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NMRMG: Nonlinear Multiple-Recursive Matrix Generator Design Approaches and Its Randomness Analysis

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Abstract

Highly random binary sequences generated by any keystream generator provides adequate service for the rapidly growing demands of cryptographic applications. Using several statistical randomness test analysis, we found that the randomness properties of binary sequences generated by multiple-recursive matrix generators are not statistically secure for cryptographic solutions. To overcome the randomness loopholes, we propose nonlinearly filtered multiple-recursive matrix generator and experimentally establish that the mentioned generator provides high-quality randomness results. We evaluate the statistical security of the proposed scheme with the help of NIST randomness test, autocorrelation test, linear complexity test, and avalanche test. Moreover, we compare our randomness test results with some of the high-quality software oriented stream ciphers like Snow, HC-128, and ZUC.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Code Availability

The code used (if any) in the manuscript is a generalized code for data analysis and is not specific to the findings of this work.

Notes

  1. \((\star )\) represents MRMG with special vector Boolean function, and \((\#)\) represents MRMG with PRESENT S-box conjecture

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Acknowledgements

The authors want to thank Prof. Subhamoy Maitra and Dr. Sartaj Ul Hasan, their comments and suggestions improved this paper’s editorial and technical quality.

Funding

This work was not funded by any agency.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, ICFAI University Tripura, Agartala, India

    Subhrajyoti Deb

  2. Applied Statistics Unit, Indian Statistical Institute, Kolkata, India

    Santu Pal

  3. Department of Information Technology, North-Eastern Hill University, Shillong, India

    Bubu Bhuyan

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  1. Subhrajyoti Deb
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  2. Santu Pal
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  3. Bubu Bhuyan
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Correspondence to Subhrajyoti Deb.

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Appendix A: Description of the MRMG

Appendix A: Description of the MRMG

1.1 A.1: Test Vectors of MRMG (I, II, III)

In this paper, we have considered three types of MRMG. Particularly, all the polynomial equations are listed in [11, B, Page No. -13]. Polynomial equation of the MRMG(I, II, III) mentioned in bellow. \(x^{16} + \sigma x^3 + \wedge \texttt {0x5437af9e} x^2 + \sigma\) [MRMG I] \(x^{16} + \wedge \texttt {0x5e8491f8} x^3 + \mathbf{L} x^6 + \mathbf{R} x^5 + 1\) [MRMG II] \(x^{16} + \wedge \texttt {0x7ceabddf} x^9 + \sqcup _{1, 1}\) [MRMG III] Notation : \(\sigma\) - Circular Rotation operation, \(\wedge\) - AND operation, \(\mathbf{L}\) - Left Rotation, \(\mathbf{R}\) - Right Rotation, \(\sqcup\) - Left Right Shift combination operation, and \(\texttt {0x5437af9e}\) - 32-bit vector. In this work, MRMG coded in C language. Further, MRMG seed value loaded by 16 blocks of Hex bits (that is each block contains 32-bit) and shown in bellow.

figure d

1.2 A.2 Hardware Structure of Special Vector Boolean Function

For the special nonlinear filter function implementation, we used the Xilinx (Virtex) FPGA, family Automotive Spartan3, device xa3s50-4-vqg100 for synthesis and simulation. Special vectorial Boolean function is presented in Fig. 4, and its synthesis results are shown in Table 9.

Fig. 4
figure 4

Synthesized hardware structure (Schematic technology) of the special nonlinear vector Boolean function

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Table 9 Synthesis results of the special nonlinear vector Boolean function using xa3s50-4-vqg100
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Deb, S., Pal, S. & Bhuyan, B. NMRMG: Nonlinear Multiple-Recursive Matrix Generator Design Approaches and Its Randomness Analysis. Wireless Pers Commun 125, 577–597 (2022). https://doi.org/10.1007/s11277-022-09566-5

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