SCHOOL

Part One

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Part Two

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The importance of lattice-based cryptography has been recently justified by the NIST post-quantum standardization process that identified two primary algorithms to be implemented for most use cases: CRYSTALS-KYBER (key-establishment) and CRYSTALS-Dilithium (digital signatures), both lattice-based. Other than for their apparent resistance to quantum attacks, lattice-based cryptographic primitives have other interesting features being supported by strong worst-case/average-case security guarantees and high asymptotic and concrete efficiency.

In this lecture, we give an introduction to the mathematical theory of lattices, describe the main tools and techniques used in lattice cryptography, and present an overview of the main post-quantum secure cryptographic primitives. (Part One)

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Introduction to the mathematical theory of lattices, description of the main tools and techniques used in lattice cryptography, and overview of the main post-quantum secure cryptographic primitives. (Part Two)

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Notion of isogeny between elliptic curves and application to cryptography. Isogeny graphs, their properties and their relation with isogeny based cryptography. (Part One)

Notion of isogeny between elliptic curves and application to cryptography. Isogeny graphs, their properties and their relation with isogeny based cryptography. (Part Two)

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Introduction to multivariate cryptography with some examples of multivariate schemes. Discussion of the Multivariate Quadratic Problem (MQP), which is the problem on which the security of multivariate cryptographic protocols relies. Discussion of the hardness of the MQP. To this extent, introducing Groebner bases and linear-algebra-based algorithms to compute them (often the most effective method to solve the MQP over finite fields). Definition of the solving degree, which controls the complexity of these algorithms.

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Discussion of the invariants which we currently use to estimate the complexity of Groebner basis algorithms. As the solving degree is in practice hard to compute, it is often estimated by using other invariants, such as the degree of regularity, the first fall degree, the last fall degree, and the Castelnuovo-Mumford regularity. Discussion on their relations and how we can bound or estimate them. Some examples of concrete situations in which we can obtain estimates using the invariants and methods introduced in the lectures, including the MinRank Problem, which is relevant in the cryptanalysis of several cryptographic schemes.

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Introduction to the area of code-based cryptography. Start with basic coding theory notions and explain the traditional approach, stemming from McEliece’s seminal 1978 paper. The talk will then cover the evolution of code-based encryption over the last 40 years, concluding with its role within NIST’s standardization protocol.

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Presentation of some topics for current and future research in code-based cryptography. In particular, the lecture will focus on the design and cryptanalysis of digital signatures, starting from the vast literature of unsuccessful attempts, and continuing with present lines of research.

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