Vectorial Boolean Functions for Cryptography

国際会議参加報告:2014 IEEE International Symposium on Information Theory

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet-Charpin-Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F2n to F2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ges 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12,20,24, they are therefore CCZ-inequivalent to any power function.

Two Classes of Quadratic APN Binomials Inequivalent to Power Functions

Following an example in [12],
we show how to change one coordinate function of an
almost perfect nonlinear
(APN) function in order to obtain new examples. It turns out that
this is a very powerful method to construct new
APN functions. In particular, we show that our approach can
be used to construct a ''non-quadratic'' APN function.
This new example
is in remarkable contrast to all recently constructed functions which
have all been quadratic.
An equivalent function has been found
 independently
by Brinkmann and Leander [8]. However, they
claimed that their function is CCZ equivalent to a quadratic one. In this
paper we give several reasons
why this new function is not equivalent to a quadratic one.

/pdf/a-new-almost-perfect-nonlinear-function-which-is-not-6xui4scplf.pdf

A new almost perfect nonlinear function which is not quadratic

Quadratic and hermitian forms over rings

New families of quadratic almost perfect nonlinear trinomials and multinomials

We present an infinite family of quadrinomial APN functions on GF(2 n ) where n is divisible by 3 but not 9. The family contains inequivalent functions, obtained by setting some coefficients equal to 0. We also discuss the inequivalence proof (by computation) which shows that these functions are new.

A few more quadratic APN functions

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

Determining the nonlinearity of a new family of APN functions

In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For base fields relevant to STBCs, these criteria are exploited, via Springer's theorem, to construct several explicit infinite families of (bi-)quaternion division algebras. These are used to obtain new $2\x 2$ and $4\x 4$ STBCs.

Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras

We propose a full-rate iterative space-time code construction, which uses an algebraic space-time code in order to design a new space-time code of double the code rank. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric multiple input double output codes. In the process, we derive an interesting way of obtaining division algebras, and study their centers and maximal subfields.

Nadya Markin

Papers

New families of quadratic almost perfect nonlinear trinomials and multinomials

A few more quadratic APN functions

Determining the nonlinearity of a new family of APN functions

Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras

Iterated Space-Time Code Constructions From Cyclic Algebras