The dawn of the disruptive quantum computing scenario marks a serious threat to the existence of traditional cryptosystems. With laws such as Shorâs, capable of factoring large integers in polynomial time, and Groverâs, able to speed up brute-force key searches, these attacks make conventional public-key infrastructures increasingly vulnerable, whereas even symmetric ciphers lose good measure of their strength. In this article, we focus on an elaborative description of a patented method for quantum-secure key generation, wherein Qabbalah (QBLH) complexity is utilized in the geometric-symbolic realm, in conjunction with magic number squares, phi/pi coordinate weighting, and tetrahedral trinary state encoding. The proposed system of TriGate QBLH Quantum-Safe Encryption converts seed inputs to multidimensional keys that resist linear algebraic attacks owing to non-linear permutations, irrational constant weighting, and topological complexity. Normally, pseudo-random number generators spatialize entropy in Euclidean geometry, as opposed to the present technique that places entropy in a completely non-Euclidean domain, where classical as well as quantum adversaries find it hard to traverse. We describe the method in detail, present its benefits over lattice- and hash-based post-quantum schemes, and walk through an example of its implementation. Consideration is also given to its potential integration with PQC standards, blockchain authentication, and decentralized finance applications. The system fuses symbolic mathematics, such as the 231 Gates of QBLH, with trinary logic mapped onto tetrahedral states to not only create encryption keys but also verifiable geometric signatures. This represents a paradigm shift toward geometric cryptography, which may be a viable method to realize scalable and trustworthy digital infrastructure in a quantum-threatened environment.
Card-based cryptography enables players to compute logical and arithmetic operations securely, such as bitwise AND and addition of integers. Several multiparty computation protocols and zero-knowledge proof protocols utilizing these secure computations have been developed as its applications. However, the realization of an efficient protocol for an arithmetic operation other than addition and subtraction remains an open problem. This paper proposes card-based protocols, based on integer commitment, for multiplication, division, and square root. Compared to general constructions for protocols for these operations based on binary integer commitment, the proposed protocols exhibit superior simplicity and efficiency. Furthermore, these protocols introduce novel applications for card-based cryptography to secure statistical data aggregation.
A zero-knowledge proof (ZKP) is an interactive proof between a prover and a verifier, with the prover's goal to convince the verifier that a solution of a computation problem exists without revealing any information about the solution itself. A card-based physical ZKP is an implementation of a ZKP using physical cards. Several metrics are used to measure the efficiency of card-based ZKPs, such as the numbers of required cards and shuffles. In this paper, we propose the first card-based ZKP for a puzzle Sudoku that uses only one shuffle. To achieve this, we use a technique of applying private operations interactively to prepare inputs for the protocol.
Julio López Fenner, Carlos Castillo-Muñoz, Francisco Escobar, Ana Bustamante-Mora · 5 authors
Privacy-preserving secure multi-party computation protocols are known to face scalability and efficiency challenges in environments where participants hold distinct attributes of the same records (vertical partitioning) or controls a subset of complete records (horizontal partitioning), as in cross-institutional health data analysis or federated IoT analytics, mostly because of communication overhead and the need to address adaptability to large scale or heterogeneous settings. This work introduces a novel MPC protocol based on the DamgĂ„rdâJurik cryptosystem and Schnorr zero-knowledge proofs (ZKP), designed to securely aggregate private data distributed across a number of parties. By combining homomorphic encryption with non-interactive ZKPâs, the protocol ensures privacy, correctness, and scalability, aligning with the principles of privacy-enhancing technologies (PETs). Our approach minimizes data exposure, allowing participants to audit results, and achieves linear O(N) communication complexity, thus making it suitable for large-scale applications in secure data analytics and collaborative computing.
This article presents a comprehensive framework for implementing privacy-preserving credit card reward systems using Secure Multi-Party Computation (SMPC) technologies. Traditional reward architectures require extensive sharing of sensitive transaction data across multiple entities, creating significant privacy risks, security vulnerabilities, and regulatory compliance challenges. It leverages cryptographic advances to enable card issuers, payment networks, and merchant partners to collaborate on reward calculations,fraud detection, and personalized offers without revealing sensitive transaction details to one another. The article explores the evolution of privacy-preserving technologies in financial systems, comparing Fully Homomorphic Encryption, Zero-Knowledge Proofs, and SMPC approaches. A detailed case study of a travel rewards program implementation demonstrates how this framework ensures data remains protected throughout the entire process while maintaining the performance characteristics necessary for production deployment. The system provides comprehensive privacy protection, enhances fraud detection capabilities through secure collaboration, and facilitates compliance with evolving privacy regulations.Performance evaluations confirm the practical viability of the article, with minimal latency impact, strong scalability characteristics, and robust security guarantees. It contributes to the growing field of privacy-enhancing technologies for financial services and offers a viable solution to balance analytical utility with privacy protection in consumer-facing applications.
