We propose a new class of post-quantum digital signature schemes that: (a) derive their security entirely from the security of symmetric-key primitives, believed to be quantum-secure, and (b) have extremely small keypairs, and, (c) are highly parameterizable. In our signature constructions, the public key is an image y=f(x) of a one-way function f and secret key x. A signature is a non-interactive zero-knowledge proof of x, that incorporates a message to be signed. For this proof, we leverage recent progress of Giacomelli et al. (USENIX'16) in constructing an efficient Σ-protocol for statements over general circuits. We improve this Σ-protocol to reduce proof sizes by a factor of two, at no additional computational cost. While this is of independent interest as it yields more compact proofs for any circuit, it also decreases our signature sizes. We consider two possibilities to make the proof non-interactive: the Fiat-Shamir transform and Unruh's transform (EUROCRYPT'12, '15,'16). The former has smaller signatures, while the latter has a security analysis in the quantum-accessible random oracle model. By customizing Unruh's transform to our application, the overhead is reduced to 1.6x when compared to the Fiat-Shamir transform, which does not have a rigorous post-quantum security analysis. We implement and benchmark both approaches and explore the possible choice of f, taking advantage of the recent trend to strive for practical symmetric ciphers with a particularly low number of multiplications and end up using Low MC (EUROCRYPT'15).

/pdf/post-quantum-zero-knowledge-and-signatures-from-symmetric-3odro7h178.pdf

Post-Quantum Zero-Knowledge and Signatures from Symmetric-Key Primitives

Using whitening keys is a well understood mean of increasing the key-length of any given cipher. Especially as it is known ever since Grover’s seminal work that the effective key-length is reduced by a factor of two when considering quantum adversaries, it seems tempting to use this simple and elegant way of extending the key-length of a given cipher to increase the resistance against quantum adversaries. However, as we show in this work, using whitening keys does not increase the security in the quantum-CPA setting significantly. For this we present a quantum algorithm that breaks the construction with whitening keys in essentially the same time complexity as Grover’s original algorithm breaks the underlying block cipher. Technically this result is based on the combination of the quantum algorithms of Grover and Simon for the first time in the cryptographic setting.

/pdf/grover-meets-simon-quantumly-attacking-the-fx-construction-4djusu8n1n.pdf

Grover Meets Simon – Quantumly Attacking the FX-construction

This paper studies indistinguishability against quantum chosen-ciphertext attacks (IND-qCCA security) of key-encapsulation mechanisms (KEMs) in quantum random oracle model (QROM). We show that the SXY conversion proposed by Saito, Yamakawa, and Xagawa (EUROCRYPT 2018) and the HU conversion proposed by Jiang, Zhang, and Ma (PKC 2019) turn a weakly-secure deterministic public-key encryption scheme into an IND-qCCA-secure KEM scheme in the QROM. The proofs are very similar to that for the IND-CCA security in the QROM, easy to understand, and as tight as the original proofs.

(Tightly) QCCA-Secure Key-Encapsulation Mechanism in the Quantum Random Oracle Model

At CRYPTO 2013, Boneh and Zhandry initiated the study of quantum-secure encryption. They proposed first indistinguishability definitions for the quantum world where the actual indistinguishability only holds for classical messages, and they provide arguments why it might be hard to achieve a stronger notion. In this work, we show that stronger notions are achievable, where the indistinguishability holds for quantum superpositions of messages. We investigate exhaustively the possibilities and subtle differences in defining such a quantum indistinguishability notion for symmetric-key encryption schemes. We justify our stronger definition by showing its equivalence to novel quantum semantic-security notions that we introduce. Furthermore, we show that our new security definitions cannot be achieved by a large class of ciphers --- those which are quasi-preserving the message length. On the other hand, we provide a secure construction based on quantum-resistant pseudorandom permutations; this construction can be used as a generic transformation for turning a large class of encryption schemes into quantum indistinguishable and hence quantum semantically secure ones. Moreover, our construction is the first completely classical encryption scheme shown to be secure against an even stronger notion of indistinguishability, which was previously known to be achievable only by using quantum messages and arbitrary quantum encryption circuits.

/pdf/semantic-security-and-indistinguishability-in-the-quantum-4mdtrv451p.pdf

Semantic Security and Indistinguishability in the Quantum World

The cryptographic community has widely acknowledged that the emergence of large quantum computers will pose a threat to most current public-key cryptography. Primitives that rely on order-finding problems, such as factoring and computing Discrete Logarithms, can be broken by Shor’s algorithm ([49]).

/pdf/an-efficient-quantum-collision-search-algorithm-and-2cu1r4j9in.pdf

An Efficient Quantum Collision Search Algorithm and Implications on Symmetric Cryptography

Due to Shor's algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.

We study applications of a quantum procedure called Simon's algorithm the simplest quantum period finding algorithm in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon's algorithm: finding a collision requires $$\varOmega 2^{n/2}$$ queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only On queries in the quantum model.

We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure with a quantum-secure PRF.

Second, we show that Simon's algorithm can also be applied to slide attacks, leading to an exponential speed-up of a classical symmetric cryptanalysis technique in the quantum model.

/pdf/breaking-symmetric-cryptosystems-usingzquantumzperiod-40xjh0ou6p.pdf

María źNaya-Plasencia

Papers

Breaking Symmetric Cryptosystems UsingźQuantumźPeriod Finding