  A new method is used to resolve a long-standing conjecture of Niho concerning
the crosscorrelation spectrum of a pair of maximum length linear recursive
sequences of length $2^{2 m}-1$ with relative decimation $d=2^{m+2}-3$, where
$m$ is even. The result indicates that there are at most five distinct
crosscorrelation values. Equivalently, the result indicates that there are at
most five distinct values in the Walsh spectrum of the power permutation
$f(x)=x^d$ over a finite field of order $2^{2 m}$ and at most five distinct
nonzero weights in the cyclic code of length $2^{2 m}-1$ with two primitive
nonzeros $\alpha$ and $\alpha^d$. The method used to obtain this result proves
constraints on the number of roots that certain seventh degree polynomials can
have on the unit circle of a finite field. The method also works when $m$ is
odd, in which case the associated crosscorrelation and Walsh spectra have at
most six distinct values.
