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Presence of dynamics of quantum dots in the digital signature using DNA alphabet and chaotic S-box

Abstract

The integrity and authenticity of the message, and it’s nonrepudiation, are provided by digital signatures. We introduce quantum digital signature schemes based on Quantum Dots, where DNA coding is used to increase the intricacy of phase space. We attain the optimal security standard by constructing a deterministic dynamic system in a finite phase space with n points and by using symbolic dynamics. Also, given the chaotic Substitution box(S-box), a confusing step has been added for greater security. The introduced quantum dynamical map is used to create procedures to resistance again the common attacks in the digital signature such as non-repudiation, unforgeability, and transferability. Its security depends on the length of the signature, directly.

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Correspondence to Sodeif Ahadpour.

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Appendix A: Calculation of Lyapunov exponents

Appendix A: Calculation of Lyapunov exponents

Entropies are basic invariants for dynamical systems. A closely related quantity to the entropy and information flow in the dynamical system is a Lyapunov exponent. The Lyapunov number is independent of the initial point, provided that the motion inside the invariant manifold is ergodic [58], defined as:

$$ \lambda=lim_{n\rightarrow\infty}\ln\mid D_{<J_{\pm(n)}>}{(J_{\pm(n+1)}}(a,a^{+},\alpha))\mid $$

In the nonchaotic area of the parameter, the Lyapunov exponent is negative, while the positive one shows the measurability of system. The variation of Lyapunov exponent for introduced quantum dynamical systems is presented with the Fig. 8a. It is obvious that in the introduced example, (5), the chaotic domain is (3.7 < α < 4). The dynamical system time series, which is shown in Fig. 8b, confirm the chaotic behavior.

Fig. 8
figure8

a The variation of the Lyapunov characteristic exponent of introduced example (5) in term of parameters α. b Time series diagram of dynamical system (5) for J+(0) = 0.3,N = 100,α = 3.8

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Hematpour, N., Ahadpour, S. & Behnia, S. Presence of dynamics of quantum dots in the digital signature using DNA alphabet and chaotic S-box. Multimed Tools Appl 80, 10509–10531 (2021). https://doi.org/10.1007/s11042-020-10059-5

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Keywords

  • Quantum digital signature
  • Quantum dots
  • Chaos
  • DNA coding
  • Substitution box(S-box)
  • Repudiation
  • Forgery