The present knowledge we had in quantum computer and the most possible architecture of a quantum computer might be able to break RSA 2048 in future. In classical computer two bits represents any one of four bit information whereas in quantum due to superposition it can be represent all four states. For ‘n’ qubits system is analogous to 2n classical bits. Quantum teleportation, quantum entanglement and other makes it possible to break present cryptosystem. Shor’s Algorithm is used for integer factorization which is polynomial time for quantum computer. This can be threat for RSA security. In this paper matlab implementation of Shor’s algorithm is presented. Used classical way for getting period of function because classical computers not engage quantum phenomena. As numbers of iterations grow, probability of getting exact factor of ‘n’ acutely increased. This paper also discusses popular methods for making qubits like Silicon based Qubits in which electron is put inside nano material which is used as a transistor. In Superconducting circuit method insulator is used as a sandwich in between two metal layers. Used by Google, IBM, Intel, Microsoft. In Flux qubits method very small size loop of superconducting metal is used. This paper also discusses Quantum Proof Algorithm like Lattice-based cryptography used concept of good and bad base. In Learning with errors method if we have more equation then variable, it is over defined system. In Code based cryptography some matrixes allow for efficient error correction (good matrix) but most matrix’s does not (bad matrix) concept is used. In Hash based signatures scheme have long signatures or keys, but they are secure. Also discuss Multivariate Quantum proof algorithm. The abstract should contain maximum of 300 words. No abbreviation should be mentioned in the abstract. Give a brief summary of your research work.
The present knowledge we had in quantum computer and the most possible architecture of a quantum computer might be able to break RSA 2048 in future. In classical computer two bits represents any one of four bit information whereas in quantum due to superposition it can be represent all four states. For ‘n’ qubits system is analogous to 2n classical bits. Quantum teleportation, quantum entanglement and other makes it possible to break present cryptosystem. Shor’s Algorithm is used for integer factorization which is polynomial time for quantum computer. This can be threat for RSA security. In this paper matlab implementation of Shor’s algorithm is presented. Used classical way for getting period of function because classical computers not engage quantum phenomena. As numbers of iterations grow, probability of getting exact factor of ‘n’ acutely increased. This paper also discusses popular methods for making qubits like Silicon based Qubits in which electron is put inside nano material which is used as a transistor. In Superconducting circuit method insulator is used as a sandwich in between two metal layers. Used by Google, IBM, Intel, Microsoft. In Flux qubits method very small size loop of superconducting metal is used. This paper also discusses Quantum Proof Algorithm like Lattice-based cryptography used concept of good and bad base. In Learning with errors method if we have more equation then variable, it is over defined system. In Code based cryptography some matrixes allow for efficient error correction (good matrix) but most matrix’s does not (bad matrix) concept is used. In Hash based signatures scheme have long signatures or keys, but they are secure. Also discuss Multivariate Quantum proof algorithm. The abstract should contain maximum of 300 words. No abbreviation should be mentioned in the abstract. Give a brief summary of your research work.
This dissertation analyzes the impact of quantum computing on the security of classical cryptographic systems, with a focus on the vulnerability of the RSA algorithm. RSA cryptography is based on the difficulty of factoring large prime numbers, a task considered computationally very difficulty for classical computers. However, with quantum computing, especially through Shor’s algorithm—this barrier can be overcome. Shor’s algorithm allows factorization in polynomial time, which would completely compromise the security of RSA.The dissertation presents the theoretical foundations of both cryptography and quantum computing, explaining key concepts such as qubits, superposition, entanglement, and quantum parallelism. It details the operation of Shor’s algorithm, from identifying the period of the modular exponential function to applying the Quantum Fourier Transform (QFT) and phase estimation to obtain the prime factors of a composite number.In addition to the theoretical component, the work includes practical implementations of the algorithm in both classical and simulated quantum environments, using IBM’s Qiskit library. Finally, it discusses post-quantum cryptography as a secure alternative in light of technological advancements, highlighting methods based on lattices, codes, hash functions, and multivariate systems.The conclusion emphasizes the urgency of adopting cryptographic systems resistant to quantum computing, as ongoing advances increasingly make the breaking of RSA a realistic threat. Post-quantum cryptography is mentioned as a necessary response to the challenge posed by quantum computing.In conclusion, although current quantum computers do not yet have sufficient power to break RSA in practice, technological progress is steadily bringing that scenario closer. Therefore, it is essential to develop and adopt quantum-resistant cryptographic systems to ensure information security in the coming decades.
