Equivalence classes of functions between finite groups.

*(English)*Zbl 1376.20030Summary: Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function \(f\), calculated from the image of its coboundary \(\partial f\). This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if \(f\) is a homomorphism, and takes its maximum value if the graph of \(f\) contains a splitting RDS.

##### MSC:

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |

20D99 | Abstract finite groups |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

PDF
BibTeX
XML
Cite

\textit{K. J. Horadam}, J. Algebr. Comb. 35, No. 3, 477--496 (2012; Zbl 1376.20030)

Full Text:
DOI

##### References:

[1] | Breveglieri, L.; Cherubini, A.; Macchetti, M.; Lee, P. J. (ed.), On the generalized linear equivalence of functions over finite fields, No. 3329, 79-91 (2004), Berlin · Zbl 1094.94028 |

[2] | Browning, K.A., Dillon, J.F., Kibler, R.E., McQuistan, M.T.: APN polynomials and related codes. J. Comb. Inf. Syst. Sci. 34, 135-159 (2009). Special Issue honoring the 75th birthday of Prof. D.K. Ray-Chaudhuri |

[3] | Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inf. Theory 52, 1141-1152 (2006) · Zbl 1177.94136 |

[4] | Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20, 205-244 (2004) · Zbl 1053.94011 |

[5] | Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125-156 (1998) · Zbl 0938.94011 |

[6] | Cavior, S.R.: Equivalence classes of functions over a finite field. Acta Arith. 10, 119-136 (1964) · Zbl 0131.25004 |

[7] | Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti Class II. Des. Codes Cryptogr. 10, 167-184 (1997) · Zbl 0872.51007 |

[8] | Elliott, J.E.H., Butson, A.T.: Relative difference sets. Ill. J. Math. 10, 517-531 (1966) · Zbl 0145.01503 |

[9] | Galati, J.C.: A group extensions approach to relative difference sets. J. Comb. Des. 12, 279-298 (2004) · Zbl 1045.05021 |

[10] | Horadam, K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton (2007) · Zbl 1145.05014 |

[11] | Horadam, K.J.: Relative difference sets, graphs and inequivalence of functions between groups. J. Comb. Des. 18, 260-273 (2010) · Zbl 1291.05221 |

[12] | Logachev, O.A., Salnikov, A.A., Yashchenko, V.V.: Bent functions on a finite abelian group. Discrete Math. Appl. 7, 547-564 (1997) · Zbl 0982.94012 |

[13] | Mullen, G.L.: Weak equivalence of functions over a finite field. Acta Arith. 35, 259-272 (1979) · Zbl 0344.12009 |

[14] | Nyberg, K., Differentially uniform mappings for cryptography, No. 765, 55-64 (1994), New York · Zbl 0951.94510 |

[15] | Pott, A.: Nonlinear functions in abelian groups and relative difference sets. Discrete Appl. Math. 138, 177-193 (2004) · Zbl 1035.05023 |

[16] | Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.