Elvira Albert, Samir Genaim, Daniel Kirchner, Enrique Martin-Martin
Program optimization is a key factor for green software. In the context of the Ethereum blockchain, optimization is particularly relevant because there is a fee to pay for each EVM (Ethereum Virtual Machine) instruction executed and also there exist bytecode-size limitations for deploying the software on the blockchain. Still, optimization of EVM code is not as widely spread as one could imagine. This is at least partly due to the lack of trust in the correctness of the tools, as security is even more relevant than efficiency in the blockchain context in which bugs may cause huge economical losses. This article develops a formal verification framework using Coq to ensure the security of EVM optimizations performed on jump-free sequences of EVM bytecode. By means of Coqâs theorem proving capabilities, we are able to automatically verify/certify that an optimized jump-free sequence of EVM opcodes is semantically equivalent to a given original one. We also present an extension to our framework that can handle inter-block optimizations that propagate global information across blocks. We have applied our tool to successfully prove the security of peephole optimizations performed by the standard Solidity compiler, and also to existing EVM superoptimization tools (namely GASOL and Superstack) in which we have found bugs that have been reported and fixed.
Recently, the practical applications of advanced cryptographic protocols, such as Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero Knowledge Proofs (ZKP), have spurred the development of a series of new symmetric encryption primitives. These novel symmetric encryption primitives, referred to as Arithmetization-Oriented (AO) ciphers, aim to minimize the number of field multiplications in large finite fields, including đœ2n or đœp. In order to evaluate the algebraic degrees of AO ciphers over đœ2n, the general monomial prediction technique was proposed by Cui et al. at ASIACRYPT 2022. However, when using their searching tool to evaluate the algebraic degrees of AO ciphers with complex affine layers, the efficiency is low, preventing solutions within a predetermined timeframe. In this study, we extend the propagation rules of monomials for field-based operations and present an automatic searching tool based on Mixed Integer Linear Programming (MILP) and Boolean Satisfiability (SAT) Problem for evaluating the upper bound of the algebraic degrees. Moreover, to accurately calculate the algebraic degrees of monomials in the SAT model, we improve the sequence encoding method, enabling it to accurately determine whether the monomials of degree d exist in the output. We apply our new searching tool to various AO ciphers, including Chaghri, MiMC, and Ciminion. For Chaghri, we compare our results with the Coefficient Grouping technique proposed by Liu et al. at EUROCRYPT 2023, and our results yield tighter upper bounds compared to Liu et al.âs findings. Additionally, we evaluate the algebraic degrees of Chaghri and MiMC with arbitrary complex affine layers and obtain tighter bounds compared to the results from Liu et al. at CRYPTO 2023. Regarding Ciminion, we have observed that starting from the 4-th round, the upper bound on the algebraic degrees for each round actually 1 degree lower than the previous bound. Our searching tool enables a more precise evaluation of the algebraic degrees of AO ciphers, contributing to a deeper understanding of the design and analysis of such primitives.
Imagine sharing that you know a secret without revealing the secret itself. This is what zero-knowledge proofs (ZKPs) aim to do. In ZKPs there is a prover who claims knowledge of something and a verifier who checks this claim. The goal of this project is to further explore current technologies revolving around ZKPs and understand possible adaptations to an everyday application beyond blockchain use cases. To explore the practical use of ZKPs, this project introduces a web puzzle application that keeps the solutions of individual users private using ZKPs. A user can solve a logic- based puzzle like Binairo or Sudoku and check the validity of their solution by sharing only a ZKP of the solution. That way, the solution never leaves the userâs device. To further strengthen the security, the following two checks are implemented: (1) making sure that the solution matches the original puzzle, and (2) integrating the userâs ID during the generation process. These checks prevent users from reusing a proof to âsolveâ other puzzles or for the proof to be stolen by another user. To implement this application, different ZKP frameworks are considered. Circom and snarkjs are selected because of their active development, clear documentation and good web development capabilities. The final result is a secure application that demonstrates how ZKPs can be applied in a realistic and practical way. This highlights their broader potential in digital security. In most applications, the impact of ZKPs is intentionally hidden, as good cybersecurity aims to operate in the background. The ZKP-Puzzles application puts the ZKPs in the spotlight and visualizes how ZKPs work.
