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On some permutation binomials and trinomials over \(\mathbb {F}_{2^n}\)

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Abstract

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.

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Notes

  1. These are polynomials of the form \(x^rf(x^{q-1\over d})\), and represent mappings of the factor group \(\mathbb {F}_{q}^{*} / C_d\) to itself, where \(C_d\) is the subgroup of \(\mathbb {F}_{q}^{*}\) of index d (see [9] for further details). We review relevant characterization results of this class of binomials in Sect. 1.2.

  2. A polynomial \(f(x)\in \mathbb {F}_q[x]\) is called complete mapping if both f(x) and \(f(x)+x\) are PPs of \(\mathbb {F}_q\), and orthomorphism if both f(x) and \(f(x)-x\) are PPs; for even characteristic both are same. Complete mappings/orthomorphisms are useful for construction of mutually orthogonal latin squares (see [9, 26]).

  3. The case of \(d=2\) was settled in [18]. However, it is relevant for fields of odd characteristic.

  4. In [22, 23], the author characterized these PBs for any d in terms of Lucas sequences. However, as we have stated before, we are interested in more explicit characterization.

  5. PBs of the form \(x^{2{q^2-1\over q-1}+1}+ax\) over \(\mathbb {F}_{q^2}\) were also characterized in the same work.

  6. This approach was taken in [24, 26] (see also [2] and references therein).

  7. Conditions (b) and (c) can be written together (see [25]) as the condition: \(x^rf(x)^{q-1\over d}\) permutes the set \(\{a \in \mathbb {F}_q: a^d=1\}.\)

References

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Acknowledgments

The authors thank anonymous reviewers for their helpful comments and corrections which improved the quality of this manuscript. The first author thanks Mr. Shashank Singh for commenting on an earlier draft.

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  1. Centre of Excellence in Cryptology, Indian Statistical Institute, Kolkata, India

    Srimanta Bhattacharya

  2. TCS Innovations Labs, Hyderabad, India

    Sumanta Sarkar

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  1. Srimanta Bhattacharya
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Correspondence to Srimanta Bhattacharya.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Bhattacharya, S., Sarkar, S. On some permutation binomials and trinomials over \(\mathbb {F}_{2^n}\) . Des. Codes Cryptogr. 82, 149–160 (2017). https://doi.org/10.1007/s10623-016-0229-0

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