RingLWEbased

Ring Learning With Errors, often abbreviated as RingLWE, is a mathematical problem that forms the foundation for various cryptographic schemes, particularly those designed for post-quantum cryptography. It is a variant of the Learning With Errors (LWE) problem, which is believed to be computationally hard for classical computers to solve. RingLWE adapts the LWE problem to work within polynomial rings, which offers several advantages, including efficiency and suitability for lattice-based cryptography.

The RingLWE problem can be described as follows: Given a polynomial ring R and a secret polynomial s, a public polynomial A is generated by multiplying a uniformly random polynomial r with A, and then adding a small "error" polynomial e. Specifically, the public information consists of pairs (A, b), where b = A * s + e. The task is to recover the secret polynomial s, given these pairs. The "error" term e is crucial, as it introduces noise that makes direct algebraic solution difficult, but is small enough for reconstruction algorithms under certain conditions.

The hardness of RingLWE stems from the difficulty of distinguishing between "noisy" linear equations and "clean" ones in polynomial rings. This difficulty is believed to be resistant to attacks from quantum computers, which is why RingLWE is a popular choice for designing quantum-resistant encryption and digital signature schemes. Cryptographic primitives based on RingLWE, such as key-encapsulation mechanisms and signature schemes, have been proposed and analyzed extensively, with some being standardized by organizations like NIST. The efficiency gained by working in polynomial rings over the standard LWE problem makes these schemes practical for real-world deployment.