Approximate algebraic structure

We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.Comment: 25 pages. Submitted to Proceedings of the ICM 2014. This version may be longer than the published one, as my submission was 4 pages too long with the official style fil

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

A Note on Non-Interactive Zero-Knowledge from CDH

We build non-interactive zero-knowledge (NIZK) and ZAP arguments for all $\mathsf{NP}$ where soundness holds for infinitely-many security parameters, and against uniform adversaries, assuming the subexponential hardness of the Computational Diffie-Hellman (CDH) assumption. We additionally prove the existence of NIZK arguments with these same properties assuming the polynomial hardness of both CDH and the Learning Parity with Noise (LPN) assumption. In both cases, the CDH assumption does not require a group equipped with a pairing. Infinitely-often uniform security is a standard byproduct of commonly used non-black-box techniques that build on disjunction arguments on the (in)security of some primitive. In the course of proving our results, we develop a new variant of this non-black-box technique that yields improved guarantees: we obtain explicit constructions (previous works generally only obtained existential results) where security holds for a relatively dense set of security parameters (as opposed to an arbitrary infinite set of security parameters). We demonstrate that our technique can have applications beyond our main results

Recovering cryptographic keys from partial information, by example

Side-channel attacks targeting cryptography may leak only partial or indirect information about the secret keys. There are a variety of techniques in the literature for recovering secret keys from partial information. In this tutorial, we survey several of the main families of partial key recovery algorithms for RSA, (EC)DSA, and (elliptic curve) Diffie-Hellman, the public-key cryptosystems in common use today. We categorize the known techniques by the structure of the information that is learned by the attacker, and give simplified examples for each technique to illustrate the underlying ideas

On ELFs, Deterministic Encryption, and Correlated-Input Security

We construct deterministic public key encryption secure for any constant number of arbitrarily correlated computationally unpredictable messages. Prior works required either random oracles or non-standard knowledge assumptions. In contrast, our constructions are based on the exponential hardness of DDH, which is plausible in elliptic curve groups. Our central tool is a new trapdoored extremely lossy function, which modifies extremely lossy functions by adding a trapdoor