A Hirzebruch genus is a ring homomorphism from the complex or oriented bordism ring to some ring $R$. Any such homomorphism can be extended in the standard way to a homomorphism from the corresponding bordism ring of manifolds with a k-dimensional torus action to the power series ring $R[[x_1, \ldots, x_k]]$. A genus is called rigid on a $T^k$-manifold $M$ if the value of its extension on the bordism class of $M$ is a constant series. The rigidity of genera has been intensively studied in the works of V. Buchstaber, T. Panov and N. Ray, and in many cases this property is equivalent to the multiplicativity of a genus with respect to fiber bundles with fiber $M$. There is a localization formula expressing the equivariant extension of a genus in terms of the fixed points data. Using this formula we proved that the universal complex genus which is rigid on SU-manifolds is the elliptic Krichever genus, and the universal oriented genus which is rigid and vanishing on the octonionic projective (Cayley) plane is the Witten genus.