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Lecture

RSA Cryptography: Primality Testing and Quadratic Residues

In course

COM-401: Cryptography and security

This course introduces the basics of cryptography. We review several types of cryptographic primitives, when it is safe to use them and how to select the appropriate security parameters. We detail how they work and sketch how they can be implemented. ...

Description

This lecture covers the basics of RSA cryptography, focusing on primality testing and quadratic residues. It explains the Fermat test, Carmichael numbers, the Miller-Rabin test, and the generation of prime numbers. The lecture also delves into the significance of the Fermat test, the Miller-Rabin criterion, and the computation of square roots in finite fields. Additionally, it discusses the Legendre and Jacobi symbols, the Goldwasser-Micali cryptosystem, and the breaking of the Decisional Diffie-Hellman assumption in Z. The presentation concludes with the application of quadratic residuosity in various cryptographic systems.

Instructor

Serge Vaudenay

Serge Vaudenay entered at the Ecole Normale Supérieure in 1989 with a major in mathematics. He earned his agrégation (secondary teaching degree) in mathematics in 1992, then a PhD in Computer Science at the University of Paris 7 - Denis Diderot in 1995. He subsequently became a senior research fellow at the CNRS, prior to being granted his habilitation à diriger des recherches (a postdoctoral degree authorizing the recipient to supervise doctoral students). In 1999, he was appointed as a Professor at the EPFL, where he created the Security and Cryptography Laboratory.

Official source

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Computer engineering

Computer security: Public-key cryptography

Related concepts (32)

Cryptography

Cryptography, or cryptology (from κρυπτός "hidden, secret"; and γράφειν graphein, "to write", or -λογία -logia, "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others.

Quadratic residue

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

Elliptic-curve cryptography

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security. Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme.

Public-key cryptography

Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security.

Quadratic reciprocity

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo .

Related lectures (30)

Prime Numbers and Primality Testing

Covers prime numbers, RSA cryptography, and primality testing, including the Chinese Remainder Theorem and the Miller-Rabin test.

Legendre and Jacobi Symbols: RSA Cryptography

Explores Legendre and Jacobi symbols, quadratic residuosity, element orders, and RSA Cryptography complexities.

Fermat's Little Theorem

Explores Fermat's Little Theorem, its extensions, primality testing algorithms, and the significance of prime numbers in cryptography.

Public-Key Cryptography: Key Exchange and Signatures

Explores public-key cryptography, key exchange, and digital signatures, discussing practical applications and security mechanisms.

Diffie-Hellman Cryptography: Key Exchange and ElGamal Encryption