On the vector subspaces of $$\mathbb {F}_{2^n}$$ over which the multiplicative inverse function sums to zero

Abstract We study the behavior of the multiplicative inverse function (which plays an important role in cryptography and in the study of finite fields), with respect to a recently introduced generalization of almost perfect nonlinearity (APNness), called k th-order sum-freedom, that extends a classic characterization of APN functions, and has also some relationship with integral attacks. This generalization corresponds to the fact that a vectorial function $$F:\mathbb {F}_2^n\mapsto \mathbb {F}_2^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>↦</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> sums to a nonzero value over every k -dimensional affine subspace of $$\mathbb {F}_2^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> </mml:math> , for some $$k\le n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> (APNness corresponds to $$k=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ). The sum of the values of the inverse function $$x\in \mathbb {F}_{2^n}\mapsto x^{2^n-2}\in \mathbb {F}_{2^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> <mml:mo>↦</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:mrow> </mml:math> over any affine subspace A of $$\mathbb {F}_{2^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:math> not containing 0 (i.e. being not a vector space) has been addressed, thanks to a simple expression of such sum, which shows that it never vanishes. We study in the present paper the case of vector (i.e. linear) subspaces, which is much less simple to handle. The sum depends on a coefficient in subspace polynomials. We study for which values of k the multiplicative inverse function can sum to nonzero values over all k -dimensional vector subspaces. We show that, for every k not co-prime with n , it sums to zero over at least one k -dimensional $$\mathbb {F}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -subspace of $$\mathbb {F}_{2^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:math> . We study the behavior of the inverse function over direct sums of vector spaces and we deduce that the property of the inverse function to be k th-order sum-free happens for k if and only if it happens for $$n-k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> . We derive several other results and we show that the set of values k such that the inverse function is not k th-order sum-free is stable when adding two values of k whose product is smaller than n (and when subtracting two values under some conditions). We clarify the case of dimension at most 4 (equivalently, of co-dimension at most 4) and this allows to address, for every n , all small enough values of k of the form $$3a+4b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> .