Isogeny formulae on extended Jacobi Quartic curves

Cryptography based on elliptic curves is endowed with efficient methods for public-key cryptography. Recent research has shown the superiority of the Montgomery and Edwards curves over the Weierstrass curves as they require fewer arithmetic operations. Using these modern curves has, however, introduced several challenges to the cryptographic algorithm's design, opening up new opportunities for optimization. Our main objective is to propose algorithmic optimizations and implementation techniques for cryptographic algorithms based on elliptic curves. In order to speed up the execution of these algorithms, our approach relies on the use of extensions to the instruction set architecture. In addition to those specific for cryptography, we use extensions that follow the Single Instruction, Multiple Data (SIMD) parallel computing paradigm. In this model, the processor executes the same operation over a set of data in parallel. We investigated how to apply SIMD to the implementation of elliptic curve algorithms. As part of our contributions, we design parallel algorithms for prime field and elliptic curve arithmetic. We also design a new three-point ladder algorithm for the scalar multiplication P+kQ, and a faster formula for calculating 3P on Montgomery curves. These algorithms have found applicability in isogeny-based cryptography. Using SIMD extensions such as SSE, AVX, and AVX2, we develop optimized implementations of the following cryptographic algorithms: X25519, X448, SIDH, ECDH, ECDSA, EdDSA, and qDSA. Performance benchmarks show that these implementations are faster than existing implementations in the state of the art. Our study confirms that using extensions to the instruction set architecture is an effective tool for optimizing implementations of cryptographic algorithms based on elliptic curves. May this be an incentive not only for those seeking to speed up programs in general but also for computer manufacturers to include more advanced extensions that support the increasing demand for cryptography.

Cryptography based on elliptic curves is endowed with efficient methods for public-key cryptography. Recent research has shown the superiority of the Montgomery and Edwards curves over the Weierstrass curves as they require fewer arithmetic operations. Using these modern curves has, however, introduced several challenges to the cryptographic algorithm’s design, opening up new opportunities for optimization. Our main objective is to propose algorithmic optimizations and implementation techniques for cryptographic algorithms based on elliptic curves. In order to speed up the execution of these algorithms, our approach relies on the use of extensions to the instruction set architecture. In addition to those specific for cryptography, we use extensions that follow the Single Instruction, Multiple Data (SIMD) parallel computing paradigm. In this model, the processor executes the same operation over a set of data in parallel. We investigated how to apply SIMD to the implementation of elliptic curve algorithms. As part of our contributions, we design parallel algorithms for prime field and elliptic curve arithmetic. We also design a new three-point ladder algorithm for the scalar multiplication P + kQ, and a faster formula for calculating 3P on Montgomery curves. These algorithms have found applicability in isogeny-based cryptography. Using SIMD extensions such as SSE, AVX, and AVX2, we develop optimized implementations of the following cryptographic algorithms: X25519, X448, SIDH, ECDH, ECDSA, EdDSA, and qDSA. Performance benchmarks show that these implementations are faster than existing implementations in the state of the art. Our study confirms that using extensions to the instruction set architecture is an effective tool for optimizing implementations of cryptographic algorithms based on elliptic curves. May this be an incentive not only for those seeking to speed up programs in general but also for computer manufacturers to include more advanced extensions that support the increasing demand for cryptography.

Cryptography based on elliptic curves is endowed with efficient methods for public-key cryptography. Recent research has shown the superiority of the Montgomery and Edwards curves over the Weierstrass curves as they require fewer arithmetic operations. Using these modern curves has, however, introduced several challenges to the cryptographic algorithm’s design, opening up new opportunities for optimization. Our main objective is to propose algorithmic optimizations and implementation techniques for cryptographic algorithms based on elliptic curves. In order to speed up the execution of these algorithms, our approach relies on the use of extensions to the instruction set architecture. In addition to those specific for cryptography, we use extensions that follow the Single Instruction, Multiple Data (SIMD) parallel computing paradigm. In this model, the processor executes the same operation over a set of data in parallel. We investigated how to apply SIMD to the implementation of elliptic curve algorithms. As part of our contributions, we design parallel algorithms for prime field and elliptic curve arithmetic. We also design a new three-point ladder algorithm for the scalar multiplication P + kQ, and a faster formula for calculating 3P on Montgomery curves. These algorithms have found applicability in isogeny-based cryptography. Using SIMD extensions such as SSE, AVX, and AVX2, we develop optimized implementations of the following cryptographic algorithms: X25519, X448, SIDH, ECDH, ECDSA, EdDSA, and qDSA. Performance benchmarks show that these implementations are faster than existing implementations in the state of the art. Our study confirms that using extensions to the instruction set architecture is an effective tool for optimizing implementations of cryptographic algorithms based on elliptic curves. May this be an incentive not only for those seeking to speed up programs in general but also for computer manufacturers to include more advanced extensions that support the increasing demand for cryptography.

