Learning With Errors

Learning With Errors (LWE) is a very important problem in cryptography and theoretical computer science, proposed by Oded Regev in 2005. The LWE problem can be described as: given a system of linear equations, each of which contains some random noise (i.e., errors), the goal is to recover the original unknown vector.

The LWE problem is considered to be equivalent to some difficult lattice problems in some cases, such as the shortest vector problem (SVP) and the shortest vector problem with errors (SIVP). Due to the difficulty of the LWE problem, it is used as an assumption in building cryptographic systems, especially public key cryptographic systems, such as LWE-based encryption schemes.

A key feature of the LWE problem is that it provides a way to base cryptographic security on computational complexity, which makes LWE-based cryptographic systems theoretically highly secure. In addition, the LWE problem has also attracted attention in the field of quantum computing. Studies have shown that there are effective quantum algorithms that can solve the LWE problem, which provides new research directions and challenges for cryptography.