  A multivariate cryptograpic instance in practice is a multivariate polynomial
system. So the security of a protocol rely on the complexity of solving a
multivariate polynomial system. In this paper there is an overview on a general
algorithm used to solve a multivariate system and the quantity to which the
complexity of this algorithm depends on: the solving degree. Unfortunately, it
is hard to compute. For this reason, it is introduced an invariant: the degree
of regularity. This invariant, under certain condition, give us an upper bound
on the solving degree. Then we speak about random polynomial systems and in
particular what "random" means to us. Finally, we give an upper bound on both
the degree of regularity and the solving degree of such random systems.

Cryptography and Security;Algebraic Geometry;Combinatorics