Cryptography based on elliptic curves is endowed with efficient methods for public-key cryptography. Recent research has shown the superiority of the Montgomery and Edwards curves over the Weierstrass curves as they require fewer arithmetic operations. Using these modern curves has, however, introduced several challenges to the cryptographic algorithm’s design, opening up new opportunities for optimization. Our main objective is to propose algorithmic optimizations and implementation techniques for cryptographic algorithms based on elliptic curves. In order to speed up the execution of these algorithms, our approach relies on the use of extensions to the instruction set architecture. In addition to those specific for cryptography, we use extensions that follow the Single Instruction, Multiple Data (SIMD) parallel computing paradigm. In this model, the processor executes the same operation over a set of data in parallel. We investigated how to apply SIMD to the implementation of elliptic curve algorithms. As part of our contributions, we design parallel algorithms for prime field and elliptic curve arithmetic. We also design a new three-point ladder algorithm for the scalar multiplication P + kQ, and a faster formula for calculating 3P on Montgomery curves. These algorithms have found applicability in isogeny-based cryptography. Using SIMD extensions such as SSE, AVX, and AVX2, we develop optimized implementations of the following cryptographic algorithms: X25519, X448, SIDH, ECDH, ECDSA, EdDSA, and qDSA. Performance benchmarks show that these implementations are faster than existing implementations in the state of the art. Our study confirms that using extensions to the instruction set architecture is an effective tool for optimizing implementations of cryptographic algorithms based on elliptic curves. May this be an incentive not only for those seeking to speed up programs in general but also for computer manufacturers to include more advanced extensions that support the increasing demand for cryptography.

SCOPUS: ed.j

A post-quantum signcryption scheme using isogeny based cryptography

Cryptography has been used from time immemorial for preserving the confidentiality of data/information in storage or transit. Thus, cryptography research has also been evolving from the classical Caesar cipher to the modern cryptosystems, based on modular arithmetic to the contemporary cryptosystems based on quantum computing. The emergence of quantum computing poses a major threat to the modern cryptosystems based on modular arithmetic, whereby even the computationally hard problems which constitute the strength of the modular arithmetic ciphers could be solved in polynomial time. This threat triggered post-quantum cryptography research to design and develop post-quantum algorithms that can withstand quantum computing attacks. This paper provides an overview of the various research directions that have been explored in post-quantum cryptography and, specifically, the various code-based cryptography research dimensions that have been explored. Some potential research directions that are yet to be explored in code-based cryptography research from the perspective of codes is a key contribution of this paper.

Isogeny-based cryptography is one of post-quantum cryptography based on the difficulty of the isogeny problem. The central object is a one-dimensional isogeny, that is, an isogeny between elliptic curves. However, in recent years, not only one-dimensional isogenies but also two-dimensional isogenies have been used to isogeny-based cryptography. Such a two-dimensional isogeny is an isogeny between products of elliptic curves, and it is computed by decomposing to prime degree isogenies. The decomposed isogenies are called a chain of isogenies. Especially, for the decomposition, the first isogeny of the chain has the domain as a product of elliptic curves E1 × E2, and a point x to compute the image is of the form of x = (x(1) , 0E2) ∈ E1 × E2 for x(1) ∈ E1. In this paper, we focus on odd prime degree isogenies with the domain as a product of elliptic curves. For such an isogeny, we propose formulas and explicit algorithms based on the formulas. As a result, the computation of the image of a point (x(1) , 0E2) is improved compared to the existing method. For the application, when we compute an odd degree isogeny chain, this result allows efficient computation of the dominant isogeny in the chain by placing the isogeny with the largest prime degree first. In addition, we implemented the proposed algorithm in SageMath and confirmed its improved efficiency over the existing algorithm by comparing running times.

