Abstract
Nowadays, we are surrounded by devices collecting and transmitting private information. Currently, the two main mathematical problems that guarantee security on the Internet are the Integer Factorization Problem and the Discrete Logarithm Problem. However, Shor’s quantum algorithm can easily solve both problems. Therefore, research into cryptographic algorithms that run in classical computers and are resistant to quantum computers is extremely necessary. This area is known as post-quantum cryptography and usually studies asymmetric cryptography. By means of asymptotic analysis, the purpose of this paper is to provide an evaluation of security and its performance for the types of cryptographic systems considered safe against quantum attacks in the second-round NIST Post-Quantum Standardization Process, namely isogeny cryptosystems based on supersingular elliptic curves, error correction code-based encryption system, and lattice-based ring learning with errors. We performed a security comparison of Key Agreements protocols based on these three post-quantum cryptographic primitives and compared them with Discrete Logarithm Problem and Integer Factorization Problem. The comparison of security and its performance is presented by security level, the former by complexity analyses to achieve theoretical minimum key sizes, and the latter by simulation to assess a practical performance comparison. In the complexity analysis, as we increase the security level and then the size of the cryptographic keys increases, techniques based on isogeny outperform all other post-quantum algorithms in relation to key sizes at practical security level. In the performance comparison, the results show that the code-based protocol presents the best results among the others.
Highlights
Google claimed having achieved Quantum Supremacy [1], using a processor with programmable superconducting qubits to create quantum states on 53 qubits
The third section of the paper is dedicated to a discussion of quantum-resistant primitives, presenting an overview of the protocols, their security and the complexity for the best know attacks to such mathematical problems. It is traced a comparison between all primitives shown before, where we focus on the evaluation of the key-length generated for each of the practical security levels recommended by the National Institute for Standards and Technology (NIST) and the performance of the referred algorithms
We were able to get the minimum key-length to achieve the practical security levels proposed by NIST
Summary
Google claimed having achieved Quantum Supremacy [1], using a processor with programmable superconducting qubits to create quantum states on 53 qubits. Considering the best-known algorithm to solve problems of primitives used in post-quantum cryptography, we compute the key-length with the same level of security used for the classical cryptographic algorithms. We present a brief overview of the non-quantum-resistant cryptographic primitives used in key agreement protocols and present algorithm complexity for the best-known attacks. Galbraith and Stolbunov [17] presented the best-known algorithm for classical computers to solve the isogeny problem [18]. As argued in [36], we can discard the attacks like BKW, because, the RLWE-based protocol described in this work use a limited number of LWE samples (m ≈ n). Previous researches [38], [39] show that the best-known algorithm to solve RLWE in the quantum-world (Q-RLWE) has complexity given by. We refer to [36] and [38] for a better understanding of how the choice of b value is made
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