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Lecture

Chinese Remainders & RSA

In course

COM-102: Advanced information, computation, communication II

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Description

This lecture covers the Chinese Remainders Theorem, RSA public-key cryptosystem, and related concepts. It explains the construction used in the proof of Lagrange's Theorem, the bijective properties of the Chinese Remainders map, and the isomorphism with respect to addition and multiplication. The lecture also delves into the proof of Fermat's Theorem, Euler's Theorem, and the application of Chinese Remainders in combining prime numbers. It concludes with the generation of RSA keys, encryption, and decryption processes, illustrating how RSA works with a practical example.

Instructors (2)

Bixio Rimoldi

Bixio Rimoldi received the Dipl. El.-Ing degree as well as the Dr. ès Sciences degree from the ETHZ, Switzerland. Since 1997, he holds a full professor position at the School of Computer and Communication Sciences at EPFL and he is the director of EPFL's Mobile Communications Laboratory (LCM). Prior to joining EPFL, he was in the faculty of the Electrical Engineering department of Washington University. In 1993, he received a US National Science Foundation Young Investigator Award. In 2000, he was elected to the grade of Fellow of the IEEE. During the period 2002-2009 he has been on the Board of Governors of the IEEE Information Theory Society, where he served in several offices including President. He was co-chairman with Bruce Hajek of the 1995 IEEE Information Theory Workshop on Information Theory, Multiple Access, and Queueing (St Louis, MO), and co-chairman with Jim Massey of the 2002 IEEE International Symposium in Information Theory (Lausanne, Switzerland). He has been a member of the editorial board of "Foundations and Trends on Communications and Information Theory," and was an editor of the European Transactions on Telecommunications. During 2005 and 2006 he was the Director of EPFL's Bachelor program in Communication Systems. His interests are in various aspects of digital communications, information theory, and software-defined radio.

Michael Christoph Gastpar

Michael Gastpar is a (full) Professor at EPFL. From 2003 to 2011, he was a professor at the University of California at Berkeley, earning his tenure in 2008. He received his Dipl. El.-Ing. degree from ETH Zürich, Switzerland, in 1997 and his MS degree from the University of Illinois at Urbana-Champaign, IL, USA, in 1999. He defended his doctoral thesis at EPFL on Santa Claus day, 2002. He was also a (full) Professor at Delft University of Technology, The Netherlands. His research interests are in network information theory and related coding and signal processing techniques, with applications to sensor networks and neuroscience. He is a Fellow of the IEEE. He is the co-recipient of the 2013 Communications Society & Information Theory Society Joint Paper Award. He was an Information Theory Society Distinguished Lecturer (2009-2011). He won an ERC Starting Grant in 2010, an Okawa Foundation Research Grant in 2008, an NSF CAREER award in 2004, and the 2002 EPFL Best Thesis Award. He has served as an Associate Editor for Shannon Theory for the IEEE Transactions on Information Theory (2008-11), and as Technical Program Committee Co-Chair for the 2010 International Symposium on Information Theory, Austin, TX.

Official source

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Algebra: Abstract algebra

Related concepts (32)

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.

Chinese remainder theorem

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). For example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23.

Euler's theorem

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and is Euler's totient function, then a raised to the power is congruent to 1 modulo n; that is In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number.

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects).

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.

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