Cryptography is an art of hiding the significant data or information with some other codes. It is a practice and study of securing information and communication. Thus, cryptography prevents third party intervention over the data communication. The cryptography technology transforms the data into some other form to enhance security and robustness against the attacks. The thrust of enhancing the security among data transfer has been emerged ever since the need of Artificial Intelligence field came into a market. Therefore, modern way of computing cryptographic algorithm came into practice such as AES, 3DES, RSA, Diffie-Hellman and ECC. These public-key encryption techniques now in use are based on challenging discrete logarithms for elliptic curves and complex factorization. However, those two difficult problems can be effectively solved with the help of sufficient large-scale quantum computer. The Post Quantum Cryptography (PQC) aims to deal with an attacker who has a large-scale quantum computer. Therefore, it is essential to build a robust and secure cryptography algorithm against most vulnerable pre-quantum cryptography methods. That is called ‘Post Quantum Cryptography’. Therefore, the present crypto system needs to propose encryption key and signature size is very large.in addition to careful prediction of encryption/decryption time and amount of traffic over the communication wire is required. The post-quantum cryptography (PQC) article discusses different families of post-quantum cryptosystems, analyses the current status of the National Institute of Standards and Technology (NIST) post-quantum cryptography standardisation process, and looks at the difficulties faced by the PQC community.

With the constantly advancing capabilities of quantum computers, conventional cryptographic systems relying on complex math problems may encounter unforeseen vulnerabilities. Unlike regular computers, which are often deemed cost-ineffective in cryptographic attacks, quantum computers have a significant advantage in calculation speed. This distinction potentially makes currently used algorithms less secure or even completely vulnerable, compelling the exploration of post-quantum cryptography (PQC) as the most reasonable solution to quantum threats. This review aims to provide current information on applications, benefits, and challenges associated with the PQC. The review employs a systematic scoping review with the scope restricted to the years 2022 and 2023; only articles that were published in scientific journals were used in this paper. The review examined the articles on the applications of quantum computing in various spheres. However, the scope of this paper was restricted to the domain of the PQC because most of the analyzed articles featured this field. Subsequently, the paper is analyzing various PQC algorithms, including lattice-based, hash-based, code-based, multivariate polynomial, and isogeny-based cryptography. Each algorithm is being judged based on its potential applications, robustness, and challenges. All the analyzed algorithms are promising for the post-quantum era in such applications as digital signatures, communication channels, and IoT. Moreover, some of the algorithms are already implemented in the spheres of banking transactions, communication, and intellectual property. Meanwhile, despite their potential, these algorithms face serious challenges since they lack standardization, require vast amounts of storage and computation power, and might have unknown vulnerabilities that can be discovered only with years of cryptanalysis. This overview aims to give a basic understanding of the current state of post-quantum cryptography with its applications and challenges. As the world enters the quantum era, this review not only shows the need for strong security methods that can resist quantum attacks but also presents an optimistic outlook on the future of secure communications, guided by advancements in quantum technology. By bridging the gap between theoretical research and practical implementation, this paper aims to inspire further innovation and collaboration in the field.

Post-Quantum Cryptography (PQC) will become soon the standard for many systems of the future. With the advent of quantum computers, all encrypted communications based on traditional asymmetric cryptography (e.g., RSA, ECC) will become insecure. The definition the PQC standards is an on going process proceeding at a fast pace, involving new and largely unexplored cryptographic primitives. For this reason, the design of hardware implementations of PQC algorithms is still under study. In this paper, we introduce the fundamentals of PQC, with a focus on lattice-based cryptography and its hardware security issues, namely side-channel and fault-based attacks. Then, we focus on isogeny-based cryptography and the SIKE algorithm. We highlight the importance of fault-tolerant design choices through the presentation of a fault attack, based on the electromagnetic injection of transient faults, targeting this cryptographic primitive. Finally, we show an interesting idea that starts from the observation that some PQC algorithms have an intrinsic probabilistic behavior. We argue that this characteristic is a clear opportunity that paves the way for the application of approximate (or inexact) computing to the implementation of PQC cryptography.

