  The spectral test of random number generators (R.R. Coveyou and R.D.
McPherson, 1967) is generalized. The sequence of random numbers is analyzed
explicitly not just via their n-tupel distributions. The generalized analysis
of many generators becomes possible due to a theorem on the harmonic analysis
of multiplicative groups of residue class rings. We find that the mixed
multiplicative generator with power of two modulus does not pass the extended
test with an ideal result. Best qualities has a new generator with the
recursion formula X(k+1)=a*X(k)+c*int(k/2) mod 2^d. We discuss the choice of
the parameters a, c for very large moduli 2^d and present an implementation of
the suggested generator with d=256, a=2^128+2^64+2^32+62181, c=(2^160+1)*11463.
