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

We consider cryptographic and physical zero-knowledge proof schemes for Sudoku, a popular combinatorial puzzle. We discuss methods that allow one party, the prover, to convince another party, the verifier, that the prover has solved a Sudoku puzzle, without revealing the solution to the verifier. The question of interest is how a prover can show: (i) that there is a solution to the given puzzle, and (ii) that he knows the solution, while not giving away any information about the solution to the verifier.

In this paper we consider several protocols that achieve these goals. Broadly speaking, the protocols are either cryptographic or physical. By a cryptographic protocol we mean one in the usual model found in the foundations of cryptography literature. In this model, two machines exchange messages, and the security of the protocol relies on computational hardness. By a physical protocol we mean one that is implementable by humans using common objects, and preferably without the aid of computers. In particular, our physical protocols utilize items such as scratch-off cards, similar to those used in lotteries, or even just simple playing cards.

The cryptographic protocols are direct and efficient, and do not involve a reduction to other problems. The physical protocols are meant to be understood by “lay-people” and implementable without the use of computers.

https://www.microsoft.com/en-us/research/wp-content/uploads/2009/01/GradwohlNPR09.pdf

Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles

http://www.wisdom.weizmann.ac.il/~naor/PAPERS/fair_carpool.pdf

On fairness in the carpool problem

A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a "qualified" subset of parties can efficiently reconstruct the secret while any "unqualified" subset of parties cannot efficiently learn anything about the secret. The collection of "qualified" subsets is defined by a monotone Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing scheme. Yao suggested a method for secret-sharing for any function that has a polynomial-size monotone circuit (a class which is strictly smaller than the class of monotone functions in $${\mathsf {P}}$$P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in $${\mathsf {NP}}$$NP: in order to reconstruct the secret a set of parties must be "qualified" and provide a witness attesting to this fact. Recently, Garg et al. (Symposium on theory of computing conference, STOC, pp 467---476, 2013) put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement $$x\in L$$xźL for a language $$L\in {\mathsf {NP}}$$LźNP such that anyone holding a witness to the statement can decrypt the message; however, if $$x\notin L$$xźL, then it is computationally hard to decrypt. Garg et al. showed how to construct several cryptographic primitives from witness encryption and gave a candidate construction. One can show that computational secret-sharing implies witness encryption for the same language. Our main result is the converse: we give a construction of a computational secret-sharing scheme for any monotone function in $${\mathsf {NP}}$$NP assuming witness encryption for $${\mathsf {NP}}$$NP and one-way functions. As a consequence we get a completeness theorem for secret-sharing: computational secret-sharing scheme for any single monotone $${\mathsf {NP}}$$NP-complete function implies a computational secret-sharing scheme for every monotone function in $${\mathsf {NP}}$$NP.

/pdf/secret-sharing-for-np-3k0e1401iy.pdf

Secret-Sharing for NP

The relative power of determinism, randomness, and nondeterminism for search problems in the Boolean decision tree model is studied. It is shown that the gaps between the nondeterministic, the randomized, and the deterministic complexities can be arbitrarily large for search problems. An interesting connection of this model to the complexity of resolution proofs is also mentioned.

/pdf/search-problems-in-the-decision-tree-model-3776f8zv3z.pdf

Search Problems in the Decision Tree Model

The relative power of determinism, randomness, and nondeterminism for search problems in the Boolean decision tree model is studied. It is shown that the CNF search problem is complete for all the variants of decision trees. It is then shown that the gaps between the nondeterministic, the randomized, and the deterministic complexities can be arbitrarily large for search problems. The special case of nondeterministic complexity is discussed. >

/pdf/search-problems-in-the-decision-tree-model-3rjttuahtq.pdf

Moni Naor

Papers

Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles

On fairness in the carpool problem

Secret-Sharing for NP

Search Problems in the Decision Tree Model

Search problems in the decision tree model