Systems and methods described herein relate to techniques in which multiple parties each generate and exchange quantities that are based on a shared secret (e.g., powers of the shared secret) without exposing the shared secret. According to a protocol, two or more parties may exchange sets of elliptic curve points generated over polynomials that can be used, by each of the two or more parties, to determine a power of a shared secret. The protocol may be utilised as part of determining parameters for a smart contract that is broadcast to a blockchain network (e.g., Bitcoin). Based on the protocol, an additional party (e.g., a third party different from the two or more parties) may perform a computational task such as execution of the smart contract.

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.

The invention relates to a cryptographic method and variants thereof based on homomorphic encryption enabling the evaluation of univariate or multivariate real-valued functions on encrypted data, in order to allow carrying out homomorphic processing on encrypted data more broadly and efficiently.

A homomorphic encryption system receives a ciphertext output of an arithmetic evaluation function. The arithmetic evaluation function is performed on a ciphertext input homomorphically encrypted from a plaintext input using a set-system including sets having an intersection property. The ciphertext output is decrypted using a summation of two or more noise-canceling party identifiers of two or more authorized parties, wherein the intersection property of the sets cancels out errors generated during the decrypting operation for the two or more authorized parties.

The present inventive concept discloses a method of designing a one-way computational system in QAP-based homomorphic encryption applied to the n-qubit encode operations of a k-qubit action M for public-key and semi-public-key schemes respectively, n≥k, wherein the method comprises: preparing a tensor-product operator =I.sub.2.sub.n-k.Math.M=.sub.1.sub.2 and decomposing it into two parts, wherein is composed of elementary gates, and let =.sub.1.sup.† and .sub.2=; providing a correction operator, =Method of Designing of Multi-Party System in QAP-Based Homomorphic Encryption