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 the role of randomness for the decisional complexity in algebraic decision (or computation) trees, i.e., the number of comparisons ignoring all other computation. Recently Ting and Yao showed that the problem of finding the maximum of n elements has decisional complexity O(log2n) (1994, Inform. Process. Lett., 49, 39?43). In contrast, Rabin showed in 1972 an ?(n) bound for the deterministic case (1972, J. Comput. System Sci., 6, 639?650). We point out that their technique is applicable to several problems for which corresponding ?(n) lower bounds hold. We show that in general the randomized decisional complexity is logarithmic in the size of the decision tree. We then turn to the question of the number of random bits needed to obtain the Ting and Yao result. We provide a deterministic O(k log n) algorithm for finding the elements which are larger than a given element, given a bound k on the number of these elements. We use this algorithm to obtain an O(log2n) random bits and O(log2n) queries algorithm for finding the maximum.

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

On the Decisional Complexity of Problems Over the Reals

If I commission a long computation, how can I check that the result is correct without re-doing the computation myself? This is the question that efficient verifiable computation deals with. In this work, we address the issue of verifying the computation as it unfolds. That is, at any intermediate point in the computation, I would like to see a proof that the current state is correct. Ideally, these proofs should be short, non-interactive, and easy to verify. In addition, the proof at each step should be generated efficiently by updating the previous proof, without recomputing the entire proof from scratch. This notion, known as incrementally verifiable computation, was introduced by Valiant [TCC 08] about a decade ago. Existing solutions follow the approach of recursive proof composition and can be based on strong and non-falsifiable cryptographic assumptions (so-called “knowledge assumptions”).

Incrementally Verifiable Computation via Incremental PCPs.

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. 
In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. 
When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist.

When Can Limited Randomness Be Used in Repeated Games

http://ieeexplore.ieee.org/iel5/2/27278/01212690.pdf

Protecting Keys: The Trace-and- Revoke Approach

We consider the problem of storing a large flle on a remote and unreliable server. To verify that the flle has not been corrupted, a user could store a small private (randomized) \flngerprint" on his own computer. This is the setting for the well-studied authentication problem in cryptography, and the required flngerprint size is well understood. We study the problem of sub-linear authentication: suppose the user would like to encode and store the flle in a way that allows him to verify that it has not been corrupted, but without reading the entire flle. If the user only wants to read t bits of the flle, how large does the size s of the private flngerprint need to be? We deflne this problem formally, and show a tight lower bound on the relationship between s and t when the adversary is not computationally bounded, namely: s £ t = ›(n), where n is the flle size. This is an easier case of the online memory checking problem, introduced by Blum et al. in 1991, and hence the same (tight) lower bound applies also to that problem. It was previously shown that when the adversary is computationally bounded, under the assumption that one-way functions exist, it is possible to construct much better online memory checkers and sub-linear authentication schemes. We show that the existence of one-way functions is also a necessary condition: even slightly breaking the s £ t = ›(n) lower bound in a computational setting implies the existence of one-way functions.

/pdf/the-complexity-of-online-memory-checking-421moj1zzz.pdf

Moni Naor

Papers

On the Decisional Complexity of Problems Over the Reals

Incrementally Verifiable Computation via Incremental PCPs.

When Can Limited Randomness Be Used in Repeated Games

Protecting Keys: The Trace-and- Revoke Approach

The Complexity of Online Memory Checking.