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Learning With Errors

Learning With Errors (LWE) is a foundational problem in lattice-based cryptography and forms the basis for many cryptographic schemes due to its hardness. The problem can be described as follows:

References:

The LWE Problem

LWE involves solving a system of noisy linear equations. More formally, the LWE problem can be defined as follows:

  1. Secret Vector: There is a secret vector \(\mathbf{s} \in \mathbb{Z}_q^n\), where \(\mathbb{Z}_q\) denotes the integers modulo \(q\).

  2. Public Information: There is a matrix \(\mathbf{A} \in \mathbb{Z}_q^{m \times n}\) and a noise vector \(\mathbf{e} \in \mathbb{Z}_q^m\). The entries of \(\mathbf{A}\) are chosen uniformly at random from \(\mathbb{Z}_q\), and the entries of \(\mathbf{e}\) are typically chosen from some error distribution (often a discrete Gaussian distribution or a uniform distribution over a small range).

Note

The LWE problem provides a set of pairs \((\mathbf{A}, \mathbf{b})\), where \(\mathbf{b} = \mathbf{A} \cdot \mathbf{s} + \mathbf{e} \pmod q\). The goal is to recover the secret vector \(\mathbf{s}\) given \(\mathbf{A}\) and \(\mathbf{b}\).

Computational vs. Decisional LWE

  • Computational LWE: Given \((\mathbf{A}, \mathbf{b})\), find the secret vector \(\mathbf{s}\).
  • Decisional LWE: Distinguish between pairs \((\mathbf{A}, \mathbf{b})\) where \(\mathbf{b} = \mathbf{A} \cdot \mathbf{s} + \mathbf{e}\) and pairs \((\mathbf{A}, \mathbf{b})\) where \(\mathbf{b}\) is uniformly random.

Applications of LWE in Cryptography

LWE’s hardness is used to construct various cryptographic primitives:

  1. Public-Key Encryption: LWE can be used to design secure public-key encryption schemes. An example is the Regev encryption scheme, which is based on the LWE problem.
  2. Homomorphic Encryption: LWE allows for the construction of fully homomorphic encryption schemes, enabling computations on encrypted data.
  3. Digital Signatures: LWE-based digital signature schemes provide secure and efficient signature generation and verification.
  4. Key Exchange Protocols: LWE can be used to design secure key exchange protocols resistant to quantum attacks.
  5. Identity-Based Encryption: LWE supports the construction of identity-based encryption schemes, where the public key can be derived from a user’s identity (e.g., email address).

Hardness and Security

The security of LWE-based schemes relies on the assumed hardness of the LWE problem. This hardness can be reduced to worst-case hardness assumptions about lattice problems:

  1. Worst-Case to Average-Case Reduction: Solving LWE in the average case is as hard as solving certain lattice problems (like the Shortest Vector Problem) in the worst case.
  2. Quantum and Classical Hardness: LWE is believed to be hard both classically and quantumly, making it a strong candidate for post-quantum cryptography.

Parameters and Trade-offs

  • Modulus \(q\): Typically a large prime number. The choice of \(q\) affects the security and efficiency of the scheme.
  • Dimension \(n\): The dimension of the secret vector. Larger \(n\) increases security but also computational complexity.
  • Number of Samples \(m\): The number of LWE samples. There is a trade-off between the number of samples and the hardness of the problem.