Cryptanalysis of the Legendre PRF and generalizations

The Legendre PRF relies on the conjectured pseudorandomness properties of the Legendre symbol with a hidden shift. Originally proposed as a PRG by Damgård at CRYPTO 1988, it was recently suggested as an efficient PRF for multiparty computation purposes by Grassi et al. at CCS 2016. Moreover, the Legendre PRF is being considered for usage in the Ethereum 2.0 blockchain. This paper improves previous attacks on the Legendre PRF and its higher-degree variant due to Khovratovich by reducing the time complexity from O(plogp/M) to O(plog^2p/M2) Legendre symbol evaluations when M≤p√4 queries are available. The practical relevance of our improved attack is demonstrated by breaking two concrete instances of the PRF proposed by the Ethereum foundation. Furthermore, we generalize our attack in a nontrivial way to the higher-degree variant of the Legendre PRF and we point out a large class of weak keys for this construction. Lastly, we provide the first security analysis of two additional generalizations of the Legendre PRF originally proposed by Damgård in the PRG setting, namely the Jacobi PRF and the power residue PRF

The Legendre Pseudorandom Function as a Multivariate Quadratic Cryptosystem: Security and Applications

Sequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs

Quantum Security of the Legendre PRF

International audienceIn this paper, we study the security of the Legendre PRF against quantum attackers, given classical queries only, and without quantum random-access memories. We give two algorithms that recover the key of a shifted Legendre symbol with unknown shift, with a complexity smaller than the exhaustive search of the key. The first one is a quantum variant of the table-based collision algorithm. The second one is an offline variant of Kuperberg's abelian hidden shift algorithm. We note that the latter, although asymptotically promising, is not currently the most efficient against practical parameters

Improved key recovery on the Legendre PRF

We give an algorithm for key recovery of the Legendre pseudorandom function that supersedes the best known algorithms so far. The expected number of operations is $O(\sqrt{p\log{\log{p}}})$ on a $\Theta(\log{p})$-bit word machine, under reasonable heuristic assumptions, and requires only $\sqrt[4]{p~{\log^2{p}}\log{\log{p}}}$ oracle queries. If the number of queries $M$ is smaller, the expected number of operations is $\frac{{p}\log{p}\log\log{p}}{M^2}$. We further show that the algorithm works in many different generalisations -- using a different character instead of the Legendre symbol, using the Jacobi symbol, or using a degree $r$ polynomial in the Legendre symbol numerator. In the latter case we show how to use Möbius transforms to lower the complexity to $O(p^{\operatorname{max}\{r-3,r/2\}}r^2\log{p})$ Legendre symbol computations, and $O(p^{\operatorname{max}\{r-4,r/2\}}r^2\log{p})$ in the case of a reducible polynomial. We also give an $O(\sqrt[3]{p})$ quantum algorithm that does not require a quantum oracle, and comments on the action of the Möbius group in the linear PRF case. On the practical side we give implementational details of our algorithm. We give the solutions of the $64, 74$ and $84$-bit prime challenges for key recovery with $M=2^{20}$ queries posed by Ethereum, out of which only the $64$ and $74$-bit were solved earlier

LegRoast: Efficient post-quantum signatures from the Legendre PRF

We introduce an efficient post-quantum signature scheme that relies on the one-wayness of the Legendre PRF. This LEGendRe One-wAyness SignaTure (LegRoast) builds upon the MPC-in-the-head technique to construct an efficient zero-knowledge proof, which is then turned into a signature scheme with the Fiat-Shamir transform. Unlike many other Fiat-Shamir signatures, the security of LegRoast can be proven without using the forking lemma, and this leads to a tight (classical) ROM proof. We also introduce a generalization that relies on the one-wayness of higher-power residue characters; the POwer Residue ChaRacter One-wAyness SignaTure (PorcRoast). LegRoast outperforms existing MPC-in-the-head-based signatures (most notably Picnic/Picnic2) in terms of signature size and speed. Moreover, PorcRoast outperforms LegRoast by a factor of 2 in both signature size and signing time. For example, one of our parameter sets targeting NIST security level I results in a signature size of 7.2 KB and a signing time of 2.8ms. This makes PorcRoast the most efficient signature scheme based on symmetric primitives in terms of signature size and signing time

