A framework for cryptographic problems from linear algebra

Journal of mathematical cryptology

article

Carl Bootland¹  Wouter Castryck²  Alan Szepieniec¹  Frederik Vercauteren¹ 

[1] ESAT/COSIC;Department of Mathematics

关键词: LWE;    SIS;    NTRU;    quotient ring;    post-quantum;   

DOI  :  10.1515/jmc-2019-0032

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We introduce a general framework encompassing the main hard problems emerging in lattice-based cryptography, which naturally includes the recently proposed Mersenne prime cryptosystem, but also problems coming from code-based cryptography. The framework allows to easily instantiate new hard problems and to automatically construct plausibly post-quantum secure primitives from them. As a first basic application, we introduce two new hard problems and the corresponding encryption schemes. Concretely, we study generalisations of hard problems such as SIS, LWE and NTRU to free modules over quotients of ℤ[ X ] by ideals of the form ( f , g ), where f is a monic polynomial and g ∈ ℤ[ X ] is a ciphertext modulus coprime to f . For trivial modules (i.e. of rank one), the case f = X n + 1 and g = q ∈ ℤ >1 corresponds to ring-LWE, ring-SIS and NTRU, while the choices f = X n – 1 and g = X – 2 essentially cover the recently proposed Mersenne prime cryptosystems. At the other extreme, when considering modules of large rank and letting deg( f ) = 1, one recovers the framework of LWE and SIS.

【 授权许可】

CC BY|CC BY-NC-ND