The emergence of quantum computing poses a serious threat to traditional communication systems, endangering the security of digital communications and critical infrastructure. Although existing cryptographic algorithms (such as RSA and ECC) rely on the inefficiency of computing large numbers or solving logarithm inequality problems, quantum algorithms (especially Shor's algorithm) can solve these problems efficiently. To mitigate this threat, the field of post-quantum cryptography (PQC) has emerged and focuses on developing cryptographic systems that are resilient to quantum attacks. The main techniques in PQC include lattice-based, hash-based, code-based, and multivariate polynomial cryptography. Lattice-based cryptography is the leading contender, providing high security and supporting good properties such as fully homomorphic encryption, but facing issues related to size and computational efficiency. Hash-based encryption enables strong digital signatures but has limitations. Code-based encryption based on error-correcting codes provides proof of security but requires small size. Multivariate polynomial cryptography has good performance but has drawbacks in some applications. While standardization efforts led by organizations such as NIST continue, PQC is at the forefront of protecting the digital future from the impact of quantum computing, ensuring the integrity and confidentiality of information in the post-quantum era.
Quantum computing endangers classical cryptography via two main avenues: Shor’s al-gorithm reduces integer factorization to polynomial time, and Grover’s algorithm yields a quadratic speed-up for unstructured key search. We develop rigorous mathematical treatment of both algorithms, enrich the literature survey with detailed complexity and resource analyses, and extend the development section to include quantum er-ror-correction overhead, resource estimates for breaking 2048-bit RSA, and advanced as-ymptotic bounds. In Discussion, we propose lattice-, code-, multivariate-, and hash-based post-quantum schemes, analyze their security margins against quantum attacks, and out-line standardized migration strategies. We include three illustrative figures and 35+ se-quential IEEE citations.
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.
Compared with electronic computers, quantum parallelism has led to a significant increase in the computing power of quantum computers.However, the computing power of quantum computers will also be limited by the laws of physics, and hence there is no doubt that the computing power of quantum computers is not unlimited.For an algorithm, we need to consider both its space complexity and time complexity.This not only requires quantum computers to achieve large-scale integration, but also requires the quantum coherence time to be sufficiently long.If either of the above two conditions is not met, the task cannot be completed.Shor's algorithm [1] reduced the computational complexity of large integer factorization to the polynomial level.This complexity is of the same order as that of the encryption and decryption for the RSA public key cryptography protocol, which leads to the insecurity of the RSA in universal quantum computing.In addition, some other quantum algorithms show different kinds of enhanced speeds for solving different problems, such as the Grover's search algorithm [2] and its improved works [ 3 , 4 ], and the HHL algorithm [5] .With the remarkable computational power of these quantum algorithms, physicists have begun to seek suitable physical systems for the implementation of quantum computing.The implementation of quantum computing includes several basic technologies, such as preparation, precise control, measurement of the physical qubits, and construction of logical qubits.At present, there are many experimental candidate quantum computing platforms, including the superconducting system [6] , trapped ions system [7] , nitrogenvacancy centers in diamonds [8] , and ultra-cold atoms [9] .Recently, Google experimentally demonstrated quantum supremacy with a superconducting quantum computer [10] , thus placing the research on quantum computing at the forefront.With the booming development of quantum computing technology, the public key cryptosystem also needs to be updated periodically.Thus, cryptographers have to constantly look for public key protocols to resist quantum computing attacks, which leads to the emergence of postquantum cryptography.It is worth noting that although the name of post-quantum cryptography contains the word 'quantum', it essentially consists of protocols of classical cryptography.The current research on post-quantum cryptography mainly includes protocols based on the following four frameworks: the protocols based on Hash function, the pro- * Corresponding author.
Introduction In an ordinary computer, information is stored in a collection of tiny circuits each of which is designed to have two stable and easily distinguishable configurations: each represents a bit . In our study of quantum cryptography, we have seen how it can be useful to express information not in ordinary bits but in qubits . Whereas a bit can have only two values, say 0 and 1, a qubit can be in any quantum superposition of ∣0〉 and ∣1〉. Moreover, a qubit can be entangled with other qubits. Thus one might wonder whether a quantum computer, in which the basic elements for storing and processing information are qubits, can outperform an ordinary (classical) computer in certain ways. This question was addressed by researchers starting in the 1980s. In terms of practical consequences, perhaps the most dramatic answer has been given by Peter Shor in his 1994 factoring algorithm for a quantum computer, an algorithm that is exponentially faster than any known classical algorithm. As we have seen in Chapter 1, the difficulty of factoring a product of two large primes is the basis of the security of the RSA cryptosystem. So if one could build a large enough quantum computer – and there is no reason in principle why this could not be done – the RSA system would be rendered ineffective. In this chapter we present the basics of quantum computation and then focus on Shor's factoring algorithm.