Quantum Resistance:

• Traditional cryptographic systems like RSA and ECC rely on the difficulty of factoring large numbers and solving discrete logarithms, respectively. Quantum algorithms, specifically Shor's algorithm, can solve these problems efficiently, rendering traditional encryption methods vulnerable.

• Lattice-based cryptography, however, is based on problems that remain hard for quantum computers to solve. Current quantum algorithms, including Grover's algorithm, do not provide a significant advantage in breaking lattice-based schemes.

Mathematical Foundations:

• Shortest Vector Problem (SVP): This problem involves finding the shortest non-zero vector in a lattice, a grid-like structure in a high-dimensional space. The difficulty of SVP grows exponentially with the dimension of the lattice, making it extremely hard for both classical and quantum computers to solve.

• Closest Vector Problem (CVP): Similar to SVP, CVP involves finding the closest vector in a lattice to a given point. This problem is also computationally intensive and resistant to quantum attacks.

NIST Endorsement:

• The National Institute of Standards and Technology (NIST) is leading the charge in developing post-quantum cryptographic standards. Their endorsement of lattice-based schemes like CRYSTALS-Kyber (for encryption) and CRYSTALS-Dilithium (for digital signatures) highlights the practical and theoretical strengths of these methods.

• These schemes have undergone rigorous evaluation and are considered to provide a high level of security while maintaining efficiency, making them suitable for widespread adoption.