On Lattices, Learning with Errors, Random Linear Codes, and Cryptography

Abstract: Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the `learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., non-quantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size $\tilde{O}(n^2)$ and encrypting a message increases its size by a factor of $\tilde{O}(n)$ (in previous cryptosystems these values are $\tilde{O}(n^4)$ and $\tilde{O}(n^2)$, respectively). In fact, under the assumption that all parties share a random bit string of length $\tilde{O}(n^2)$, the size of the public key can be reduced to $\tilde{O}(n)$.

• The paper shows a reduction from worst-case lattice problems (SVP/SIVP) to the Learning with Errors problem, bridging theoretical and practical cryptography.
• It presents a public-key cryptosystem with reduced key sizes and message expansion, based on the quantum hardness of lattice problems.
• The work reveals an equivalence between LWE and decoding random linear codes, highlighting key implications for cryptanalysis and post-quantum security.

An Essay on "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography"

The paper "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography" by Oded Regev explores the relationship between fundamental lattice problems, learning problems with errors, and cryptographic constructs. The paper presents key theoretical advances that draw significant connections between these domains, and it introduces practical cryptographic schemes based on these theoretical underpinnings.

Main Contributions

1. Reduction from Lattice Problems to Learning Problems: The core contribution of the paper is a reduction from worst-case lattice problems, such as the decision Shortest Vector Problem (SVP) and the Shortest Independent Vectors Problem (SIVP), to the Learning with Errors (LWE) problem. LWE is a natural extension of the Learning from Parity with Error (LPN) problem to higher moduli. This result implies that finding efficient algorithms for LWE under certain parameter choices would advance solutions for SVP and SIVP in the quantum setting.
2. Public-Key Cryptosystem Based on LWE: The paper proposes a public-key cryptosystem whose security is provably based on the hardness of LWE, and by reduction, on the worst-case quantum hardness of SVP and SIVP. This cryptosystem demonstrates enhanced efficiency compared to previous lattice-based schemes, exhibiting a reduction in both public key size and message expansion factor.
3. Implications for Random Linear Codes: The equivalence between the LWE problem and decoding random linear codes is highlighted as an important facet of understanding the difficulty of these problems. Specifically, the paper shows that solving LWE implies a solution for decoding in the context of random linear codes, providing evidence for the inherent hardness of such decoding tasks.

Strong Numerical Results and Claims

• The cryptosystem proposed by Regev achieves a public key size of $\tilde{O}(n^2)$ and an encryption expansion factor of $\tilde{O}(n)$, which is significantly better than the $\tilde{O}(n^4)$ and $\tilde{O}(n^2)$ respective factors achieved by prior lattice-based cryptosystems.
• For the specified parameters, such as $\alpha = 1/(\sqrt{n} \log^2 n)$ and $p = O(n^2)$, the probability of decision errors in the cryptosystem is shown to be negligible, ensuring both correctness and security.

Quantum and Classical Reductions

A remarkable aspect of the work is the quantum nature of the reduction. Specifically, the reduction from SVP and SIVP to LWE is quantum. While this raises questions about whether such quantum reductions could be made classical, proving such would significantly strengthen the foundational assumptions of the cryptosystem. Furthermore, it implies that breakthroughs in quantum computing could potentially disrupt the presumed hardness of these problems.

Implications and Future Directions

Regev's research reveals profound implications for cryptography, particularly in constructing efficient, secure cryptographic systems that rely on hard lattice problems. The equivalence drawn between LWE and problems in coding theory further opens avenues for cross-pollination of techniques and insights between these fields.

Practically, the instantiated cryptosystem provides a promising direction for developing post-quantum cryptographic schemes that resist attacks even in the advent of quantum computers.

Theoretically, this paper suggests several future directions:

• Investigating whether the quantum reduction from SVP and SIVP to LWE can be rendered classical.
• Exploring tighter bounds and reductions between other learning problems and lattice problems, potentially uncovering more efficient cryptographic primitives.
• Understanding the full implications of LWE's hardness on random linear codes, which may influence both cryptanalysis and the development of robust cryptographic protocols.

Conclusion

The paper "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography" lays a comprehensive foundation for the interplay between worst-case lattice problems and average-case LWE problems. Oded Regev's contributions extend from theoretical reductions to practical cryptographic applications, setting a significant milestone in the field of cryptographic research and offering a robust framework for future exploration in both quantum and classical settings.