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.

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Identity-Based Encryption from the Weil Pairing

Mechanism Design is the algorithmic component of Game Theory, the synthesis of protocols for selfish parties to achieve certain properties. A protocol is a method to aggregate the preferences of the parties in order to decide on some "social choice," where typical examples include: deciding whether a community should build a bridge, how to route packets in a network and deciding who wins an auction. Each party has a utility function which expresses how much it values each possible outcome of the protocol. The goal is to design a protocol where the winning strategies achieve the social choice. Recently Mechanism Design has received attention by computer scientists in light of the above applications, see [19, 20].

Cryptography and mechanism design

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 Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing schemes. 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 P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in 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. (STOC 2013) put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement "x in L" for a language L in NP such that anyone holding a witness to the statement can decrypt the message, however, if x is not in L, 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 NP assuming witness encryption for NP and one-way functions. As a consequence we get a completeness theorem for secret-sharing: computational secret-sharing scheme for any single monotone NP-complete function implies a computational secret-sharing scheme for every monotone function in NP.

Secret-Sharing for NP from Indistinguishability Obfuscation.

We study the sample complexity of learning threshold functions under the constraint of differential privacy. It is assumed that each labeled example in the training data is the information of one individual and we would like to come up with a generalizing hypothesis $h$ while guaranteeing differential privacy for the individuals. Intuitively, this means that any single labeled example in the training data should not have a significant effect on the choice of the hypothesis. This problem has received much attention recently; unlike the non-private case, where the sample complexity is independent of the domain size and just depends on the desired accuracy and confidence, for private learning the sample complexity must depend on the domain size $X$ (even for approximate differential privacy). Alon et al. (STOC 2019) showed a lower bound of $\Omega(\log^*|X|)$ on the sample complexity and Bun et al. (FOCS 2015) presented an approximate-private learner with sample complexity $\tilde{O}\left(2^{\log^*|X|}\right)$. In this work we reduce this gap significantly, almost settling the sample complexity. We first present a new upper bound (algorithm) of $\tilde{O}\left(\left(\log^*|X|\right)^2\right)$ on the sample complexity and then present an improved version with sample complexity $\tilde{O}\left(\left(\log^*|X|\right)^{1.5}\right)$. 
Our algorithm is constructed for the related interior point problem, where the goal is to find a point between the largest and smallest input elements. It is based on selecting an input-dependent hash function and using it to embed the database into a domain whose size is reduced logarithmically; this results in a new database, an interior point of which can be used to generate an interior point in the original database in a differentially private manner.

Privately Learning Thresholds: Closing the Exponential Gap

Motivated by the challenging task of designing “secure” vote storage mechanisms, we study information storage mechanisms that operate in extremely hostile environments. In such environments, the majority of existing techniques for information storage and for security are susceptible to powerful adversarial attacks. We propose a mechanism for storing a set of at most K elements from a large universe of size N on write-once memories in a manner that does not reveal the insertion order of the elements. We consider a standard model for write-once memories, in which the memory is initialized to the all 0’s state, and the only operation allowed is flipping bits from 0 to 1. Whereas previously known constructions were either inefficient (required Θ(K) memory), randomized, or employed cryptographic techniques which are unlikely to be available in hostile environments, we eliminate each of these undesirable properties. The total amount of memory used by the mechanism is linear in the number of stored elements and poly-logarithmic in the size of the universe of elements. We also demonstrate a connection between secure vote storage mechanisms and one of the classical distributed computing problems: conflict resolution in multiple-access channels. By establishing a tight connection with the basic building block of our mechanism, we construct the first deterministic and non-adaptive conflict resolution algorithm whose running time is optimal up to poly-logarithmic factors. A preliminary version of this work appeared in Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP), pages 303–315, 2007. ∗Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Email: {tal.moran,moni.naor,gil.segev}@weizmann.ac.il. Research supported in part by a grant from the Israel Science Foundation. †Incumbent of the Judith Kleeman Professorial Chair.

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Deterministic History-Independent Strategies for Storing Information on Write-Once Memories

Faced with the threats posed by man-in-the-middle attacks, messaging platforms rely on “out-of-band” authentication, assuming that users have access to an external channel for authenticating one short value. For example, assuming that users recognizing each other’s voice can authenticate a short value, Telegram and WhatApp ask their users to compare 288-bit and 200-bit values, respectively. The existing protocols, however, do not take into account the plausible behavior of users who may be “lazy” and only compare parts of these values (rather than their entirety).

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Moni Naor

Papers

Cryptography and mechanism design

Secret-Sharing for NP from Indistinguishability Obfuscation.

Privately Learning Thresholds: Closing the Exponential Gap

Deterministic History-Independent Strategies for Storing Information on Write-Once Memories

The Security of Lazy Users in Out-of-Band Authentication.