Open access
graph theory and CDMA systems
Graph Labeling and Dimension Problems
Advanced Steganography and Watermarking Techniques
The advent of 6G networks places very high demands on ultra-low latency, high throughput, and quantum-secure communication to power Industry 5.0 use cases. Traditional blockchain architectures, given their decentralized and secure nature, often fall short in meeting the performance and security requirements of such an ecosystem. In this paper, we present a post-quantum blockchain architecture that employs CRYSTALS-Dilithium and SPHINCS+ for digital signatures and block and transaction verification, respectively, along with zk-STARKs to facilitate scalable zero-knowledge proof-based privacy, and a DPoS+VDFs consensus protocol to satisfy fairness and efficiency. We prototyped and evaluated the proposed framework with a benchmarking setup composed of Python, PQClean, liboqs, and Google Benchmark tools. Experimental results demonstrate that the system achieves a 40% reduction in latency, a 35% increase in transaction throughput, and a 25% reduction in computational overhead due to the integration of zk-STARK. Furthermore, finality time for consensus was reduced by 30% by using the hybrid DPoS-VDF consensus approach. Comparative studies with various lattice-, hash-, and code-based quantum cryptographic primitives have shown that CRYSTALS-Dilithium and SPHINCS+ outperform others in key generation, signing, and verification performance indicators, and thus qualify as optimal solutions for edge-centric 6G infrastructures. Conversely, zk-Starks showed near-optimal timeliness and verification effectiveness among the several examined zero-knowledge proof schemes. These findings validate the proposed framework as an efficient, scalable, and performance-enhanced blockchain solution for securing industrial ecosystems with latency sensitivity in a 6G-enabled environment.
Aleksandra Szczegielniak-Rekiel, Krzysztof Kanciak, Jan M. Kelner
This study explored the diverse applications of zero-knowledge proofs (ZKPs) in next-generation network technologies, particularly in fifth-generation (5G) and emerging sixth-generation (6G) systems. ZKPs are cryptographic methods that enable one party to prove the validity of a statement without revealing the statement itself, thereby offering significant advantages in privacy-preserving authentication and authorization. Given these properties, ZKPs have garnered increasing research attention in contexts such as the Internet of Things (IoT), vehicular communications, and telecommunication protocols. To the best of our knowledge, this is the first study to provide a comprehensive, taxonomy-driven analysis of ZKP applications specifically designed for 5G and beyond. We categorize existing solutions according to the type of application, the underlying cryptographic technology, maturity level, and relevance to 6G. Furthermore, this paper examines how ZKPs can help mitigate various cybersecurity threats, such as distributed denial-of-service (DDoS) attacks, man-in-the-middle attacks, and location tracking. We also assess recent advancements in ZKP acceleration techniques and highlight the key implementation challenges. Finally, this study outlines promising directions for future research in this rapidly evolving field.
Open access
2 source records
Physical Unclonable Functions (PUFs) and Hardware Security
The introduction of non-fungible tokens (NFTs) has revolutionised digital ownership and asset management in the quickly changing environment of blockchain technology. NFTs are a kind of token that cannot be exchanged for another token. On the other hand, as the market for NFTs continues to grow, customers are becoming more and more interested in interoperability across various blockchain networks. Cross-chain non-fungible token markets have been developed as a result of this necessity. These marketplaces make it possible for different blockchain ecosystems to engage in transactions and interactions with one another. LayerZero and Chainlink are two technologies that are very significant in addressing the difficulty of interoperability across different chains.
Abstract A zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.
Abstract A zero-knowledge proof protocol is a cryptographic protocol in which a prover, who knows the witness to a statement, can convince a verifier that the statement is true without revealing any information about the witness. Although zero-knowledge proof protocols are typically executed on electronic computers, there is a line of research to design zero-knowledge proof protocols based on physical objects (e.g., a deck of cards). This is called physical zero-knowledge proof. In this paper, we construct a physical zero-knowledge proof protocol for a logical puzzle called Sukoro. Sukoro has many cells on the puzzle board, like Sudoku, where each cell must be empty or filled with a number from one to four, and each number must match the number of adjacent filled cells, and the same numbers must not be adjacent to each other. In addition, it has a rule that all filled cells must be connected, which is called the connectivity condition. Although some existing protocols deal with the connectivity condition, all existing methods are interactive , which requires the proverâs knowledge to determine how the cards are manipulated during the execution of the protocols. In this paper, we give a new method for verifying the connectivity condition in the non-interactive setting, which means that the protocol can be executed without the proverâs knowledge, and construct a physical zero-knowledge proof protocol for Sukoro.
