Lecture

Homomorphic Encryption: BGV Cryptosystem

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In course

CS-523: Advanced topics on privacy enhancing technologies

This advanced course will provide students with the knowledge to tackle the design of privacy-preserving ICT systems. Students will learn about existing technologies to prect privacy, and how to evaluate the protection they provide. ...

Description

This lecture covers the BGV cryptosystem, focusing on security, correctness, encrypted addition, noise growth, and encrypted multiplication. It explains the key generation, encryption, and decryption processes, emphasizing noise management and the challenges of homomorphic operations. The lecture also discusses practical considerations, such as generalized operations, relinearization, and post-quantum resistance. It explores the transition from somewhat to fully homomorphic encryption using bootstrapping and the Lattigo library. Practical implementations of lattice-based cryptosystems and noise management techniques are highlighted, along with references to seminal works in the field.

Instructors (2)

Carmela González Troncoso

SPRING Lab Our research focuses on building secure and privacy-preserving systems. We aim to develop tools and methodologies to help engineers embedding strong security and privacy guarantees in their designs, as well as techniques that help them to systematically quantifying the information that an adversary can infer from data she may have access to.More information about our activities here: https://spring.epfl.ch/ Teaching & PhD PhD Students Saiid El Hajj Chehade, Eric Jollès, Christian Knabenhans, Boy

Alessandro Chiesa

Alessandro Chiesa is a faculty member in computer science. He conducts research in complexity theory, cryptography, and security, with a focus on the theoretical foundations and practical implementations of cryptographic proofs that are short and easy to verify. He is a co-author of several zkSNARK libraries, and is a co-inventor of the Zerocash protocol. He has co-founded Zcash and StarkWare Industries. He is a recipient of a Sloan Research Fellowship (2021), an Okawa Foundation Research Grant (2020), and Google Faculty Research Awards (2018 and 2017). He was included in MIT Technology Review's "35 Innovators Under 35" list in 2018.

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Related concepts (25)

Post-quantum cryptography

In cryptography, post-quantum cryptography (PQC) (sometimes referred to as quantum-proof, quantum-safe or quantum-resistant) refers to cryptographic algorithms (usually public-key algorithms) that are thought to be secure against a cryptanalytic attack by a quantum computer. The problem with currently popular algorithms is that their security relies on one of three hard mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem.

Ideal lattice

In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient.

Learning with errors

In cryptography, Learning with errors (LWE) is a mathematical problem that is widely used in cryptography to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. In more technical terms, it refers to the computational problem of inferring a linear -ary function over a finite ring from given samples some of which may be erroneous.

Ring learning with errors

In post-quantum cryptography, ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption. Public-key cryptography relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known.

Lattice problem

In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: Lattice problems are an example of NP-hard problems which have been shown to be average-case hard, providing a test case for the security of cryptographic algorithms. In addition, some lattice problems which are worst-case hard can be used as a basis for extremely secure cryptographic schemes.

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