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International Conference on Information Security and Cryptology

ICISC 2013: Information Security and Cryptology -- ICISC 2013 pp 311–322Cite as

A Family of Cryptographically Significant Boolean Functions Based on the Hidden Weighted Bit Function

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8565))

Abstract

Based on the hidden weighted bit function, we propose a family of cryptographically significant Boolean functions. We investigate its algebraic degree and use Schur polynomials to study its algebraic immunity. For a subclass of this family, we deduce a lower bound on its nonlinearity. Moreover, we give an infinite class of balanced functions with very good cryptographic properties: optimum algebraic degree, optimum algebraic immunity, high nonlinearity (higher than the Carlet-Feng function and the function proposed by [25]) and a good behavior against fast algebraic attacks. These functions seem to have the best cryptographic properties among all currently known functions.

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Acknowledgment

The first author would like to thank the financial support from the National Natural Science Foundation of China (Grant No. 61202463) and Shanghai Key Laboratory of Intelligent Information Processing, China (Grant No. IIPL-2011-005).

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Authors and Affiliations

  1. Temasek Laboratories, National University of Singapore, Singapore, 117411, Singapore

    Qichun Wang, Chik How Tan & Timothy Foo

Authors
  1. Qichun Wang
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  2. Chik How Tan
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  3. Timothy Foo
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Corresponding author

Correspondence to Qichun Wang .

Editor information

Editors and Affiliations

  1. EWHA Womans University, Seoul, Korea, Republic of (South Korea)

    Hyang-Sook Lee

  2. Kookmin University, Seoul, Korea, Republic of (South Korea)

    Dong-Guk Han

Appendix

Appendix

The truth table of \({f}\) in Example 1:

$$ \begin{array}{cccccccc} 7076 &{} C0E7 &{} 6DE7 &{} 1AE6 &{} C208 &{} 2149 &{} EC73 &{} 03D3 \\ 2F63 &{} 777B &{} 91F4 &{} 471C &{} 0309 &{} A268 &{} 76F6 &{} 2A79 \\ 7204 &{} E561 &{} C9E6 &{} C229 &{} 18ED &{} 19E9 &{} CB48 &{} 7D2B \\ F6B0 &{} BBA4 &{} AAB5 &{} E13E &{} 845A &{} 88C7 &{} D7BB &{} C93A \\ A0EA &{} 6FC9 &{} 03C5 &{} 53D8 &{} 9F7C &{} 45D2 &{} 87D5 &{} B3B5 \\ BEAE &{} 9348 &{} DEAC &{} 1B5A &{} 8672 &{} CD36 &{} 516E &{} F3B0 \\ 01C7 &{} 707F &{} A956 &{} 315E &{} 3533 &{} 398E &{} 6575 &{} E901 \\ A78D &{} 8E22 &{} A75B &{} 6BC9 &{} 6851 &{} B036 &{} 6D34 &{} F033 \\ ABBF &{} 01B2 &{} 5A24 &{} 49B5 &{} 99F1 &{} 16CE &{} 8878 &{} 057D \\ 2406 &{} E617 &{} DEEC &{} 287A &{} 7B50 &{} 1DEC &{} 14F9 &{} A98A \\ 7202 &{} 2303 &{} 1EF0 &{} CF7D &{} 8567 &{} 3107 &{} 6C10 &{} CCBB \\ 3BD2 &{} 5891 &{} 090D &{} 7881 &{} D5B8 &{} A782 &{} 10B6 &{} 50E7 \\ 66FE &{} 6BD8 &{} D89D &{} 8480 &{} 3325 &{} D481 &{} F06D &{} 5C03 \\ 6088 &{} D479 &{} 703C &{} 7D8B &{} 875D &{} C0AC &{} 6124 &{} 67DC \\ 7741 &{} 2E0E &{} 7B9B &{} EA91 &{} BAE4 &{} 8CB8 &{} D250 &{} 29BC \\ C3AE &{} 449C &{} 74EF &{} E8A3 &{} 4A0E &{} B446 &{} 81D9 &{} D479 \\ 3B70 &{} 1690 &{} 7FBC &{} 52E2 &{} 8EA2 &{} F3ED &{} EFAD &{} B0A8 \\ BD1E &{} 406C &{} F483 &{} 6D8A &{} 3BA7 &{} 5939 &{} D954 &{} 404A \\ B5C7 &{} E6A7 &{} 0153 &{} 65A2 &{} 8527 &{} 4586 &{} 725B &{} A023 \\ E438 &{} 44B9 &{} 9A8C &{} 1A4F &{} 6DCF &{} 6624 &{} 3A38 &{} 1B3A \\ 48F3 &{} 9972 &{} F2B4 &{} F9E1 &{} FC0A &{} 11B9 &{} 2B81 &{} D04E \\ 2FEE &{} C341 &{} 60BE &{} DF50 &{} A39F &{} FC9F &{} 2B87 &{} B876 \\ F021 &{} 1DD7 &{} AE3F &{} 3C9E &{} FF58 &{} B914 &{} C879 &{} 3FE5 \\ 0EDC &{} 205F &{} 11D1 &{} 37B1 &{} 65FD &{} 145A &{} DC9F &{} 6758 \\ 83C3 &{} 8967 &{} C721 &{} 043B &{} 0C7D &{} 3E15 &{} A68B &{} 7BF6 \\ 9678 &{} FAEA &{} 6857 &{} B6BE &{} 19FB &{} E594 &{} C40F &{} 9BE9 \\ D2FB &{} 3936 &{} 1DAB &{} A634 &{} C06E &{} F84B &{} A01C &{} A610 \\ 3E84 &{} 889C &{} 07BB &{} 5386 &{} C1F4 &{} F2AB &{} E1D9 &{} 1C09 \\ E640 &{} 4EFA &{} D18A &{} 7E23 &{} C912 &{} 3A38 &{} 8213 &{} 7CE4 \\ 32BB &{} 8A8B &{} 7D16 &{} B97F &{} 9DF1 &{} EA67 &{} 733A &{} 309F \\ 0FD2 &{} 759A &{} 4EA9 &{} 7E17 &{} A05E &{} 1B00 &{} 032E &{} AAB0 \\ C611 &{} DB0A &{} B98F &{} ED5B &{} 7F6D &{} 0C18 &{} 0CDF &{} D03A \end{array} $$

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Wang, Q., Tan, C.H., Foo, T. (2014). A Family of Cryptographically Significant Boolean Functions Based on the Hidden Weighted Bit Function. In: Lee, HS., Han, DG. (eds) Information Security and Cryptology -- ICISC 2013. ICISC 2013. Lecture Notes in Computer Science(), vol 8565. Springer, Cham. https://doi.org/10.1007/978-3-319-12160-4_19

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  • DOI: https://doi.org/10.1007/978-3-319-12160-4_19

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