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

Polynomial IOPs for Linear Algebra Relations

This paper proposes new Polynomial IOPs for arithmetic circuits. They rely on the monomial coefficient basis to represent the matrices and vectors arising from the arithmetic constraint satisfaction system, and build on new protocols for establishing the correct computation of linear algebra relations such as matrix-vector products and Hadamard products. Our protocols give rise to concrete proof systems with succinct verification when compiled down with a cryptographic compiler whose role is abstracted away in this paper. Depending only on the compiler, the resulting SNARKs are either transparent or rely on a trusted setup

Short Solutions to Nonlinear Systems of Equations

This paper presents a new hard problem for use in cryptography, called Short Solutions to Nonlinear Equations (SSNE). This problem generalizes the Multivariate Quadratic (MQ) problem by requiring the solution be short; as well as the Short Integer Solutions (SIS) problem by requiring the underlying system of equations be nonlinear. The joint requirement causes common solving strategies such as lattice reduction or Gröbner basis algorithms to fail, and as a result SSNE admits shorter representations of equally hard problems. We show that SSNE can be used as the basis for a provably secure hash function. Despite failing to find public key cryptosystems relying on SSNE, we remain hopeful about that possibility

New Techniques for Electronic Voting

This paper presents a novel unifying framework for electronic voting in the universal composability model that includes a property which is new to universal composability but well-known to voting systems: universal verifiability. Additionally, we propose three new techniques for secure electronic voting and prove their security and universal verifiability in the universal composability framework. 1. A tally-hiding voting system, in which the tally that is released consists of only the winner without the vote count. Our proposal builds on a novel solution to the millionaire problem which is of independent interest. 2. A self-tallying vote, in which the tally can be calculated by any observer as soon as the last vote has been cast --- but before this happens, no information about the tally is leaked. 3. Authentication of voting credentials, which is a new approach for electronic voting systems based on anonymous credentials. In this approach, the vote authenticates the credential so that it cannot afterwards be used for any other purpose but to cast that vote. We propose a practical voting system that instantiates this high-level concept

Quantum LLL with an Application to Mersenne Number Cryptosystems

In this work we analyze the impact of translating the well-known LLL algorithm for lattice reduction into the quantum setting. We present the first (to the best of our knowledge) quantum circuit representation of a lattice reduction algorithm in the form of explicit quantum circuits implementing the textbook LLL algorithm. Our analysis identifies a set of challenges arising from constructing reversible lattice reduction as well as solutions to these challenges. We give a detailed resource estimate with the Toffoli gate count and the number of logical qubits as complexity metrics. As an application of the previous, we attack Mersenne number cryptosystems by Groverizing an attack due to Beunardeau et. al that uses LLL as a subprocedure. While Grover\u27s quantum algorithm promises a quadratic speedup over exhaustive search given access to a oracle that distinguishes solutions from non-solutions, we show that in our case, realizing the oracle comes at the cost of a large number of qubits. When an adversary translates the attack by Beunardeau et al. into the quantum setting, the overhead of the quantum LLL circuit may be as large as $2^{52}$ qubits for the text-book implementation and $2^{33}$ for a floating-point variant

A framework for cryptographic problems from linear algebra

We introduce a general framework encompassing the main hard problems emerging in lattice-based cryptography, which naturally includes the recently proposed Mersenne prime cryptosystem, but also problems coming from code-based cryptography. The framework allows to easily instantiate new hard problems and to automatically construct plausibly post-quantum secure primitives from them. As a first basic application, we introduce two new hard problems and the corresponding encryption schemes. Concretely, we study generalisations of hard problems such as SIS, LWE and NTRU to free modules over quotients of Z[X] by ideals of the form (f,g), where f is a monic polynomial and g∈Z[X] is a ciphertext modulus coprime to f. For trivial modules (i.e. of rank one), the case f=Xn+1 and g=q∈Z>1 corresponds to ring-LWE, ring-SIS and NTRU, while the choices f=Xn−1 and g=X−2 essentially cover the recently proposed Mersenne prime cryptosystems. At the other extreme, when considering modules of large rank and letting deg(f)=1, one recovers the framework of LWE and SIS

Mutator Sets and their Application to Scalable Privacy

A mutator set is a cryptographic data structure for authenticating operations on a changing set of data elements called items. Informally: - There is a short commitment to the set. - There are succinct membership proofs for elements of the set. - It is possible to update the commitment as well as the membership proofs with minimal effort as new items are added to the set or as existing items are removed from it. - Items cannot be removed before they were added. - It is difficult to link an item\u27s addition to the set to its removal from the set, except when using information available only to the party that generated it. This paper formally defines the notion, motivates its existence with an application to scalable privacy in the context of cryptocurrencies, and proposes an instantiation inspired by Merkle mountain ranges and Bloom filters

Lattice-Based Cryptography in Miden VM

This note discusses lattice-based cryptography over the field with $p= 2^{64} - 2^{32} + 1$ elements, with an eye to supporting lattice-based cryptography operations in virtual machines such as Miden VM that operate natively over this field. It discusses how to support Dilithium and Falcon, two lattice-based signature scheme recently selected by the NIST PQC project; and proposes parameters for efficient public key encryption and publicly re-randomizable commitments modulo $p$

SoK: Gröbner Basis Algorithms for Arithmetization Oriented Ciphers

Many new ciphers target a concise algebraic description for efficient evaluation in a proof system or a multi-party computation. This new target for optimization introduces algebraic vulnerabilities, particularly involving Gröbner basis analysis. Unfortunately, the literature on Gröbner bases tends to be either purely mathematical, or focused on small fields. In this paper, we survey the most important algorithms and present them in an intuitive way. The discussion of their complexities enables researchers to assess the security of concrete arithmetization-oriented ciphers. Aside from streamlining the security analysis, this paper helps newcomers enter the field

Key Encapsulation from Noisy Key Agreement in the Quantum Random Oracle Model

A multitude of post-quantum key encapsulation mechanisms (KEMs) and public key encryption (PKE) schemes implicitly rely on a protocol by which Alice and Bob exchange public messages and converge on secret values that are identical up to some small noise. By our count, 24 out of 49 KEM or PKE submissions to the NIST Post-Quantum Cryptography Standardization project follow this strategy. Yet the notion of a noisy key agreement (NKA) protocol lacks a formal definition as a primitive in its own right. We provide such a formalization by defining the syntax and security for an NKA protocol. This formalization brings out four generic problems, called A and B State Recovery, Noisy Key Search and Noisy Key Distinguishing, whose solutions must be hard in the quantum computing model. Informally speaking, these can be viewed as noisy, quantum-resistant counterparts of the problems arising from the classical Diffie-Hellman type protocols. We show that many existing proposals contain an NKA component that fits our formalization and we reveal the induced concrete hardness assumptions. The question arises whether considering NKA as an independent primitive can help provide modular designs with improved efficiency and/or proofs. As the second contribution of this paper, we answer this question positively by presenting a generic transform from a secure NKA protocol to an IND-CCA secure KEM in the quantum random oracle model, with a security bound tightly related to the NKD problem. This transformation is essentially the same as that of the NIST candidate Ramstake. While establishing the security of Ramstake was our initial objective, the collection of tools that came about as a result of this journey is of independent interest