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Design of Grover’s Algorithm over 2, 3 and 4-Qubit Systems in Quantum Programming Studio

Diana Jingle Shylu Sam Mano Paul Ananth Jude Daniel Selvaraj
International Journal of Electronics and Telecommunications | 2022 | vol. 68 | No 1 | 77-82 | DOI: 10.24425/ijet.2022.139851
Keywords quantum search Oracle Qubit Hadamard transform phase shift
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Abstract

In this paper, we design and analyse the Circuit for Grover’s Quantum Search Algorithm on 2, 3 and 4-qubit systems, in terms of number of gates, representation of state vectors and measurement probability for the state vectors. We designed, examined and simulated the quantum circuit on IBM Q platform using Quantum Programming Studio. We present the theoretical implementation of the search algorithm on different qubit systems. We observe that our circuit design for 2 and 4-qubit systems are precise and do not introduce any error while experiencing a small error to our design of 3-qubit quantum system.
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Authors and Affiliations

Diana Jingle
1
Shylu Sam
2
Mano Paul
3
Ananth Jude
4
Daniel Selvaraj
4
  1. Christ University, Bangalore, India
  2. Karunya Institute of Technology and Sciences, Coimbatore, India
  3. Alliance University, Bangalore, India
  4. Sri Krishna College of Engineering and Technology, Coimbatore, India

Binary Tree Based Forward Secure Signature Scheme in the Random Oracle Model

Mariusz Jurkiewicz
International Journal of Electronics and Telecommunications | 2021 | vol. 67 | No 4 | 717-726 | DOI: 10.24425/ijet.2021.137868
Keywords forward secure digital signature scheme bilinear pairing of Type 3 random-oracle model bilinear Diffie-Hellman inversion problem
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Abstract

In this paper we construct and consider a new group-based digital signature scheme with evolving secret key, which is built using a bilinear map. This map is an asymmetric pairing of Type 3, and although, for the reason of this paper, it is treated in a completely abstract fashion it ought to be viewed as being actually defined over E(Fqn)[p] × E(Fqnk )[p] → Fqnk [p]. The crucial element of the scheme is the key updater algorithm. With the adoption of pairings and binary trees where a number of leaves is the same as a number of time periods, we are assured that an updated secret key can not be used to recover any of its predecessors. This, in consequence, means that the scheme is forward-secure. To formally justify this assertion, we conduct analysis in fu-cma security model by reducing the security of the scheme to the computational hardness of solving the Weak ℓ-th Bilinear Diffie-Hellman Inversion problem type. We define this problem and explain why it can be treated as a source of security for cryptographic schemes. As for the reduction itself, in general case, it could be possible to make only in the random oracle model.
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Bibliography

