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

Let ψ be a 2-DNF formula on boolean variables x1,...,xn ∈ {0,1}. We present a homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn. In other words, given the encryption of the bits x1,...,xn, anyone can create the encryption of ψ(x1,...,xn). More generally, we can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set. We present a number of applications of the system:

In a database of size n, the total communication in the basic step of the Kushilevitz-Ostrovsky PIR protocol is reduced from $\sqrt{n}$ to $\sqrt[3]{n}$.

An efficient election system based on homomorphic encryption where voters do not need to include non-interactive zero knowledge proofs that their ballots are valid. The election system is proved secure without random oracles but still efficient.

A protocol for universally verifiable computation.

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Evaluating 2-DNF formulas on ciphertexts

Tables of integrals

We present a somewhat homomorphic encryption scheme that is both very simple to describe and analyze, and whose security (quantumly) reduces to the worst-case hardness of problems on ideal lattices. We then transform it into a fully homomorphic encryption scheme using standard "squashing" and "bootstrapping" techniques introduced by Gentry (STOC 2009).

One of the obstacles in going from "somewhat" to full homomorphism is the requirement that the somewhat homomorphic scheme be circular secure, namely, the scheme can be used to securely encrypt its own secret key. For all known somewhat homomorphic encryption schemes, this requirement was not known to be achievable under any cryptographic assumption, and had to be explicitly assumed. We take a step forward towards removing this additional assumption by proving that our scheme is in fact secure when encrypting polynomial functions of the secret key.

Our scheme is based on the ring learning with errors (RLWE) assumption that was recently introduced by Lyubashevsky, Peikert and Regev (Eurocrypt 2010). The RLWE assumption is reducible to worstcase problems on ideal lattices, and allows us to completely abstract out the lattice interpretation, resulting in an extremely simple scheme. For example, our secret key is s, and our public key is (a, b = as+2e), where s, a, e are all degree (n - 1) integer polynomials whose coefficients are independently drawn from easy to sample distributions.

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Fully homomorphic encryption from ring-LWE and security for key dependent messages

We establish the following, quite unexpected, result: replication of data for the computational private information retrieval problem is not necessary. More specifically, based on the quadratic residuosity assumption, we present a single database, computationally private information retrieval scheme with O(n/sup /spl epsiv//) communication complexity for any /spl epsiv/>0.

Replication is not needed: single database, computationally-private information retrieval

Algorithmic number theory

Preface 1. Stringology 2. Number theory and algebra 3. Numeration systems 4. Finite automata and other models of computation 5. Automatic sequences 6. Uniform morphisms and automatic sequences 7. Morphic sequences 8. Frequency of letters 9. Characteristic words 10. Subwords 11. Cobham's theorem 12. Formal power series 13. Automatic real numbers 14. Multidimensional automatic sequences 15. Automaticity 16. k-regular sequences 17. Physics Appendix. Hints, references and solutions for selected exercises Bibliography Index.

Automatic Sequences: Theory, Applications, Generalizations

We discuss a well-known binary sequence called the Thue-Morse sequence, or the Prouhet-Thue-Morse sequence This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921 However, it was already implicit in an 1851 paper of Prouhet The Prouhet-Thue-Morse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences

The Ubiquitous Prouhet-Thue-Morse Sequence

The ring of k -regular sequences, II

Intended for graduate students and advanced undergraduates in computer science, A Second Course in Formal Languages and Automata Theory treats topics in the theory of computation not usually covered in a first course. After a review of basic concepts, the book covers combinatorics on words, regular languages, context-free languages, parsing and recognition, Turing machines, and other language classes. Many topics often absent from other textbooks, such as repetitions in words, state complexity, the interchange lemma, 2DPDAs, and the incompressibility method, are covered here. The author places particular emphasis on the resources needed to represent certain languages. The book also includes a diverse collection of more than 200 exercises, suggestions for term projects, and research problems that remain open.

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Jeffrey Shallit

Papers

Algorithmic number theory

Automatic Sequences: Theory, Applications, Generalizations

The Ubiquitous Prouhet-Thue-Morse Sequence

The ring of k -regular sequences, II

A Second Course in Formal Languages and Automata Theory