On Cryptographic Properties of Random Boolean Functions

Boolean functions used in cryptographic applications have to satisfy various cryptographic criteria. Although the choice of the criteria depends on the cryptosystem in which they are used, there are some properties (balancedness, nonlinearity, high algebraic degree, correlation immunity, propagation criteria) which a cryptographically strong Boolean function ought to have. We study the above mentioned properties in the set of all Boolean functions (all balanced Boolean functions) and prove that almost every Boolean function (almost every balanced Boolean function) satisfies all above mentioned criteria on levels very close to optimal and therefore can be considered to be cryptographically strong