic, in the domain of Farey rationals that is induced by a ring isomorphism, such that said encoded integers preserve inverses, and additive and multiplicative homomorphic properties. This encoding permits MPC systems to perform arithmetic more efficiently and accurately, particularly for division.

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 homomorphic compatible, p-adic arithmetic based encoding and decoding of rational numbers to integers and back to rational numbers for use with existing Fully Homomorphic Encryption (FHE) systems. Embodiments support both Approximate Greatest Common Devisor (AGCD) systems such as those with an Integer—Dijk, Gentry, Halevi, and Vaikuntanathan (IDGHV) scheme, and Ring Learning With Error (RLWE) systems such as a Fan and Vercauteren (FV) scheme modified for encrypting integers (ModFV). Encoded integers are provided to an FHE system on a source device that may optionally deliver the encrypted ciphertext to an intermediary device for performance of homomorphic algebra operations, and, the resultant and/or original ciphertext is delivered to a destination device for decryption of the ciphertext, and decoding of the decrypted integer back to a rational number.

Abstract: Disclosed are methods and systems to encrypt data with SomeWhat Homomorphic Encryption (SWHE) properties for submission to a distributed ledger/blockchain that allows further open operations retained in the distributed ledger/blockchain on the encrypted data that will be properly reflected when the encrypted result is decrypted by the data owner. The somewhat homomorphic properties include addition and scalar division. Also disclosed is an ability to update a secret key applied for a ciphertext such that a single piece of data may be provided on the distributed ledger/blockchain by a data owner to a new data owner without also exposing other data encrypted with the original secret key of the original data owner.

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.