Notice of Removal: On the Inverses and Their Hamming Weights of Known APN, 4-Differentially Uniform and CPP Exponents Over F₂ⁿ

Abstract

This article was initially motivated by an interesting work of Kyureghyan and Suder, in 2014, whereas an open problem it was stated to express the Hamming weights of the inverses of Kasami exponent 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2<i>k</i></sup> - 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i></sup> + 1 modulo 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup> - 1 in terms of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> when it exists. In 2020, Kölsch solved the open problem by developing the modular add-with-carry approach first formally introduced in 2001. By these two nice works, the Hamming weights of inverses of all known APN and 4-differentially uniform exponents over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>n</i></sub> have been determined. However, their common difficulty is guessing the inverse related structures by performing some numerical experiments. As also mentioned by Fu in [Discrete Math. 345, 112658, 2022], where inversions of some exponents are revisited, it is unclear how their approaches can be generalized to others except that they studied. In this paper, we first present a new way to find the Hamming weights of inverses of Kasami exponents, which might be generalized to other exponents which are independent of the field size. Although this provides an alternative simpler solution to the aforementioned open problem and the found representations for inverses are fundamentally different from ones found by Kölsch, the representations still do not express the least positive residues of inverses in a closed form. Then, we present a method to find explicitly the least positive residue of the modular inverse without a prior guess and experiment when the exponent depends on the field size. With this method, we determine the inverses and their Hamming weights of all known APN, 4-differentially uniform and complete permutation polynomial (CPP) exponents over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>n</i></sub> , which solves a research problem raised in [Des. Codes, Cryptogr. 88, 2597-2621, 2020].