Higher Order $c$-Differentials

In [9], the notion of c-differentials was introduced as a potential expansion of differential cryptanalysis against block ciphers utilizing substitution boxes. Drawing inspiration from the technique of higher order differential cryptanalysis, in this paper we propose the notion of higher order c-derivatives and differentials and investigate their properties. Additionally, we consider how several classes of functions, namely the multiplicative inverse function and the Gold function, perform under higher order c-differential uniformity.

[1]  Sihem Mesnager,et al.  Bent Functions , 2016, Springer International Publishing.

[2]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[3]  Pantelimon Stanica,et al.  C-Differentials, Multiplicative Uniformity, and (Almost) Perfect c-Nonlinearity , 2019, IEEE Transactions on Information Theory.

[4]  Sihem Mesnager,et al.  Solving Xq+1+X+a=0 over Finite Fields , 2019, IACR Cryptol. ePrint Arch..

[5]  Antonia W. Bluher,et al.  On xq+1+ax+b , 2004, Finite Fields Their Appl..

[6]  P. Alam.  ‘L’ , 2021, Composites Engineering: An A–Z Guide.

[7]  Sihem Mesnager,et al.  Bent Functions: Fundamentals and Results , 2016 .

[8]  Lilya Budaghyan,et al.  Construction and Analysis of Cryptographic Functions , 2015, Springer International Publishing.

[9]  Sihem Mesnager,et al.  Investigations on c-(Almost) Perfect Nonlinear Functions , 2021, IEEE Transactions on Information Theory.

[10]  Xiangyong Zeng,et al.  New PcN and APcN functions over finite fields , 2021, Designs, Codes and Cryptography.

[11]  S. Mesnager.  Subclasses of Bent Functions: Hyper-Bent Functions , 2016 .

[12]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[13]  Sihem Mesnager,et al.  Solving X q+1 +X+a=0 over Finite Fields. , 2019 .

[14]  Deng Tang,et al.  Further Cryptographic Properties of the Multiplicative Inverse Function , 2020, IACR Cryptol. ePrint Arch..

[15]  Eli Biham,et al.  Differential cryptanalysis of DES-like cryptosystems , 1990, Journal of Cryptology.

[16]  P. Alam.  ‘Z’ , 2021, Composites Engineering: An A–Z Guide.

[17]  Claude Carlet,et al.  Boolean Functions for Cryptography and Coding Theory , 2020 .

[18]  Daniele Bartoli,et al.  On construction and (non)existence of c-(almost) perfect nonlinear functions , 2020, Finite Fields Their Appl..

[19]  Tor Helleseth,et al.  On the equation x2l+1+x+a=0 over GF(2k) , 2008, Finite Fields Their Appl..

[20]  Lars R. Knudsen,et al.  Truncated and Higher Order Differentials , 1994, FSE.

[21]  L. Budaghyan.  Construction of Planar Functions , 2014 .

[22]  Marie Henderson,et al.  A note on the roots of trinomials over a finite field , 2004, Bulletin of the Australian Mathematical Society.

[23]  Timo Neumann,et al.  BENT FUNCTIONS , 2006 .

[24]  Lei Hu,et al.  Some classes of power functions with low c-differential uniformity over finite fields , 2020, Designs, Codes and Cryptography.

[25]  Xuejia Lai.  Higher Order Derivatives and Differential Cryptanalysis , 1994 .