A Zero-Knowledge Proof (ZKP) protocol allows a participant to prove the knowledge of some secret without revealing any information about it. While such protocols are typically executed by computers, there exists a line of research proposing physical instances of ZKP protocols. Up to now, many card-based ZKP protocols for pen-and-pencil puzzles, like Sudoku, have been designed. Those games, mostly edited by Nikoli, have simple rules, yet designing them in card-based ZKP protocols is non-trivial. In this work, we propose a card-based ZKP protocol for Usowan, a Nikoli game. In Usowan, for each room of a puzzle instance, there is exactly one piece of false information. The goal of the game is to detect this wrong data amongst the correct data and also to satisfy the other rules. Designing a card-based ZKP protocol to deal with the property of detecting a liar has never been done. In some sense, we propose a physical ZKP for hiding of a liar. This work extends a previous paper appearing in Ref. [1]. In this extension, we propose two other protocols, for Herugolf and Five Cells. The puzzles are specifically chosen because each of those three puzzles shares a common constraint, connectivity. However, showing the connected configuration cannot be done with generic approach and brings new construction to the existing connectivity ZKP protocol. Indeed, in Herugolf, the connectivity is handled with a given length of cell which is decremental (i.e., the length of each connected cell decreases by one at each step). For Five Cells, there is an additional step in the setup allowing to encode all the information needed to ensure a valid ZKP protocol.
Open access
graph theory and CDMA systems
Graph Labeling and Dimension Problems
Advanced Steganography and Watermarking Techniques
Pseudorandom Correlation Functions (PCFs) allow two parties, given correlated evaluation keys, to locally generate arbitrarily many pseudorandom correlated strings, e.g. Oblivious Transfer (OT) correlations, which can then be used by the two parties to jointly run secure computation protocols. In this work, we provide a novel and simple approach for constructing PCFs for OT correlation, by relying on constrained pseudorandom functions for a class of constraints containing a weak pseudorandom function (wPRF). We then show that tweaking the Naor-Reingold pseudorandom function and relying on low-complexity pseudorandom functions allow us to instantiate our paradigm. We further extend our ideas to obtain efficient public-key PCFs, which allow the distribution of correlated keys between parties to be non-interactive: each party can generate a pair of public/secret keys, and any pair of parties can locally derive their correlated evaluation key by combining their secret key with the other partyâs public key. In addition to these theoretical contributions, we detail various optimizations and provide concrete instantiations of our paradigm relying on the Boneh-Ishai-PasselĂšgue-Sahai-Wu wPRF and the Goldreich-Applebaum-Raykov wPRF. Putting everything together, we obtain public-key PCFs with a throughput of 15kâ40k OT/s, which is of a similar order of magnitude to the state-of-the-art interactive PCFs and about 4 orders of magnitude faster than state-of-the art public-key PCFs. As a side result, we also show that public-key PCFs can serve as a building block to construct reusable designated-verifier non-interactive zero-knowledge proofs (DV-NIZK) for NP. Combined with our instantiations, this yields simple and efficient reusable DV-NIZKs for NP in pairing-free groups.
Abstract This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K -algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t -designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341â345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21â23, 1983) of the SauerâShelahâPerles Lemma, we produce a new one based only in duality in the $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -algebra $$\mathbb {Q}[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set $$[n]:=\{1,2,\ldots , n\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>âŠ</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . All results are equally true if we replace $$\mathbb {Q}[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> by $$K[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , where K is any perfect field of characteristics $$\not =2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>â </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.
The rise of modern cryptographic protocols such as Zero-Knowledge proofs and secure Multi-party Computation has led to an increased demand for a new class of symmetric primitives. Unlike traditional platforms such as servers, microcontrollers, and desktop computers, these primitives are designed to be implemented in arithmetical circuits. In terms of security evaluation, arithmetization-oriented primitives are more complex compared to traditional symmetric cryptographic primitives. The arithmetization-oriented permutation Grendel employs the Legendre Symbol to increase the growth of algebraic degrees in its nonlinear layer. To analyze the security of Grendel thoroughly, it is crucial to investigate its resilience against algebraic attacks. This paper presents a preimage attack on the sponge hash function instantiated with the complete rounds of the Grendel permutation, employing algebraic methods. A technique is introduced that enables the elimination of two complete rounds of substitution permutation networks (SPN) in the sponge hash function without significant additional cost. This method can be combined with univariate root-finding techniques and Gröbner basis attacks to break the number of rounds claimed by the designers. By employing this strategy, our attack achieves a gain of two additional rounds compared to the previous state-of-the-art attack. With no compromise to its security margin, this approach deepens our understanding of the design and analysis of such cryptographic primitives.
