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Quantum Information and Computation

Quantum binary field inversion

AmentoBrittanney, RöttelerMartin, SteinwandtRainer

Published Jan 1, 2013• 0 citations
Publisher

Abstract

Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the...
DOI
10.5555/2481591.2481598

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THE EVOLUTION OF LAW CONCEPT OF INFORMATION: FROM PHILOSOPHICAL ABSTRACTION TO QUANTUM INFORMATION AND CRYPTOGRAPHY IN THE CONTEXT OF GROWING CYBER THREATS AND THE TRANSITION TO POST-QUANTUM SECURITY METHOD

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The article examines the significant transformation of the concept of information and its impact on modern security technologies. Until the middle of the 20th century, the concept of information was considered as a philosophical category related to the transfer of knowledge and meaning. With the advent of Claude Shannon's communication theory, information was reinterpreted as a measurable and transferable quantity, which opened up new horizons for the development of fields such as coding theory and cybernetics. The introduction of the bit as a unit of information measurement allowed for a more accurate analysis of data transmission processes. Landauer and Bennett's research has shown that information has a physical nature, closely related to energy and entropy. In recent decades, the concept of information has begun to play an important role in quantum theory, which has given rise to such areas as quantum information and quantum cryptography. In the face of increasing cybersecurity threats, quantum cryptography can become a reliable basis for protecting critical information systems. Existing post-quantum algorithms, although they have larger keys, can serve as a substitute for traditional security systems. The development of new standards and the transition to post-quantum security methods are becoming important steps to ensure data security in the near future. Quantum cryptography offers not only adaptation to new challenges, but also a fundamentally new level of security previously inaccessible to classical cryptographic systems, which makes it a key element for data protection in the future. Conclusion. The presented work demonstrates the evolution of the concept of information from a philosophical abstraction to a practical and measurable entity in the field of computer science, physics and cryptography. The article highlights the importance of quantum cryptography as a means of protection against future threats in the field of cybersecurity, offering both a theoretical and practical basis for protecting critical information. Taking into account the further development of quantum information science, its impact on data security and protection will have a significant impact on the formation of future cybersecurity strategies in various industries.

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Finite fields of the form F2m{\mathbb F}_{2^m}F2m​ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth $\bigO(m\log(m))$. To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields F2m{\mathbb F}_{2^m}F2m​ based on an elementary gate set that is useful for fault-tolerant implementation.
In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over the general linear group with those for non-binary LDPC codes over finite field transmitted over the q-ary memoryless symmetric channel under belief propagation decoding. To analyze non-binary LDPC codes defined over both general linear group GL(m, F2) and finite field F2m, we investigate non-binary LDPC codes defined over GL(m3, F2m4). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the optimized non-binary LDPC codes defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels in the edges which satisfy the condition for the optimization.
The article examines the significant transformation of the concept of information and its impact on modern security technologies. Until the middle of the 20th century, the concept of information was considered as a philosophical category related to the transfer of knowledge and meaning. With the advent of Claude Shannon's communication theory, information was reinterpreted as a measurable and transferable quantity, which opened up new horizons for the development of fields such as coding theory and cybernetics. The introduction of the bit as a unit of information measurement allowed for a more accurate analysis of data transmission processes. Landauer and Bennett's research has shown that information has a physical nature, closely related to energy and entropy. In recent decades, the concept of information has begun to play an important role in quantum theory, which has given rise to such areas as quantum information and quantum cryptography. In the face of increasing cybersecurity threats, quantum cryptography can become a reliable basis for protecting critical information systems. Existing post-quantum algorithms, although they have larger keys, can serve as a substitute for traditional security systems. The development of new standards and the transition to post-quantum security methods are becoming important steps to ensure data security in the near future. Quantum cryptography offers not only adaptation to new challenges, but also a fundamentally new level of security previously inaccessible to classical cryptographic systems, which makes it a key element for data protection in the future. Conclusion. The presented work demonstrates the evolution of the concept of information from a philosophical abstraction to a practical and measurable entity in the field of computer science, physics and cryptography. The article highlights the importance of quantum cryptography as a means of protection against future threats in the field of cybersecurity, offering both a theoretical and practical basis for protecting critical information. Taking into account the further development of quantum information science, its impact on data security and protection will have a significant impact on the formation of future cybersecurity strategies in various industries.