Lecture

Discrete Exponentiation Complexity

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COM-102: Advanced information, computation, communication II

Text, sound, and images are examples of information sources stored in our computers and/or communicated over the Internet. How do we measure, compress, and protect the informatin they contain? ...

Description

This lecture covers the complexity of discrete exponentiation, cyclic groups, and a summary of the Crypto Chapter. It starts by explaining the complexity of exponentiation in modular arithmetic, then delves into the concept of cyclic groups and their generators. The lecture further explores the properties of cyclic groups, including their orders and isomorphism. It also discusses discrete logarithms in cyclic groups and the process of finding inverses. Additionally, the lecture provides insights into practical cryptography, symmetric-key, and public-key systems. It concludes with a summary of popular cryptographic algorithms like Diffie-Hellman, ElGamal, and RSA, highlighting their key features and applications.

Instructors (2)

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.

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.

Official source

Ontological neighbourhood

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.

Key (cryptography)

A key in cryptography is a piece of information, usually a string of numbers or letters that are stored in a file, which, when processed through a cryptographic algorithm, can encode or decode cryptographic data. Based on the used method, the key can be different sizes and varieties, but in all cases, the strength of the encryption relies on the security of the key being maintained. A key's security strength is dependent on its algorithm, the size of the key, the generation of the key, and the process of key exchange.

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).

Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

Symmetric-key algorithm

Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between the two keys. The keys, in practice, represent a shared secret between two or more parties that can be used to maintain a private information link. The requirement that both parties have access to the secret key is one of the main drawbacks of symmetric-key encryption, in comparison to public-key encryption (also known as asymmetric-key encryption).

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Cryptographic Security Fundamentals

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