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

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

PROBLEM TO BE SOLVED: To solve the problem, wherein it is impossible for an electronic content information provider to provide commercially secure and effective method, for a configurable general-purpose electronic commercial transaction/distribution control system. SOLUTION: In this system, having at least one protected processing environment for safely controlling at least one portion of decoding of digital information, a secure content distribution method comprises a process for encapsulating digital information in one or more digital containers; a process for encrypting at least a portion of digital information; a process for associating at least partially secure control information for managing interactions with encrypted digital information and/or digital container; a process for delivering one or more digital containers to a digital information user; and a process for using a protected processing environment, for safely controlling at least a portion of the decoding of the digital information. COPYRIGHT: (C)2006,JPO&NCIPI

https://apps.dtic.mil/dtic/tr/fulltext/u2/a423264.pdf

Systems and Methods for Secure Transaction Management and Electronic Rights Protection

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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

A secure function evaluation protocol allows two parties to jointly compute a function f(x,y) of their inputs in a manner not leaking more information than necessary. A major result in this field is: “any function f that can be computed using polynomial resources can be computed securely using polynomial resources” (where “resources” refers to communication and computation). This result follows by a general transformation from any circuit for f to a secure protocol that evaluates f. Although the resources used by protocols resulting from this transformation are polynomial in the circuit size, they are much higher (in general) than those required for an insecure computation of f.We propose a new methodology for designing secure protocols, utilizing the communication complexity tree (or branching program) representation of f. We start with an efficient (insecure) protocol for f and transform it into a secure protocol. In other words, ``any function f that can be computed using communication complexity c can be can be computed securely using communication complexity that is polynomial in c and a security parameter''. We show several simple applications of this new methodology resulting in protocols efficient either in communication or in computation. In particular, we exemplify a protocol for the Millionaires problem, where two participants want to compare their values but reveal no other information. Our protocol is more efficient than previously known ones in either communication or computation.

Communication preserving protocols for secure function evaluation

We describe new computationally secure protocols of 1-out-of-N oblivious transfer, k-out-of-N oblivious transfer, and oblivious transfer with adaptive queries. The protocols are very efficient compared with solutions based on generic two-party computation or on information-theoretic security. The 1-out-of-N oblivious transfer protocol requires only log N executions of a 1-out-of-2 oblivious transfer protocol. The k-out-of-N protocol is considerably more efficient than k repetitions of 1-out-of-N oblivious transfer, as is the construction for oblivious transfer with adaptive queries. The efficiency of the new oblivious transfer protocols makes them useful for many applications. A direct corollary of the 1-out-of-N oblivious transfer protocol is an efficient transformation of any Private Information Retrieval protocol to a Symmetric PIR protocol.

/pdf/computationally-secure-oblivious-transfer-1qqbupd90b.pdf

Computationally Secure Oblivious Transfer

``Perfect zero-knowledge arguments'' is a cryptographic primitive which allows one polynomial-time player to convince another polynomial-time player of the validity of an NP statement, without revealing any additional information (in the information-theoretic sense). Here the security achieved is on-line: in order to cheat and validate a false theorem, the prover must break a cryptographic assumption on-line during the conversation, while the verifier cannot find (ever) any information unconditionally. Despite their practical and theoretical importance, it was only known how to implement zero-knowledge arguments based on specific algebraic assumptions.

In this paper we show a general construction which can be based on any one-way permutation. The result is obtained by a construction of an information-theoretic secure bit-commitment protocol. The protocol is efficient (both parties are polynomial time) and can be based on any one-way permutation.

http://www.wisdom.weizmann.ac.il/%7Enaor/PAPERS/novy.pdf

Perfect Zero-Knowledge Arguments for NP Using Any One-Way Permutation

We present a new cryptographic primitive called pseudo-random synthesizer and show how to use it in order to get a parallel construction of a pseudo-random function. We show an NC/sup 1/ implementation of pseudo-random synthesizers based on the RSA or the Diffie-Hellman assumptions. This yields the first parallel (NC/sup 2/) pseudo-random function and the only alternative to the original construction of Goldreich, Gold-wasser and Micali (GGM). The security of our constructions is similar to the security of the underling assumptions. We discuss the connection with problems in computational learning theory.

Synthesizers and their application to the parallel construction of pseudo-random functions

A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication, and dissemination of information.
Given a strategy to pick quorums, the load LS is the minimal access probability of the busiest element, minimizing over the strategies. The capacity \capS\ is the highest quorum accesses rate that cS can handle, so $\capS=1/\LS$.
The availability of a quorum system cS is the probability that at least one quorum survives, assuming that each element fails independently with probability p. A tradeoff between LS and the availability of cS is shown.
We present four novel constructions of quorum systems, all featuring optimal or near optimal load, and high availability. The best construction, based on paths in a grid, has a load of $O(1/\sqn)$, and a failure probability of $\exp(-\Omega(\sqn))$ when the elements fail with probability $p<\half$. Moreover, even in the presence of faults, with exponentially high probability the load of this system is still $O(1/\sqn)$. The analysis of this scheme is based on percolation theory.

Moni Naor

Papers

Communication preserving protocols for secure function evaluation

Computationally Secure Oblivious Transfer

Perfect Zero-Knowledge Arguments for NP Using Any One-Way Permutation

Synthesizers and their application to the parallel construction of pseudo-random functions

The Load, Capacity, and Availability of Quorum Systems