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Elliptic Curve Cryptography and Quantum Computing

Emily Alderson

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

In the year 2007, a slightly nerdy girl fell in love with all things math. Even though she only was exposed to a small part of the immense field of mathematics, she knew that math would always have a place in her heart. Ten years later, that passion for math is still burning inside. She never thought she would be interested in anything other than strictly mathematics. However, she discovered a love for computer science her sophomore year of college. Now, she is graduating college with a double major in both mathematics and computer science.\nThis nerdy girl is me. It was my first semester, freshman year at Ouachita Baptist University when I discovered an interest in cryptography. Discrete Mathematics had a small portion of the class that discussed various types of cryptosystems. These were the simplified versions that were easier to comprehend so that we could truly grasp the underlying concepts of cryptography. However, I craved more. I wanted to delve further into the advanced world of cryptography. It was not until my Honors Directed Study that I was able to do so.\nDuring my directed study, I was able to research the history of cryptography and what it has transformed into today. While it was fascinating to look at all the various cryptosystems that have been created throughout time, one cryptosystem caught my eye in particular—Elliptic Curve Cryptography, commonly referred to as ECC. I had heard of a few cryptosystems but never the ECC. As such, my interest was piqued. Having the opportunity to research Elliptic Curve Cryptography, I was ready to try and understand a new mathematical concept. I was ready to further expand my love and passion for mathematics, and I was not disappointed. Learning even the basics of elliptical curves proved challenging. The math behind ECC is immense and quite frankly, intimidating. Nevertheless, I do not back down from a challenge…at least not a mathematical one.\nMy research of elliptic curves, the basis of Elliptic Curve Cryptography, opened up my eyes to an entirely new field of mathematics. The challenges that arose during my studies, while frustrating at times, are well worth the wealth of knowledge I was able to acquire. The research in my study of elliptical curve cryptography only increased the ten-year passion for mathematics that is still inside the slightly nerdy girl.\nThis paper is the culmination of all my research over elliptic curves. It reflects the knowledge that I was able to acquire while studying elliptic curve cryptography and quantum computers.

