There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum Computation and Quantum Information

From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

We propose a fully functional identity-based encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem. Our system is based on the Weil pairing. We give precise definitions for secure identity based encryption schemes and give several applications for such systems.

/pdf/identity-based-encryption-from-the-weil-pairing-sfqdyadlbb.pdf

Identity-Based Encryption from the Weil Pairing

This paper investigates a novel computational problem, namely the Composite Residuosity Class Problem, and its applications to public-key cryptography. We propose a new trapdoor mechanism and derive from this technique three encryption schemes : a trapdoor permutation and two homomorphic probabilistic encryption schemes computationally comparable to RSA. Our cryptosystems, based on usual modular arithmetics, are provably secure under appropriate assumptions in the standard model.

/pdf/public-key-cryptosystems-based-on-composite-degree-1zopwb21ba.pdf

Public-key cryptosystems based on composite degree residuosity classes

The problem of generating a shared secret key S by two parties knowing dependent random variables X and Y, respectively, but not sharing a secret key initially, is considered. An enemy who knows the random variable Z, jointly distributed with X and Y according to some probability distribution P/sub XYZ/, can also receive all messages exchanged by the two parties over a public channel. The goal of a protocol is that the enemy obtains at most a negligible amount of information about S. Upper bounds on H(S) as a function of P/sub XYZ/ are presented. Lower bounds on the rate H(S)/N (as N to infinity ) are derived for the case in which X=(X/sub 1/, . . ., X/sub N/), Y=(Y/sub 1/, . . ., Y/sub N/) and Z=(Z/sub 1/, . . ., Z/sub N/) result from N independent executions of a random experiment generating X/sub i/, Y/sub i/ and Z/sub i/ for i=1, . . ., N. It is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels, even when the enemy's channel is superior to the other two channels. >

Secret key agreement by public discussion from common information

This paper, provides a general treatment of privacy amplification by public discussion, a concept introduced by Bennett, Brassard, and Robert for a special scenario. Privacy amplification is a process that allows two parties to distil a secret key from a common random variable about which an eavesdropper has partial information. The two parties generally know nothing about the eavesdropper's information except that it satisfies a certain constraint. The results have applications to unconditionally secure secret-key agreement protocols and quantum cryptography, and they yield results on wiretap and broadcast channels for a considerably strengthened definition of secrecy capacity.

/pdf/generalized-privacy-amplification-diymb2a026.pdf

Generalized privacy amplification

One of the basic problems in cryptography is the generation of a common secret key between two parties, for instance in order to communicate privately. In this paper we consider information-theoretically secure key agreement. Wyner and subsequently Csiszar and Korner described and analyzed settings for secret-key agreement based on noisy communication channels. Maurer as well as Ahlswede and Csiszar generalized these models to a scenario based on correlated randomness and public discussion. In all these settings, the secrecy capacity and the secret-key rate, respectively, have been defined as the maximal achievable rates at which a highly-secret key can be generated by the legitimate partners. However, the privacy requirements were too weak in all these definitions, requiring only the ratio between the adversary's information and the length of the key to be negligible, but hence tolerating her to obtain a possibly substantial amount of information about the resulting key in an absolute sense. We give natural stronger definitions of secrecy capacity and secret-key rate, requiring that the adversary obtains virtually no information about the entire key. We show that not only secret-key agreement satisfying the strong secrecy condition is possible, but even that the achievable key-generation rates are equal to the previous weak notions of secrecy capacity and secret-key rate. Hence the unsatisfactory old definitions can be completely replaced by the new ones. We prove these results by a generic reduction of strong to weak key agreement. The reduction makes use of extractors, which allow to keep the required amount of communication negligible as compared to the length of the resulting key.

/pdf/information-theoretic-key-agreement-from-weak-to-strong-vwx51hoyb4.pdf

Information-theoretic key agreement: from weak to strong secrecy for free

We show that verifiable secret sharing (VSS) and secure multi-party computation (MPC) among a set of n players can efficiently be based on any linear secret sharing scheme (LSSS) for the players, provided that the access structure of the LSSS allows MPC or VSS at all. Because an LSSS neither guarantees reconstructability when some shares are false, nor verifiability of a shared value, nor allows for the multiplication of shared values, an LSSS is an apparently much weaker primitive than VSS or MPC.

Our approach to secure MPC is generic and applies to both the information-theoretic and the cryptographic setting. The construction is based on 1) a formalization of the special multiplicative property of an LSSS that is needed to perform a multiplication on shared values, 2) an efficient generic construction to obtain from any LSSS a multiplicative LSSS for the same access structure, and 3) an efficient generic construction to build verifiability into every LSSS (always assuming that the adversary structure allows for MPC or VSS at all).

The protocols are efficient. In contrast to all previous information-theoretically secure protocols, the field size is not restricted (e.g, to be greater than n). Moreover, we exhibit adversary structures for which our protocols are polynomial in n while all previous approaches to MPC for non-threshold adversaries provably have super-polynomial complexity.

https://link.springer.com/content/pdf/10.1007%2F3-540-45539-6_22.pdf

General secure multi-party computation from any linear secret-sharing scheme

The goals of this paper are two-fold. First we introduce and motivate a generalization of the fundamental concept of the indistinguishability of two systems, called indifferentiability. This immediately leads to a generalization of the related notion of reducibility of one system to another. In contrast to the conventional notion of indistinguishability, indifferentiability is applicable in settings where a possible adversary is assumed to have access to additional information about the internal state of the involved systems, for instance the public parameter selecting a member from a family of hash functions.

/pdf/indifferentiability-impossibility-results-on-reductions-and-2690px0jq1.pdf

Ueli Maurer

Papers

Secret key agreement by public discussion from common information

Generalized privacy amplification

Information-theoretic key agreement: from weak to strong secrecy for free

General secure multi-party computation from any linear secret-sharing scheme

Indifferentiability, Impossibility Results on Reductions, and Applications to the Random Oracle Methodology