Cryptology (DIAMANT)

Cryptology deals with mathematical techniques for design and analysis of algorithms and protocols for digital security in the presence of malicious adversaries. For example, encryption and digital signatures are used to construct private and authentic communication channels,which are instrumental to secure Internet transactions. Another example of increasing importance is secure computation, which in principle enables an arbitrary computation to be distributed among the processors in a network so that computations remain secret and are performed correctly, even if a certain quorum of the network is under full control by an adversary. Advancing our understanding of secure communications and secure computation are among the primary goals in cryptology.
It is fascinating and promising that the connection between cryptology and fields such as algebra, number theory, geometry, complexity theory, quantum physics and information theory is in the process of becoming still deeper than ever before.

This introductory course focuses on mathematical aspects of cryptology and consists of a *selection* from the following topics. Unilateral Security: Secure and Authentic Channels. Number theoretic intractability assumptions, public-key cryptography, security by reduction, sequences of games. Public-key encryption, chosen ciphertext security, Decision Diffie-Hellman Assumption, Cramer-Shoup cryptosystem. Digital signatures, chosen message attacks, Factoring Assumption, Goldwasser-Micali-Rivest signature scheme. Identity-based cryptography and pairings on elliptic curves. Applications from information theory to cryptography. Privacy amplification... or how to mod out the adversary's information. Constructions of entropy extractors from discrete mathematics. Secret key exchange by public discussion. Bounded-storage model.
Bi- and Multi-Lateral Security: Cryptographic Protocols.
Convincing a sceptical verifier without giving away the proof of your theorem: zero-knowledge proofs and Sigma protocols with example applications in electronic voting, and, more generally, secure computation based on threshold homomorphic cryptosystems. Blind signatures with example applications in electronic cash and credential mechanisms. Applications from algebraic number theory to (universal) secret sharing. Algebraic coding theory, verifiable secret sharing and information theoretically secure multi-party computation. Algebraic combinatorics and the efficiency of multi-party computation secure against general adversaries. Secure distributed linear algebra. Quantum Cryptography: Cryptography by Means of Quantum Physics.
Quantum information, entanglement, quantum teleportation, entropic uncertainty relations, quantum privacy amplification, quantum key-distribution, (im)possibility of quantum commitments and quantum oblivious-transfer.

Handouts

We may use parts of the following books:

1 - Ivan Damgaard (Ed.). Lectures on Data Security - Modern Cryptolology in Theory and Practice. Springer Verlag, 1999.

2 - Oded Goldreich. Modern Cryptography, Probabilistic Proofs and Pseudo-Randomness. Springer Verlag, 1999.

3 - Oded Goldreich. Foundations of Cryptography. Cambridge University Press, 2001.

4 - Victor Shoup. A Computational Introduction to Number Theory and Algebra. Cambridge University Press, 2005.