## Symmetric Boolean functions

Anne Canteaut and Marion Videau

INRIA, projet CODES

` {Anne.Canteaut,Marion.Videau}@inria.fr`
Regular paper in IEEE Trans. Inform. Theory, To appear.

### Abstract

We present an extensive study of symmetric Boolean functions,
especially of their cryptographic properties.

Our main result
establishes the link between the periodicity of the simplified value
vector of a symmetric

Boolean function and its degree. Besides the
reduction of the amount of memory required for representing

a
symmetric function, this property has some consequences from a
cryptographic point of view. For instance,

it leads to a
new general bound on the order of resiliency of symmetric functions,
which improves Siegenthaler's bound.

The
propagation characteristics of these functions are also addressed and the
algebraic normal forms of all their

derivatives are given.
We finally detail the characteristics of the symmetric
functions of degree at most *7*,

for any number of variables. Most
notably, we determine all balanced symmetric functions of degree
less than or equal to*7*.
**KEYWORDS**

Symmetric functions, Boolean functions, degree,
correlation-immunity, resiliency, propagation criterion, derivation.