The multivariate method strikes again: New power functions with low differential uniformity in odd characteristic

Abstract Let f ( x ) = x d be a power mapping over $\mathbb {F}_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $\mathcal {U}_{d}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> </mml:math> the maximum number of solutions $x\in \mathbb {F}_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> of ${\Delta }_{f,c}(x):=f(x+c)-f(x)=a\text {, where }c,a\in \mathbb {F}_{n}\text { and } c\neq 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>Δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mo>)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mtext>, where</mml:mtext> <mml:mspace /> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mspace /> <mml:mtext>and</mml:mtext> <mml:mspace /> <mml:mi>c</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:math> . f is said to be differentially k -uniform if $\mathcal {U}_{d} =k$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> </mml:math> . The investigation of power functions with low differential uniformity over finite fields $\mathbb {F}_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> of odd characteristic has attracted a lot of research interest since Helleseth, Rong and Sandberg started to conduct extensive computer search to identify such functions. These numerical results are well-known as the Helleseth-Rong-Sandberg tables and are the basis of many infinite families of power mappings $x^{d_{n}},n \in \mathbb {N},$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℕ</mml:mi> <mml:mo>,</mml:mo> </mml:math> of low uniformity (see e.g. Dobbertin et al. Discret. Math. 267 , 95–112 2003; Helleseth et al. IEEE Trans. Inform Theory, 45 , 475–485 1999; Helleseth and Sandberg AAECC, 8 , 363–370 1997; Leducq Amer. J. Math. 1 (3) 115–123 1878; Zha and Wang Sci. China Math. 53 (8) 1931–1940 2010). Recently the crypto currency IOTA and Cybercrypt started to build computer chips around base-3 logic to employ their new ternary hash function Troika, which currently increases the cryptogrpahic interest in such families. Especially bijective power mappings are of interest, as they can also be employed in block- and stream ciphers. In this paper we contribute to this development and give a family of power mappings $x^{d_{n}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> with low uniformity over $\mathbb {F}_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> , which is bijective for p ≡ 3 mod 4. For p = 3 this yields a family $x^{d_{n}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> with $3\leq \mathcal {U}_{d_{n}}\leq 4,$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:msub> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> </mml:math> where the family of inverses has a very simple description. These results explain “open entries” in the Helleseth-Rong-Sandberg tables. We apply the multivariate method to compute the uniformity and thereby give a self-contained introduction to this method. Moreover we will prove for a related family of low uniformity introduced in Helleseth and Sandberg ( AAECC , 8 363–370 1997) that it yields permutations.