Modern cryptographic protocols such as zero-knowledge proofs and secure multi-party computation have increased the demand for a novel category of symmetric primitives. These primitives are not optimized for traditional platforms such as servers, microcontrollers, and desktop computers but rather for their ability to be implemented in arithmetic circuits. To enable efficient arithmetic operations, they define operations over larger finite fields and use low-degree invertible functions to construct their non-linear layers. Grendel is an arithmetization-oriented permutation that leverages the Legendre Symbol to enhance the growth of algebraic degrees in its non-linear layer. In this paper, we present a preimage attack on the sponge hash function instantiated with the full rounds of the Grendel permutation using algebraic methods. We introduce a technique that allows us to eliminate two full rounds of substitution permutation networks (SPN) in the sponge hash function with minimal or no additional cost. This method can be combined with univariate root-finding techniques and Gröbner basis attacks to break the number of rounds claimed by the designers. By utilizing this strategy, our attack achieves an improvement of two additional rounds compared to the previous state-of-the-art attack. While not breaking its security margin, it allows us to further understand the design and analysis of such cryptographic primitives.
Motivated by new applications such as secure Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero-Knowledge proofs (ZK), many MPC-, FHE- and ZK-friendly symmetric-key primitives that minimize the< number of multiplications over Fp for a large prime p have been recently proposed in the literature. These symmetric primitives are usually defined via invertible functions, including (i) Feistel and Lai-Massey schemes and (ii) SPN constructions instantiated with invertible non-linear S-Boxes. However, the âinvertibilityâ property is actually never required in any of the mentioned applications.In this paper, we discuss the possibility to set up MPC-/FHE-/ZK-friendly symmetric primitives instantiated with non-invertible bounded surjective functions. In contrast to one-to-one functions, each output of a l-bounded surjective function admits at most l pre-images. The simplest example is the square map x â x2 over Fp for a prime p â„ 3, which is (obviously) 2-bounded surjective. When working over Fnp for n â„ 2, we set up bounded surjective functions by re-considering the recent results proposed by Grassi, Onofri, Pedicini and Sozzi at FSE/ToSC 2022 as starting points. Given a quadratic local map F : Fmp â Fp for m â {1, 2, 3}, they proved that the shift-invariant non-linear function over Fnp defined as SF (x0, x1, . . . , xnâ1) = y0â„y1â„ . . . â„ynâ1 where yi := F(xi, xi+1) is never invertible for any n â„ 2 · m â 1. Here, we prove that âą the quadratic function F : Fmp â Fp for m â {1, 2} that minimizes the probability of having a collision for SF over Fnp is of the form F(x0, x1) = x20 + x1 (or equivalent);âą the function SF over Fnp defined as before via F(x0, x1) = x20 +x1 (or equivalent) is 2n-bounded surjective.As concrete applications, we propose modified versions of the MPC-friendly schemes MiMC, HadesMiMC, and (partially of) Hydra, and of the FHE-friendly schemes Masta, Pasta, and Rubato. By instantiating them with the bounded surjective quadratic functions proposed in this paper, we are able to improve the security and/or the performances in the target applications/protocols.
In recent years, cryptocurrencies, cryptoassets, electronic coins, tokens, non-fungible tokens, and other various terms for electronic assets have gained prodigious attention in the financial world. From the spike (and subsequent drop) in value of Bitcoin, to people spending millions of dollars on pixelated pictures of punks, the market for these assets has been extremely active despite its ups and downs. However, in addition to potential financial success via crypto markets, the development of crypto technology has allowed for a transformation of how individuals and institutions think of currency, financial security, and access to information Part I of this Comment explains what a cryptoasset is, as well as the current tax regime applicable to them. Part II defines like-kind exchanges and provides the historical context for the nonrecognition event. It also considers the IRSâs recent guidance pertaining to crypto like-kind exchanges. Part III puts forth this Commentâs main arguments for allowing crypto-for-crypto exchanges to qualify as like-kind exchanges.