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Journal of Mathematical Cryptology

A Ring-LWE-based digital signature inspired by Lindner–Peikert scheme

Javad Sharafi, Hassan Daghigh

Published Jan 1, 2022• 1 citation
Publisher

Abstract

Abstract In this article, we give a digital signature by using Lindner–Peikert cryptosystem. The security of this digital signature is based on the assumptions about hardness of Ring-LWE and Ring-SIS problems, along with providing public key and signature of compact (1–1.5 kilobytes) size. We prove the security of our signature scheme in the Quantum Random Oracle Model. Our cryptanalysis has been done based on methods of Aggarwal et al. and Chen et al.

DOI
10.1515/jmc-2021-0013

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Abstract In this article, we give a digital signature by using Lindner–Peikert cryptosystem. The security of this digital signature is based on the assumptions about hardness of Ring-LWE and Ring-SIS problems, along with providing public key and signature of compact (1–1.5 kilobytes) size. We prove the security of our signature scheme in the Quantum Random Oracle Model. Our cryptanalysis has been done based on methods of Aggarwal et al. and Chen et al.
We propose a variant version of ring learning with errors (R-LWE) assumption. Under the modified slightly assumption which is reducible to the worst-case problems on ideal lattice, we present a construction of digital signatures. So the scheme is provably secure based on the hardness of lattice problems (such as approximating the length of the shortest vector within a fixed poly(n) factor). Compared with some existing typical lattice-based signature schemes, the construction enjoys many advantages, including simple and efficient key generation algorithm, signing algorithm and verification algorithm to improve the efficiency of the scheme as well as to reduce the overhead. The sizes of the secret and public keys and the signature are almost linear O(n log n) (up to poly-logarithmic factors) in the dimension n of the lattice.
Abstract. We provide an alternative method for constructing lattice-based digital signatures which does not use the “hash-and-sign ” methodology of Gentry, Peikert, and Vaikuntanathan (STOC 2008). Our resulting signature scheme is secure, in the random oracle model, based on the worst-case hardness of the Õ(n1.5)-SIVP problem in general lattices. The secret key, public key, and the signature size of our scheme are smaller than in all previous instantiations of the hash-and-sign signature, and our signing algorithm is also quite simple, requiring just a few matrix-vector multiplications and rejection samplings. We then also show that by slightly changing the parameters, one can get even more efficient signatures that are based on the hardness of the Learning With Errors problem. Our construction naturally transfers to the ring setting, where the size of the public and secret keys can be significantly shrunk, which results in the most practical to-date provably secure signature scheme based on lattices. 1
In the present scenario where thousands and millions of digital documents are transferred every day, the security of data has become a topic of global concern. Along with other cryptographic primitives, the proxy signature scheme is also widely used in various applications. The main part of our paper is a lattice-based proxy signature scheme. A Proxy Signature scheme is an arrangement that allows the transfer of singing rights from one entity or person to another on behalf of the original signer. Our scheme is based on the Ring-Learning with Error(R-LWE) problem which provides strong security with the worst-case to average-case reduction problem. Key construction and operations in our scheme are based on a ring of polynomials due to which both public and private keys are reduced and the size of the proxy signature also shrunk, while the security of our proposed scheme is based on the Short Integer Solution (SIS) problem.
Among post-quantum alternatives, latticebased cryptography is the most promising one, due to its simple operations, reduction from aver-age-case to worstcase hardness, and supporting of rich functionalities. Ring signature enables a user to sign anonymously on behalf of an adaptively chosen group, and has multiple applications in anonymous e-voting, anonymous authentication, whistle blowing etc. However, most lattice-based ring signature schemes were constructed in the random oracle model from lattice basis delegation and they suffer large verification key sizes as a common disadvantage. This work proposes an efficient ring signature scheme from lattice basis delegation without random oracle based on the extended split-SIS problem, whose security is approximately as hard as the worst-case SIVP problem. Our scheme is proved to be anonymous and existentially unforgeable under latticebased assumptions. Finally, the verification key size is significantly reduced to a small constant, instead of increasing linearly with the number of ring members.
Quantum computers are moving closer to reality, and with that comes a serious concern for today’s public-key cryptography. Many of the encryption and digital signature systems we rely on now, such as RSA and ECC, may no longer be secure in the future. Because of this, researchers are actively looking for new methods that can protect digital communication in the post-quantum era. One of the most promising directions is lattice-based cryptography. This paper gives a simple and focused review of lattice-based signature schemes. It brings together the main ideas from the existing literature and explains how these schemes are built, why they are considered secure, and what makes them practical. The review mainly discusses schemes based on hard lattice problems such as SIS and LWE. To make the comparison clearer, a table is also included that compares different signature schemes in terms of key size, signature size, security level, and computational efficiency. Overall, this study shows why lattice-based signature schemes are becoming an important part of modern cryptographic research and why they may play a major role in securing digital systems against future quantum threats.
On the basis of the signatures scheme without trapdoors from lattice, which was proposed by Vadim Lyubashevsky in 2012, we present a new ring signature scheme from lattice. The proposed ring signature scheme is an extension of the signatures scheme without trapdoors. We prove that our scheme is strongly unforgeable against adaptive chosen message in the random oracle model, and prove that the security of our scheme can be reduced to the hardness of the small integer solution (SIS) problem by rejection samplings. Compared with the existing lattice-based ring signature schemes, our new scheme is more efficient and with shorter signature length.