Definition ∞ Ring arithmetic refers to mathematical operations performed within an algebraic structure known as a ring, which consists of a set equipped with two binary operations, typically addition and multiplication. These operations adhere to specific properties, such as associativity, distributivity, and the existence of additive and multiplicative identities. In cryptography, this form of arithmetic is fundamental for constructing various encryption schemes and zero-knowledge proofs. It provides the mathematical foundation for computations on integers modulo a certain number or polynomials.
Context ∞ Ring arithmetic is a core component in the construction of advanced cryptographic schemes, including fully homomorphic encryption (FHE) and lattice-based cryptography, which are critical for privacy-preserving computations on blockchains. Discussions often focus on optimizing the efficiency of these arithmetic operations to reduce the computational overhead in FHE systems. Future research aims to develop more efficient algorithms and hardware accelerators for ring arithmetic, making privacy-preserving digital asset transactions more practical and scalable.
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