Periodic Structure of the Exponential Pseudorandom Number Generator

We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$

On Improving Communication Complexity in Cryptography

Cryptography grew to be much more than "the study of secret writing". Modern cryptography is concerned with establishing properties such as privacy, integrity and authenticity in protocols for secure communication and computation. This comes at a price: Cryptographic tools usually introduce an overhead, both in terms of communication complexity (that is, number and size of messages transmitted) and computational efficiency (that is, time and memory required). As in many settings communication between the parties involved is the bottleneck, this thesis is concerned with improving communication complexity in cryptographic protocols. One direction towards this goal is scalable cryptography: In many cryptographic schemes currently deployed, the security degrades linearly with the number of instances (e.g. encrypted messages) in the system. As this number can be huge in contexts like cloud computing, the parameters of the scheme have to be chosen considerably larger - and in particular depending on the expected number of instances in the system - to maintain security guarantees. We advance the state-of-the-art regarding scalable cryptography by constructing schemes where the security guarantees are independent of the number of instances. This allows to choose smaller parameters, even when the expected number of instances is immense. - We construct the first scalable encryption scheme with security against active adversaries which has both compact public keys and ciphertexts. In particular, we significantly reduce the size of the public key to only about 3% of the key-size of the previously most efficient scalable encryption scheme. (Gay,Hofheinz, and Kohl, CRYPTO, 2017) - We present a scalable structure-preserving signature scheme which improves both in terms of public-key and signature size compared to the previously best construction to about 40% and 56% of the sizes, respectively. (Gay, Hofheinz, Kohl, and Pan, EUROCRYPT, 2018) Another important area of cryptography is secure multi-party computation, where the goal is to jointly evaluate some function while keeping each party’s input private. In traditional approaches towards secure multi-party computation either the communication complexity scales linearly in the size of the function, or the computational efficiency is poor. To overcome this issue, Boyle, Gilboa, and Ishai (CRYPTO, 2016) introduced the notion of homomorphic secret sharing. Here, inputs are shared between parties such that each party does not learn anything about the input, and such that the parties can locally evaluate functions on the shares. Homomorphic secret sharing implies secure computation where the communication complexity only depends on the size of the inputs, which is typically much smaller than the size of the function. A different approach towards efficient secure computation is to split the protocol into an input-independent preprocessing phase, where long correlated strings are generated, and a very efficient online phase. One example for a useful correlation are authenticated Beaver triples, which allow to perform efficient multiplications in the online phase such that privacy of the inputs is preserved and parties deviating the protocol can be detected. The currently most efficient protocols implementing the preprocessing phase require communication linear in the number of triples to be generated. This results typically in high communication costs, as the online phase requires at least one authenticated Beaver triple per multiplication. We advance the state-of-the art regarding efficient protocols for secure computation with low communication complexity as follows. - We construct the first homomorphic secret sharing scheme for computing arbitrary functions in NC 1 (that is, functions that are computably by circuits with logarithmic depth) which supports message spaces of arbitrary size, has only negligible correctness error, and does not require expensive multiplication on ciphertexts. (Boyle, Kohl, and Scholl, EUROCRYPT, 2019) - We introduce the notion of a pseudorandom correlation generator for general correlations. Pseudorandom correlation generators allow to locally extend short correlated seeds into long pseudorandom correlated strings. We show that pseudorandom correlation generators can replace the preprocessing phase in many protocols, leading to a preprocessing phase with sublinear communication complexity. We show connections to homomorphic secret sharing schemes and give the first instantiation of pseudorandom correlation generators for authenticated Beaver triples at reasonable computational efficiency. (Boyle, Couteau, Gilboa, Ishai, Kohl, and Scholl, CRYPTO, 2019