The advent of fault-tolerant quantum computing precipitates a foundational threat to the security of global digital infrastructure by rendering obsolete the mathematical assumptions underlying classical public-key cryptography. Widely deployed algorithms, including RSA, ECDSA, and Diffie-Hellman, which rely on the computational intractability of integer factorization and discrete logarithm problems, are vulnerable to polynomial-time attacks via Shor's algorithm. Concurrently, Grover's algorithm imposes a quadratic reduction in the security strength of symmetric primitives. In response, Post-Quantum Cryptography (PQC) has emerged as a critical field of research, dedicated to developing cryptographic systems secure against both classical and quantum attacks, while remaining deployable on existing classical hardware. This paper presents a comprehensive and in-depth examination of PQC, analyzing the five principal families: lattice-based, code-based, multivariate, hash-based, and isogeny-based cryptography. Each family is scrutinized through rigorous mathematical exposition, conceptual analysis, comparative performance evaluations, and contemporary security assessments. The study situates PQC within the evolving global threat landscape, provides a detailed analysis of the National Institute of Standards and Technology (NIST) PQC standardization process, and addresses critical implementation challenges such as constrained environments, migration strategies, hybrid cryptographic modes, and the imperative for cryptographic agility. The paper concludes by delineating essential future research directions vital for constructing a robust, quantum-resilient global cryptographic infrastructure.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

This work expands the machinery we have for isogeny-based cryptography in genus 2 by developing a toolbox of several essential algorithms for Kummer surfaces, the dimension-2 analogue of x-only arithmetic on elliptic curves. Kummer surfaces have been suggested in hyper-elliptic curve cryptography since at least the 1980s and recently these surfaces have reappeared to efficiently compute (2,2)-isogenies. We construct several essential analogues of techniques used in one-dimensional isogeny-based cryptography, such as pairings, deterministic point sampling and point compression and give an overview of (2,2)-isogenies on Kummer surfaces. We furthermore show how Scholten's construction can be used to transform isogeny-based cryptography over elliptic curves over Fp2 into protocols over Kummer surfaces over Fp. As an example of this approach, we demonstrate that SQIsign verification can be performed completely on Kummer surfaces, and, therefore, that one-dimensional SQIsign verification can be viewed as a two-dimensional isogeny between products of elliptic curves,

Virtually all asymmetric cryptographic schemes currently in use are threatened by the potential development of powerful quantum computers. Although there is currently no definite answer and it is very unclear when or even if CRQC will ever be built and the gap between modern quantum computers and the envisioned CRQC is huge, the risk of creating CRQC means that currently deployed public key cryptography must be replaced by quantum-resistant ones alternatives. For example, information encrypted using modern public key cryptography can be recorded by cryptanalysts and then attacked if a QRQC can be created. The potential harm that CRQC could cause is the basis of the motivation to seek countermeasures, even though we have uncertainties about when and if these computers can be built. Deployed systems that use public key cryptography can also take years to update. Post-quantum cryptography is one way to combat quantum computer threats. Its security is based on the complexity of mathematical problems that are currently considered unsolvable efficiently – even with the help of quantum computers. Post-quantum cryptography deals with the development and research of asymmetric cryptosystems, which, according to current knowledge, cannot be broken even by powerful quantum computers. These methods are based on mathematical problems for the solution of which neither efficient classical algorithms nor efficient quantum algorithms are known today. Various approaches to the implementation of post-quantum cryptography are used in modern research, including: code-based cryptography, lattice-based cryptography, hashing-based cryptography, isogeny-based cryptography, and multidimensional cryptography. The purpose of this work is to review the computational model of quantum computers; quantum algorithms, which have the greatest impact on modern cryptography; the risk of creating cryptographically relevant quantum computers (CRQC); security of symmetric cryptography and public key cryptography in the presence of CRQC; NIST PQC standardization efforts; transition to quantum-resistant public-key cryptography; relevance, views and current state of development of quantum-resistant cryptography in the European Union. It also highlights the progress of the most important effort in the field: NIST's standardization of post-quantum cryptography.