An academic and engineering deep-dive into the algebraic ring structure, error distributions, and hardness reductions that secure ML-KEM against quantum and classical adversaries.
The security of ML-KEM rests on the hardness of solving the Module Learning with Errors problem over the cyclotomic ring R_q = Z_q[X]/(X^256 + 1). The specific prime modulus q = 3329 was chosen because 3329 = 1 mod 512, which ensures that X^256 + 1 splits completely into 128 linear factors over Z_q, enabling high-speed Number Theoretic Transforms.
Secrets and error polynomials are sampled from a centered binomial distribution CBD_eta, where eta in {2, 3}. CBD_eta produces small coefficients that mimic discrete Gaussians with significantly lower computational complexity and zero risk of timing leakage during rejection sampling.
We review the formal reduction from worst-case lattice problems—such as the Shortest Independent Vectors Problem (SIVP) and GapSVP—to average-case M-LWE. Against the best known quantum sieve algorithms (e.g., BKZ 2.0 with block size beta >= 400), ML-KEM-768 provides an insurmountable security margin exceeding 128 quantum bits.