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[3] D. Boneh and X. Boyen, ”Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles”, in Advances in Cryptology - EUROCRYPT 2004, C. Cachin and J.L. Camenisch, Eds. 2004, pp. 223- 238.
[4] D. Boneh, X. Boyen and E.-J. Goh, ”Hierarchical Identity Based Encryption with Constant Size Ciphertext”, Cryptology ePrint Archive, Report 2005/015. [Online]. Available: https://eprint.iacr.org/2005/015.pdf.
[5] X. Boyen, H. Shacham, E. Shen and B. Waters, ”Forward Secure Signatures with Untrusted Update”, in Proceedings of CCS 2006, W. Rebecca Ed. 2006, pp. 191–200.
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[7] J. Camenisch and M. Koprowski, ”Fine-grained Forward-secure Signature Schemes without Random Oracles”, Discrete Applied Mathematics, vol. 154, no. 2, pp. 175–188, Feb. 2006, doi: 10.1016/j.dam.2005.03.028.
[8] R. Canetti, S. Halevi, J. Katz, ”A Forward-Secure Public-Key Encryption Scheme”, in Advances in Cryptology - EUROCRYPT 2003, E. Biham, Ed. 2003, pp. 255–271.
[9] Y. Cui, E. Fujisaki, G. Hanaoka, H. Imai and R. Zhang, ”Formal Security Treatments for Signatures from Identity-Based Encryption”, in Provable Security, W. Susilo, J. K. Liu, Y. Mu, Eds. 2007, pp. 218–227.
[10] A. Fiat and A. Shamir, ”How to Prove Yourself: Practical Solutions to Identification and Signature Problems”, in Conference on the theory and application of cryptographic techniques, 1986, pp. 186–194.
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[12] S. Goldwasser S. Micali and R. L. Rivest, ”A Digital Signature Scheme Secure Against Adaptive Chosen-Message Attacks”, SIAM Journal on Computing, vol. 17, no. 2, pp. 281–308, 1988, doi: 10.1137/0217017.
[13] S. Hohenberger and B.Waters, ”New Methods and Abstractions for RSA-Based Forward Secure Signatures”, in International Conference on Applied Cryptography and Network Security, M. Conti, J. Zhou, E. Casalicchio and Angelo Spognardi, Eds. 2020, pp. 292–312.
[14] G. Itkis, and L. Reyzin, ”Forward-secure Signatures with Optimal Signing and Verifying”, in Advances in Cryptology - CRYPTO ’01, 21st Annual International Cryptology Conference, J. Kilian, Ed. 2001, pp. 332–354.
[15] M. Jurkiewicz, ”Improving Security of Existentially Unforgeable Signature Schemes”, International Journal of Electronics and Telecommunications, vol. 66, no. 3, pp. 473–480, 2020, doi: 10.24425/ijet.2020.131901.
[16] H. Krawczyk, ”Simple Forward-secure Signatures from any Signature Scheme”, in Proceedings of the 7th ACM conference on Computer and Communications Security, P. Samarati, Ed. 2000, pp. 108–115, doi: 10.1145/352600.352617.
[17] S. Mitsunari, R. Sakai and M. Kasahara, ”A new traitor tracing”, IEICE transactions on fundamentals of electronics, communications and computer sciences, vol. 85, no. 2, pp. 481–484, Feb. 2002.
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Authors and Affiliations

Mariusz Jurkiewicz
1
  1. Faculty of Cybernetics, Military University of Technology, Warsaw, Poland

The Computational and Pragmatic Approach to the Dynamics of Science

Witold Marciszewski
Filozofia i Nauka | 2020 | Tom 8 | Część 1 | 31-67
Keywords algorithm behavioral (vs declarative) knowledge computability corroboration innate knowledge intuition invention logic gates oracle pragmatic (vs classical) rationalism problem-solving reasoning symbolic logic Turing machine
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Abstract

Science means here mathematics and those empirical disciplines which avail themselves of mathematical models. The pragmatic approach is conceived in Karl R. Popper’s The Logic of Scientific Discovery (p. 276) sense: a logical appraisal of the success of a theory amounts to the appraisal of its corroboration. This kind of appraisal is exemplified in section 6 by a case study—on how Isaac Newton justified his theory of gravitation. The computational approach in problem-solving processes consists in considering them in terms of computability: either as being performed according to a model of computation in a narrower sense, e.g., the Turing machine, or in a wider perspective—of machines associated with a non-mechanical device called “oracle” by Alan Turing (1939). Oracle can be interpreted as computer theoretic representation of intuition or invention. Computational approach in another sense means considering problem-solving processes in terms of logical gates, supposed to be a physical basis for solving problems with a reasoning.

Pragmatic rationalism about science, seen at the background of classical rationalism (Descartes, Gottfried Leibniz etc.), claims that any scientific idea, either in empirical theories or in mathematics, should be checked through applications to problem-solving processes. Both the versions claim the existence of abstract objects, available to intellectual intuition. The difference concerns the dynamics of science: (i) the classical rationalism regards science as a stationary system that does not need improvements after having reached an optimal state, while (ii) the pragmatical version conceives science as evolving dynamically due to fertile interactions between creative intuitions, or inventions, with mechanical procedures.

The dynamics of science is featured with various models, like Derek J. de Solla Price’s exponential and Thomas Kuhn’s paradigm model (the most familiar instances). This essay suggests considering Turing’s idea of oracle as a complementary model to explain most adequately, in terms of exceptional inventiveness, the dynamics of mathematics and mathematizable empirical sciences.

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Authors and Affiliations

Witold Marciszewski

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