CRYSTALS-Kyber is a key encapsulation mechanism (KEM) that is designed to be resistant against quantum computer attacks. It is based on the learning with errors (LWE) problem, which is a well-studied mathematical problem in lattice-based cryptography. Let me dive into the details of how CRYSTALS-Kyber works under the hood.
1. Mathematical Foundation:
- CRYSTALS-Kyber is built upon the hardness of the LWE problem, which involves finding a secret vector given a matrix of "noisy" linear equations.
- The security of Kyber relies on the presumed difficulty of solving the LWE problem, even with the presence of a quantum computer.
2. Key Generation:
- Kyber operates over a polynomial ring Rq = Zq[X] / (X^n + 1), where n is a power of 2 and q is a prime modulus.
- The secret key is a vector of polynomials s = (s1, s2) chosen from a centered binomial distribution over Rq.
- The public key is computed as (t, ρ), where t = As + e, A is a matrix of randomly chosen polynomials from Rq, and e is an error vector sampled from a discrete Gaussian distribution.
- The matrix A is generated using an extendable output function (XOF) applied to a seed ρ, which allows for compact public key representation.
3. Encapsulation (Encryption):
- To encapsulate a shared secret, the sender generates a random vector r and an error vector e1, both sampled from a centered binomial distribution.
- The sender computes u = AT r + e1 and v = tT r + e2 + Decode(m), where e2 is another error vector and m is the message to be encrypted.
- The ciphertext is (u, v), and the shared secret is derived from the high-order bits of v.
4. Decapsulation (Decryption):
- Upon receiving the ciphertext (u, v), the recipient computes w = vT - sT u.
- The shared secret is obtained by decoding w and extracting the high-order bits.
- The decoding process involves a rounding function that maps elements of Rq to the message space.
5. Parameter Selection:
- CRYSTALS-Kyber defines multiple parameter sets (Kyber512, Kyber768, Kyber1024) that offer different levels of security and performance trade-offs.
- The choice of parameters, such as the dimension n, modulus q, and the error distribution, affects the security level and efficiency of the scheme.
6. Security Analysis:
- The security of CRYSTALS-Kyber is based on the hardness of the LWE problem over module lattices.
- Rigorous security proofs have been provided to demonstrate the reduction of Kyber's security to the LWE problem.
- Kyber has undergone extensive cryptanalysis and has been shown to resist various known attacks, including those using quantum algorithms.
7. Implementation Aspects:
- CRYSTALS-Kyber is designed to be efficient and portable across different platforms and architectures.
- It employs various optimization techniques, such as the use of the Number Theoretic Transform (NTT) for polynomial multiplication and the use of centered binomial distributions for efficient sampling.
- Side-channel resistance is also considered in the implementation, with techniques like constant-time operations and masking to prevent information leakage.
CRYSTALS-Kyber has been selected as a finalist in the NIST Post-Quantum Cryptography Standardization process and has undergone thorough scrutiny by the cryptographic community. Its strong security guarantees, efficiency, and versatility make it a promising candidate for post-quantum key encapsulation.
It's worth noting that the exact details and parameters of CRYSTALS-Kyber may evolve based on ongoing research and analysis. The final specification will be determined through the standardization process and collaboration within the cryptographic community.
