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Analisys of Lattice-based Public Key Cryptosystems

A. R. Orlov, E. B. Makhovenko

Published Mar 1, 2011• 0 citations
OpenAlex

Abstract

Mathematical problems for lattice-based public key cryptosystem design are analyzed. A method of constructing theoretically strong against quantum attacks cryptosystem is proposed.

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Abstract Future assaults from quantum computers will be able to resist most contemporary public key cryptosystems. Post-quantum encryption has attracted much attention recently because of the mathematical tools it provides to guard communications and data against these types of attacks. Based on the challenging mathematical difficulties in high-dimensional lattice theory, lattice-based encryption—which boasts great efficiency, reasonable size, and robust security—may be a kind of post-quantum encryption. With any chance, this survey will address all the foundations of lattice-based cryptography. The writers start by summarizing the fundamental lattice problems in mathematics. They then provide the theoretical foundation for lattice-based encryption as well as the essential cryptanalytic methods.

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Abstract Future assaults from quantum computers will be able to resist most contemporary public key cryptosystems. Post-quantum encryption has attracted much attention recently because of the mathematical tools it provides to guard communications and data against these types of attacks. Based on the challenging mathematical difficulties in high-dimensional lattice theory, lattice-based encryption—which boasts great efficiency, reasonable size, and robust security—may be a kind of post-quantum encryption. With any chance, this survey will address all the foundations of lattice-based cryptography. The writers start by summarizing the fundamental lattice problems in mathematics. They then provide the theoretical foundation for lattice-based encryption as well as the essential cryptanalytic methods.
The standard public-key cryptosystems used today relies mathematical problems that require a lot of computing force to solve, so much that, with the right parameters, they are computationally unsolvable. But there are quantum algorithms that are able to solve these problems in much shorter time. These quantum algorithms have been known for many years, but have only been a problem in theory because of the lack of quantum computers. But with recent development in the building of quantum computers, the cryptographic world is looking for quantum-resistant replacements for today’s standard public-key cryptosystems. Public-key cryptosystems based on lattices are possible replacements. This thesis presents several possible candidates for new standard public-key cryptosystems, mainly NTRU and ring-LWE-based systems. The latticebased cryptosystems are shown to be very fast and have strong, provable security against quantum computers, but are a lot more complicated than RSA and Diffie-Hellman.
Lattice-Based Cryptosystems and Quantum CryptanalysisQuantum computers are probably coming, though we don't know when-and when they arrive, they will, most likely, be able to break our standard public-key cryptography algorithms.In anticipation of this possibility, cryptographers have been working on quantum-resistant public-key algorithms.The National Institute for Standards and Technology (NIST) has been hosting a competition a since 2017, and there already are several proposed standards.b Most of these are based on lattice problems.The mathematics of lattice cryptography revolve around combining sets of vectors-that's the lattice-in a multi-dimensional space.These lattices are filled with multi-dimensional periodicities.The hard problem c that's used in cryptography is to find the shortest periodicity in a large, random-looking lattice.This can be turned into d a public-key cryptosystem in a variety of different ways.Research has been ongoing since 1996, and there has been some really great work since then-including many practical public-key algorithms.On April 10, Yilei Chen from Tsinghua University in Beijing posted a paper describing a new quantum attack on that shortest-path lattice problem.It's a very dense mathematical paper-63 pages long-and my guess is that only a few cryptographers are able to understand all of its details.(I was not one of them.)But the conclusion was pretty devastating, breaking essentially all of the lattice-based fully homomorphic encryption schemes and coming significantly closer to attacks against the recently proposed (and NIST-approved) lattice key-exchange and signature schemes.However, there was a small but critical mistake in the paper, on the bottom of page 37.It was independently discovered by Hongxun Wu from Berkeley and Thomas Vidick from the Weizmann Institute in Israel eight days later.The attack algorithm in its current form doesn't work.This was discussed last week at the Cryptographers' Panel e at the RSA Conference.Adi Shamir, the "S" in RSA and a 2002 recipient of ACM's A.M. Turing award, described the result as psychologically significant because it shows that there is still a lot to be discovered about quantum
Abstract The current widely used public‐key cryptosystems are vulnerable to quantum attacks. To prepare for cybersecurity in the quantum era, some projects have been launched to call for post‐quantum alternatives. Due to solid security and desirable performance, lattice‐based cryptosystems are viewed as promising candidates in the upcoming standardisation of post‐quantum cryptography. This study surveys the lattice‐based cryptosystems in the post‐quantum standardisation processes including the NIST Post‐Quantum Cryptography Standardisation and the Chinese Cryptographic Algorithm Design Competition, from both design and security aspects. We present generic design paradigms of lattice‐based schemes and describe several representative proposals and recent progress. We also recap some main cryptanalytic results and methods for estimating the concrete security of lattice‐based schemes.
The security of classical public-key cryptosystems, which operate based on the computational intractability of integer factorization and discrete logarithm problems, is profoundly threatened by the rapid development of quantum computing. Post-Quantum Cryptography (PQC) has been proposed to improve security against quantum attacks. Lattice-based cryptography is widely regarded as one of the most promising PQC method because of its proven security, efficient implementation, and functional versatility, and it has achieved significant progress in both algorithmic design and engineering implementation in recent years. This paper presents a systematic survey of the status of research on lattice-based cryptographic algorithms. First, the fundamental theory of lattices and their core hard problems are discussed. Next, the technical characteristics and performance of encryption and key encapsulation mechanisms, digital signature schemes, and key exchange protocols constructed based on lattice hard problems are reviewed and compared. Subsequently, the principles and optimization paths of two key technologies for lattice cryptography implementation, namely polynomial multiplication using the fast Number Theoretic Transform (NTT) and pre-image sampling algorithms, are discussed. Finally, considering the PQC standardization process of the National Institute of Standards and Technology (NIST), an outlook on future research directions for lattice-based cryptography is presented. The research directions include the development of more "succinct" fundamental hard problems, strengthening security proofs, improving algorithm efficiency, and countering novel attacks. This review serves as a reference for promoting the practical deployment and further research of lattice-based cryptography.