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 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 P). Around 1990 Rudich raised the possibility of obtaining secretsharing 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 ∈ L for a language L ∈ NP such that anyone holding a witness to the statement can decrypt the message, however if x / ∈ 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. ∗Weizmann Institute of Science. Email: {ilan.komargodski,moni.naor,eylon.yogev}@weizmann.ac.il. Research supported in part by a grant from the Israel Science Foundation, the I-CORE Program of the Planning and Budgeting Committee, BSF and the Israeli Ministry of Science and Technology. Moni Naor is the incumbent of the Judith Kleeman Professorial Chair.

/pdf/secret-sharing-for-np-32q0ta16el.pdf

Secret-Sharing for NP

We present a new concern when collecting data from individuals that arises from the attempt to mitigate privacy leakage in multiple reporting: tracking of users participating in the data collection via the mechanisms added to provide privacy. We present several definitions for untrackable mechanisms, inspired by the differential privacy framework. 
Specifically, we define the trackable parameter as the log of the maximum ratio between the probability that a set of reports originated from a single user and the probability that the same set of reports originated from two users (with the same private value). We explore the implications of this new definition. We show how differentially private and untrackable mechanisms can be combined to achieve a bound for the problem of detecting when a certain user changed their private value. 
Examining Google's deployed solution for everlasting privacy, we show that RAPPOR (Erlingsson et al. ACM CCS, 2014) is trackable in our framework for the parameters presented in their paper. 
We analyze a variant of randomized response for collecting statistics of single bits, Bitwise Everlasting Privacy, that achieves good accuracy and everlasting privacy, while only being reasonably untrackable, specifically grows linearly in the number of reports. For collecting statistics about data from larger domains (for histograms and heavy hitters) we present a mechanism that prevents tracking for a limited number of responses. 
We also present the concept of Mechanism Chaining, using the output of one mechanism as the input of another, in the scope of Differential Privacy, and show that the chaining of an $\varepsilon_1$-LDP mechanism with an $\varepsilon_2$-LDP mechanism is $\ln\frac{e^{\varepsilon_1+\varepsilon_2}+1}{e^{\varepsilon_1}+e^{\varepsilon_2}}$-LDP and that this bound is tight.

Can Two Walk Together: Privacy Enhancing Methods and Preventing Tracking of Users

We consider Primary-Secondary-Resolver Membership Proof Systems (PSR for short) and show different constructions of that primitive. A PSR system is a 3-party protocol, where we have a primary, which is a trusted party which commits to a set of members and their values, then generates public and secret keys in order for secondaries (provers with knowledge of both keys) and resolvers (verifiers who only know the public key) to engage in interactive proof sessions regarding elements in the universe and their values. The motivation for such systems is for constructing a secure Domain Name System (DNSSEC) that does not reveal any unnecessary information to its clients. We require our systems to be complete, so honest executions will result in correct conclusions by the resolvers, sound, so malicious secondaries cannot cheat resolvers, and zero-knowledge, so resolvers will not learn additional information about elements they did not query explicitly. Providing proofs of membership is easy, as the primary can simply precompute signatures over all the members of the set. Providing proofs of non-membership, i.e. a denial-of-existence mechanism, is trickier and is the main issue in constructing PSR systems. We provide three different strategies to construct a denial of existence mechanism. The first uses a set of cryptographic keys for all elements of the universe which are not members, which we implement using hierarchical identity based encryption and a tree based signature scheme. The second construction uses cuckoo hashing with a stash, where in order to prove non-membership, a secondary must prove that a search for it will fail, i.e. that it is not in the tables or the stash of the cuckoo hashing scheme. The third uses a verifiable “random looking” function which the primary evaluates over the set of members, then signs the values lexicographically and secondaries then use those signatures to prove to resolvers that the value of the non-member was not signed by the primary. We implement this function using a weaker variant of verifiable random/unpredictable functions and pseudorandom functions with interactive zero

/pdf/primary-secondary-resolver-membership-proof-systems-3gq9lr1a96.pdf

Primary-Secondary-Resolver Membership Proof Systems.

/pdf/out-of-band-authenticated-group-key-exchange-from-strong-4h43nkso7j.pdf

Out-of-Band Authenticated Group Key Exchange: From Strong Authentication to Immediate Key Delivery.

We introduce the Holiday Gathering Problem which models the difficulty in scheduling non-interfering transmissions in (wireless) networks. Our goal is to schedule transmission rounds so that the antennas that transmit in a given round will not interfere with each other, i.e. all of the other antennas that can interfere will not transmit in that round, while minimizing the number of consecutive rounds in which antennas do not transmit.Following a long tradition in Computer Science, we introduce the problem by an intuitive story. Assume we live in a perfect world where families enjoy being together. Consequently, parents, whose children are in a monogamous relation, would like to have all their children at home for the holiday meal (i.e. there is a special pleasure gained by hosting all the children simultaneously and they wish to have this event occur as frequently as possible). However, the conflict is that the in-laws would also be happiest if all their children come to them. Our goal can be described as scheduling an infinite sequence of "guest lists" in a distributed setting so that each child knows where it will spend the holiday. The holiday gathering problem is closely related to several classical problems in computer science, such as the dining philosophers problem on a general graph and periodic scheduling.The process of the scheduling should be done with no further communication after initialization, by using a small amount of local data. The result should minimize the number of consecutive holidays where the family is not together. In a good sequence this number depends on local properties of the parents (e.g., their number of children). Furthermore, solutions that are periodic, i.e. a gathering occurs every fixed number of rounds, are useful for maintaining a small amount of information at each node and reducing the amount of ongoing communication and computation.Our algorithmic techniques show interesting connections between periodic scheduling, coloring, and universal prefix free encodings. We develop a coloring-based construction where the period of each node colored with the c is at most 21+log*c ⋅ prodi=0log*c log(i)c (where log(i) means iterating the log function i times). This is achieved via a connection with prefix-free encodings. We prove that this is the best possible for coloring-based solutions. We also show a construction with period at most 2d for a node of degree d.

https://dl.acm.org/doi/pdf/10.1145/2935764.2935788

Moni Naor

Papers

Secret-Sharing for NP

Can Two Walk Together: Privacy Enhancing Methods and Preventing Tracking of Users

Primary-Secondary-Resolver Membership Proof Systems.

Out-of-Band Authenticated Group Key Exchange: From Strong Authentication to Immediate Key Delivery.

The Family Holiday Gathering Problem or Fair and Periodic Scheduling of Independent Sets