Hardness of Structured Lattice Problems for Post-Quantum Cryptography

Ravinder Kaur

Andrea Lesavourey

Lattice-based cryptography is an excellent candidate for post-quantum cryptography, i.e. cryptosystems which are resistant to attacks run on quantum computers. For efficiency reason, most of the constructions explored nowadays are based on structured lattices, such as module lattices or ideal lattices. The security of most constructions can be related to the hardness of retrieving a short element in such lattices, and one does not know yet to what extent these additional structures weaken the cryptosystems. A related problem – which is an extension of a classical problem in computational number theory – called the Short Principal Ideal Problem (or SPIP), consists of finding a short generator of a principal ideal. Its assumed hardness has been used to build some cryptographic schemes. However it has been shown to be solvable in quantum polynomial time over cyclotomic fields, through an attack which uses the Log-unit lattice of the field considered. Later, practical results showed that multiquadratic fields were also weak to this strategy.\nThe main general question that we study in this thesis is To what extent can structured lattices be used to build a post-quantum cryptography?

Thor Tunge

In this thesis we will discuss hard computational problems in lattice theory and relate them to cryptographic constructions. Many of these problems enjoy average-case hardness which makes them attractive for cryptography. In addition, lattice based cryptography is a candidate for post-quantum cryptography, as there is no known quantum algorithm which breaks various hardness theorems.\n\nWe build a foundation in algebraic number theory to have the required background to discuss schemes based on discrete algebraic structures. These structures are free Z-modules which permits unique factorization in prime ideals. We relate this algebraic number theory to lattices in R^n so we can use the theory from algebra to our advantage.\n\nWe then define some standard hard computational lattice problems and show how many of these are related to each other. We prove that these problems are at least as hard as finding the shortest vector of a lattice, which we conjecture is computationally infeasible. We then prove a quantum reduction the learning with errors problem, a problem in machine learning . We also show that there is a similar reduction for a variant of this problem over more general rings.

G. Sunil, B. Girirajan, Manda Sai Anjani

Johannes A Buchmann, Richard Lindner, Markus Rückert +1 more

Jaskari, Mikko

The security of conventional public-key cryptography schemes is based on mathematical problems which are considered to be infeasible to solve by classical computers in a reasonable amount of time. However, the rise of quantum computers might break this security in the future. Post-quantum cryptography (PQC) aims to achieve secure schemes which are resistant against attacks assisted by quantum computers. Public-key cryptography refers to asymmetric cryptography in which encryption is done with a public key and decryption is done with a private key. Characteristic properties of public-key cryptography include the possibility to derive the private key from public information by solving some presumably hard problem. The hardness of the problem constitutes the basis for the system's security. In the post-quantum era, the security must be based on problems that are too hard to solve even for quantum computers. In this thesis, we will study lattice-based cryptography which is believed to serve as a good candidate for secure post-quantum public-key cryptography. Our main attention is in the problem of learning with errors, which is in the base of some lattice-based schemes and is linked to the hard problem of finding short vectors in a lattice.