Abstract
Cryptography is necessary to the functioning of modern society, yet
the design of symmetrickey ciphers is with some justification
considered a Black Art. Certainly the ``blackest" part of such
design is the construction of the nonlinear substitution (or Sbox)
function that provides the bulk of a cipher's resistance against
attack. Up until very recently, these functions were designed
primarily as random lookup tables and tested against known attacks.
This is hardly ideal.
However, a new method of designing Sboxes has been advanced that
represents cipher components as functions on finite fields of the
form . Using a wide array of tools suitable security
criteria can be defined, and several categories of functions can be
proven to satisfy the criteria to maximal or nearmaximal extents.
Among the tools are the field trace and Walsh transform
. Representing the criteria will be measures of
nonlinearity , and the concept of a function being
differentially uniform, which tracks correlation between
input and output differences. Finally, functions of the form
and will be paid special attention.
This stuff is not just of theoretic interest the cipher Rijndael
was developed using this process, and in 2000 was chosen from a
field of very qualified candidates to be the Advanced Encryption
Standard by NIST, in large part due to its clean,
mathematicallyderived design. Mathematics trumps black magic.
