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Construction of RSBFs with improved cryptographic properties to resist differential fault attack on grain family of stream ciphers

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Abstract

In recent literature, the differential fault analysis (DFA) on Grain family of stream ciphers has been shown to exploit the low algebraic degree of the derivative of the nonlinear combining function h of the stream cipher, h(x) ⊕ h(x ⊕ α). The low algebraic degree allows the DFA adversary to create a linearly independent system of equations generated from the faulty and fault-free keystreams and use these equations to extract the initial state of the NFSR and LFSR stages in the stream cipher. In this paper, we propose a construction scheme for rotation symmetric Boolean functions (RSBFs) h(x) along with an orbit-tuple flip based iterative hill-climbing based construction algorithm for balanced RSBFs with high nonlinearity, low absolute indicator value of global avalanche characteristics (GAC), and high algebraic degree of h(x) ⊕ h(x ⊕ α). The construction algorithm is scalable for higher input variables like n = 9,10,11 as shown in the paper. We find some interesting autocorrelation spectra and Walsh spectra properties for the class of RSBFs and then use them in the construction of RSBFs with improved cryptographic properties. We present the cryptographic properties of the RSBFs constructed for high input variables which can be used to make DFA attack harder using the existing techniques.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, IIT Kharagpur, Kharagpur, India

    Bodhisatwa Mazumdar, Debdeep Mukhopadhyay & Indranil Sengupta

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  1. Bodhisatwa Mazumdar
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  2. Debdeep Mukhopadhyay
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Correspondence to Bodhisatwa Mazumdar.

Appendix:

Appendix:

Table 7 g(b) equations for different values of α for a 5 × 1 RSBF
Full size table

Some instances of constructed RSBFs for higher values of n :

  1. 1.

    RSTT of Output RSBF (n = 9): 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1.

  2. 2.

    RSTT of Output RSBF (n = 10): 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1.

  3. 3.

    RSTT of Output RSBF (n = 11): 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1.

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Mazumdar, B., Mukhopadhyay, D. & Sengupta, I. Construction of RSBFs with improved cryptographic properties to resist differential fault attack on grain family of stream ciphers. Cryptogr. Commun. 7, 35–69 (2015). https://doi.org/10.1007/s12095-014-0108-3

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