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In the year 2007, a slightly nerdy girl fell in love with all things math. Even though she only was exposed to a small part of the immense field of mathematics, she knew that math would always have a place in her heart. Ten years later, that passion for math is still burning inside. She never thought she would be interested in anything other than strictly mathematics. However, she discovered a love for computer science her sophomore year of college. Now, she is graduating college with a double major in both mathematics and computer science.\nThis nerdy girl is me. It was my first semester, freshman year at Ouachita Baptist University when I discovered an interest in cryptography. Discrete Mathematics had a small portion of the class that discussed various types of cryptosystems. These were the simplified versions that were easier to comprehend so that we could truly grasp the underlying concepts of cryptography. However, I craved more. I wanted to delve further into the advanced world of cryptography. It was not until my Honors Directed Study that I was able to do so.\nDuring my directed study, I was able to research the history of cryptography and what it has transformed into today. While it was fascinating to look at all the various cryptosystems that have been created throughout time, one cryptosystem caught my eye in particular—Elliptic Curve Cryptography, commonly referred to as ECC. I had heard of a few cryptosystems but never the ECC. As such, my interest was piqued. Having the opportunity to research Elliptic Curve Cryptography, I was ready to try and understand a new mathematical concept. I was ready to further expand my love and passion for mathematics, and I was not disappointed. Learning even the basics of elliptical curves proved challenging. The math behind ECC is immense and quite frankly, intimidating. Nevertheless, I do not back down from a challenge…at least not a mathematical one.\nMy research of elliptic curves, the basis of Elliptic Curve Cryptography, opened up my eyes to an entirely new field of mathematics. The challenges that arose during my studies, while frustrating at times, are well worth the wealth of knowledge I was able to acquire. The research in my study of elliptical curve cryptography only increased the ten-year passion for mathematics that is still inside the slightly nerdy girl.\nThis paper is the culmination of all my research over elliptic curves. It reflects the knowledge that I was able to acquire while studying elliptic curve cryptography and quantum computers.
The emergence of quantum computing poses significant challenges and opportunities in the realm of cryptography. Traditional cryptographic algorithms, such as RSA and ECC (Elliptic Curve Cryptography), rely on the computational difficulty of certain mathematical problems, like factoring large integers and solving discrete logarithms. However, quantum computers leverage quantum mechanics to solve these problems in polynomial time, which could render many of today’s encryption techniques obsolete. This article examines the potential impact of quantum computing on modern cryptographic algorithms and explores quantum-safe cryptography alternatives that are being developed to withstand quantum attacks. We also discuss the readiness of quantum computers for real-world applications, their current limitations, and the future directions of quantum cryptography research.
In this work, we present the theory behind Elliptic Curve Cryptography (ECC) and Isogeny-based cryptography. We start with a quick introduction to Complexity Theory and Cryptography. We give a detailed description of elliptic curves and discuss how they are used to create secure protocols. We present the strength of this scheme and also its weaknesses regarding quantum computation. To show the necessity for other protocols, we review some basic concepts of Quantum Computing and present how Shor's Quantum Algorithm breaks ECC. Then, we survey the rich theory of isogenies between ordinary and supersingular elliptic curves to end with cryptographic protocols, that use these isogenies to create post-quantum cryptographic systems.
Recent advances in quantum computing have made existing public key cryptosystems vulnerable to enormous security threats. Therefore, numerous efforts have been exploring post-quantum cryptographic techniques to address the emergence of quantum computing. We focus on one promising post-quantum cryptography known as “isogeny-based cryptography,” first by reviewing some concepts of the elliptic curve isogeny, and then, we present three major methods for constructing isogeny-based difficult problems. Then, we present some existing isogeny-based cryptographic primitives, such as signature and key exchange, and analyze their advantages and disadvantages. Finally, we discuss a few major challenges of isogeny research that, we hope, will attract more attention to isogeny-based cryptography.
The rapid advancement of quantum computing has posed a significant threat to traditional public-key cryptographic systems, particularly those based on elliptic curve cryptography (ECC). Although ECC has long been valued for its strong classical security and computational efficiency, quantum algorithms capable of solving the elliptic curve discrete logarithm problem would render these systems vulnerable in the near future. This study investigated how secure elliptic-curve-based cryptographic systems could be redesigned for the post-quantum era by integrating post-quantum cryptographic (PQC) primitives into hybrid security architectures. A design-oriented and analytical research approach was employed to compare classical ECC, standalone PQC, and hybrid ECC–PQC models using quantitative measures of security strength, computational cost, and communication overhead. The results demonstrated that while ECC provided excellent performance, it lacked quantum resistance, whereas PQC offered strong quantum-safe security at the cost of higher computational and bandwidth requirements. The hybrid ECC–PQC approach achieved the most balanced outcome, combining high classical efficiency with robust quantum resistance and strong long-term security. Numerical performance evaluation showed that hybrid systems increased execution time and handshake size moderately compared to PQC alone, yet delivered significantly improved security guarantees. These findings confirmed that hybrid cryptographic architectures represent a practical and reliable pathway for migrating existing elliptic-curve infrastructures toward quantum-resilient security. The study contributed a mathematically grounded and deployment-oriented framework that supports secure, gradual transition strategies for protecting digital communications in a quantum-enabled future.
Traditionally, information security needed encryption, authentication, key management, non-repudiation and authorization which were being met using several techniques. Standardization of algorithms by National Institute of Standards and Technology (NIST) has facilitated international communication for banking and information transfer using these standards. Encryption can be carried out using Advanced Encryption Standard (AES) using variable block lengths (128, 192 or 256 bits) and variable key lengths (128, 192 or 256 bits). Solutions for light weight applications such as those for Internet of Things (IoT) are also being standardized. Message integrity is possible using host of hash algorithms such as SHA-1, SHA-2 etc., and more recently using SHA-3 algorithm. Authentication is possible using well known Rivest-Shamir-Adleman (RSA) algorithm needing 2048/4096 bit operations. Elliptic Curve Cryptography (ECC) is also quite popular and used in several practical systems such as WhatsApp, Blackberry etc. Key exchange is possible using Diffie-Hellman algorithm and its variations. Digital Signatures can be carried out using RSA algorithm or Elliptic Curve Digital Signature Algorithm (ECDSA) or DSA (Digital Signature Algorithm). All these algorithms derive security from difficulty in solving some mathematical problems such as factorization problem or discrete logarithm problem. Though published literature gives evidence of solving factorization problem upto 768 bits only, it is believed that using Quantum computers, these problems could be solved by the end of this decade. This is due to availability of the pioneering work of Shor and Grover [1]. For factoring an integer of N bits, Shor’s algorithm takes quantum gates. As such, there is ever growing interest in being ready for the next decade with algorithms that may resist attacks in the quantum computer era. NIST has foreseen this need and has invited proposals from researchers all over the world. In the first round, about 66 submissions were received which have been scrutinized for completeness of submissions , novelty of the approach and security and 25 of these were promote to second round to improve based on the comments received on the first round submission. These will be analyzed for security and some will be selected for final recommendation for use by industry. These are for encryption/decryption, key agreement, hashing and Digital Signatures for both hardware and software implementations. In this paper, we present a brief survey of the state of the art in post-Quantum Cryptography (PQC) followed by study of one of technique referred to as Learning With Errors (LWE) in some detail.