10 M. R. SCHROEDER
fraction of all possible permutations. Not counting cyclical shifts and
reflections, there are (p - 2) !/2 "inequivalent" permutations. For p = 17,
we have thus 653 837 184 000 such permutations, of which ((( 16) = 8 (about
one in
1011)
have the flat Fourier property. Questions: are there other
permutations that, when used instead of
gn
in (5), have this property?
5. Forbidding property of the Fermat primes
Since Gauss's great discovery of 30 March 1796 concerning the
factorization of
xp
- 1, where p is a Fermat prime,
p =
22n
+ 1 ,
these primes have enjoyed great esteem, spreading from the factoring factories
of the experts to the sweatshops of the amateur. The
4
geometrical"
constructions of regular p-gons rests on the fact that p - 1 has only one
prime factor, namely 2. Now, in the context of wave diffraction, this same
property - instead of permitting a particular construction -forbids a desirable
application, namely the construction of two (or higher) dimensional phase
arrays with the flat Fourier property mentioned above.
The two-dimensional generalization of the periodic sequence (5) is the
periodic array rkt with
k =
(n)modK
and = (n)
m o d L
,
where K and L (in best Chinese remainder fashion) are two coprime factors of
p - 1. Thus there are no n-dimensional (n 1) phase gratings based on the
known Fermat primes 3, 5, 17, 257, 65 537. (The smallest prime for a three-
dimensional array is 31, having a 2 3 5 "unit cell.'*)
6. Euler totients and cryptography
One of the most spectacular applications of number theory in recent times
is public-key cryptography in which each potential recipient of a secret
message publishes his encryption key, thereby avoiding the (often substantial)
problems of secure secret-key distribution. But how can a key be public and
yet produce secret messages? The answer is based on Euler's totient function
|(r) and the role it plays in inverting modular exponentiation. The public key
consists of a modulus r and an exponent s, coprime to |(r). The message is
represented by an integer M, 1 M r, and the encrypted message E is
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