Efficient Scalable Constant-Round MPC via Garbled Circuits

In the setting of secure multiparty computation, a set of mutually distrustful parties carry out a joint computation of their inputs, without revealing anything but the output. Over recent years, there has been tremendous progress towards making secure computation practical, with great success in the two-party case. In contrast, in the multiparty case, progress has been much slower, even for the case of semi-honest adversaries. In this paper, we consider the case of constant-round multiparty computation, via the garbled circuit approach of BMR (Beaver et al., STOC 1990). In recent work, it was shown that this protocol can be efficiently instantiated for semi-honest adversaries (Ben-Efraim et al., ACM CCS 2016). However, it scales very poorly with the number of parties, since the cost of garbled circuit evaluation is quadratic in the number of parties, per gate. Thus, for a large number of parties, it becomes expensive. We present a new way of constructing a BMR-type garbled circuit that can be evaluated with only a constant number of operations per gate. Our constructions use key-homomorphic pseudorandom functions (one based on DDH and the other on Ring-LWE) and are concretely efficient. In particular, for a large number of parties (e.g., 100), our new circuit can be evaluated faster than the standard BMR garbled circuit that uses only AES computations. Thus, our protocol is an important step towards achieving concretely efficient large-scale multiparty computation for Internet-like settings (where constant-round protocols are needed due to high latency)

Two-sources Randomness Extractors for Elliptic Curves

This paper studies the task of two-sources randomness extractors for elliptic curves defined over finite fields $K$, where $K$ can be a prime or a binary field. In fact, we introduce new constructions of functions over elliptic curves which take in input two random points from two differents subgroups. In other words, for a ginven elliptic curve $E$ defined over a finite field $\mathbb{F}_q$ and two random points $P \in \mathcal{P}$ and $Q\in \mathcal{Q}$, where $\mathcal{P}$ and $\mathcal{Q}$ are two subgroups of $E(\mathbb{F}_q)$, our function extracts the least significant bits of the abscissa of the point $P\oplus Q$ when $q$ is a large prime, and the $k$-first $\mathbb{F}_p$ coefficients of the asbcissa of the point $P\oplus Q$ when $q = p^n$, where $p$ is a prime greater than $5$. We show that the extracted bits are close to uniform. Our construction extends some interesting randomness extractors for elliptic curves, namely those defined in \cite{op} and \cite{ciss1,ciss2}, when $\mathcal{P} = \mathcal{Q}$. The proposed constructions can be used in any cryptographic schemes which require extraction of random bits from two sources over elliptic curves, namely in key exchange protole, design of strong pseudo-random number generators, etc

On the Non-Existence of Blockwise 2-Local PRGs with Applications to Indistinguishability Obfuscation

Lin and Tessaro (Eprint 2017/250) recently proposed indistinguishability obfuscation and functional encryption candidates and proved their security based on a standard assumption on bilinear maps and a non-standard assumption on ``Goldreich-like'' pseudorandom generators (PRG). In a nutshell, they require the existence of pseudo-random generators $G:\Sigma^n \to \{0,1\}^m$ for some $\mathsf{poly}(n)$-size alphabet $\Sigma$ where each output bit depends on at most two input alphabet symbols, and which achieve sufficiently large stretch. We show a polynomial-time attack against such generators. Our attack uses tools from the literature on two-source extractors (Chor and Goldreich, SICOMP 1988) and efficient refutation of 2-CSPs over large alphabets (Allen, O'Donnell and Witmer, FOCS 2015). Finally, we propose new ways to instantiate the Lin-Tessaro construction that do not immediately fall to our attacks. While we cannot say with any confidence that these modifications are secure, they certainly deserve further cryptanalysis

On Cryptographic Building Blocks and Transformations

Cryptographic building blocks play a central role in cryptography, e.g., encryption or digital signatures with their security notions. Further, cryptographic building blocks might be constructed modularly, i.e., emerge out of other cryptographic building blocks. Essentially, one cryptographically transforms the underlying block(s) and their (security) properties into the emerged block and its properties. This thesis considers cryptographic building blocks and new cryptographic transformations