Legendre PRF (Multiple) Key Attacks and the Power of Preprocessing

Due to its amazing speed and multiplicative properties the Legendre PRF recently finds widespread applications e.g. in Ethereum 2.0, multiparty computation and in the quantum-secure signature proposal LegRoast. However, its security is not yet extensively studied. The Legendre PRF computes for a key $k$ on input $x$ the Legendre symbol $L_k(x) = \left( \frac {x+k} {p} \right)$ in some finite field \F_p. As standard notion, PRF security is analysed by giving an attacker oracle access to $L_k(\cdot)$. Khovratovich\u27s collision-based algorithm recovers $k$ using $L_k(\cdot)$ in time $\sqrt{p}$ with constant memory. It is a major open problem whether this birthday-bound complexity can be beaten. We show a somewhat surprising wide-ranging analogy between the discrete logarithm problem and Legendre symbol computations. This analogy allows us to adapt various algorithmic ideas from the discrete logarithm setting. More precisely, we present a small memory multiple-key attack on $m$ Legendre keys $k_1, \ldots, k_m$ in time $\sqrt{mp}$, i.e. with amortized cost $\sqrt{p/m}$ per key. This multiple-key attack might be of interest in the Ethereum context, since recovering many keys simultaneously maximizes an attacker\u27s profit. Moreover, we show that the Legendre PRF admits precomputation attacks, where the precomputation depends on the public $p$ only -- and not on a key $k$. Namely, an attacker may compute e.g. in precomputation time $p^{\frac 2 3}$ a hint of size $p^{\frac 1 3}$. On receiving access to $L_k(\cdot)$ in an online phase, the attacker then uses the hint to recover the desired key $k$ in time only $p^{\frac 1 3}$. Thus, the attacker\u27s online complexity again beats the birthday-bound. In addition, our precomputation attack can also be combined with our multiple-key attack. We explicitly give various tradeoffs between precomputation and online phase. E.g. for attacking $m$ keys one may spend time $mp^{\frac 2 3}$ in the precomputation phase for constructing a hint of size $m^2 p^{\frac 1 3}$. In an online phase, one then finds {\em all $m$ keys in total time} only $p^{\frac 1 3}$. Precomputation attacks might again be interesting in the Ethereum 2.0 context, where keys are frequently changed such that a heavy key-independent precomputation pays off

Security, Scalability and Privacy in Applied Cryptography

In the modern digital world, cryptography finds its place in countless applications. However, as we increasingly use technology to perform potentially sensitive tasks, our actions and private data attract, more than ever, the interest of ill-intentioned actors. Due to the possible privacy implications of cryptographic flaws, new primitives’ designs need to undergo rigorous security analysis and extensive cryptanalysis to foster confidence in their adoption. At the same time, implementations of cryptographic protocols should scale on a global level and be efficiently deployable on users’ most common devices to widen the range of their applications. This dissertation will address the security, scalability and privacy of cryptosystems by presenting new designs and cryptanalytic results regarding blockchain cryptographic primitives and public-key schemes based on elliptic curves. In Part I, I will present the works I have done in regards to accumulator schemes. More precisely, in Chapter 2, I cryptanalyze Au et al. Dynamic Universal Accumulator, by showing some attacks which can completely take over the authority who manages the accumulator. In Chapter 3, I propose a design for an efficient and secure accumulator-based authentication mechanism, which is scalable, privacy-friendly, lightweight on the users’ side, and suitable to be implemented on the blockchain. In Part II, I will report some cryptanalytical results on primitives employed or considered for adoption in top blockchain-based cryptocurrencies. In particular, in Chapter 4, I describe how the zero-knowledge proof system and the commitment scheme adopted by the privacy-friendly cryptocurrency Zcash, contain multiple subliminal channels which can be exploited to embed several bytes of tagging information in users’ private transactions. In Chapter 5, instead, I report the cryptanalysis of the Legendre PRF, employed in a new consensus mechanism considered for adoption by the blockchain-based platform Ethereum, and attacks for further generalizations of this pseudo-random function, such as the Higher-Degree Legendre PRF, the Jacobi Symbol PRF, and the Power-Residue PRF. Lastly, in Part III, I present my line of research on public-key primitives based on elliptic curves. In Chapter 6, I will describe a backdooring procedure for primes so that whenever they appear as divisors of a large integer, the latter can be efficiently factored. This technique, based on elliptic curves Complex Multiplication theory, enables to eventually generate non-vulnerable certifiable semiprimes with unknown factorization in a multi-party computation setting, with no need to run a statistical semiprimality test common to other protocols. In Chapter 7, instead, I will report some attack optimizations and specific implementation design choices that allow breaking a reduced-parameters instance, proposed by Microsoft, of SIKE, a post-quantum key-encapsulation mechanism based on isogenies between supersingular elliptic curves

HOLMES: Efficient Distribution Testing for Secure Collaborative Learning

Using secure multiparty computation (MPC), organizations which own sensitive data (e.g., in healthcare, finance or law enforcement) can train machine learning models over their joint dataset without revealing their data to each other. At the same time, secure computation restricts operations on the joint dataset, which impedes computation to assess its quality. Without such an assessment, deploying a jointly trained model is potentially illegal. Regulations, such as the European Union\u27s General Data Protection Regulation (GDPR), require organizations to be legally responsible for the errors, bias, or discrimination caused by their machine learning models. Hence, testing data quality emerges as an indispensable step in secure collaborative learning. However, performing distribution testing is prohibitively expensive using current techniques, as shown in our experiments. We present HOLMES, a protocol for performing distribution testing efficiently. In our experiments, compared with three non-trivial baselines, HOLMES achieves a speedup of more than 10x for classical distribution tests and up to 10^4x for multidimensional tests. The core of HOLMES is a hybrid protocol that integrates MPC with zero-knowledge proofs and a new ZK-friendly and naturally oblivious sketching algorithm for multidimensional tests, both with significantly lower computational complexity and concrete execution costs

On the Use of the Legendre Symbol in Symmetric Cipher Design

This paper proposes the use of Legendre symbols as component gates in the design of ciphers tailored for use in cryptographic proof systems. Legendre symbols correspond to high-degree maps, but can be evaluated much faster. As a result, a cipher that uses Legendre symbols can offer the same security as one that uses high-degree maps but without incurring the penalty of a comparatively slow evaluation time. After discussing the design considerations induced by the use of Legendre symbol gates, we present a concrete design that follows this strategy, along with an elaborate security analysis thereof. This cipher is called Grendel