The notion of multi-output rotation symmetric functions was introduced.Firstly

the properties of generalized Walsh spectrum and the generalized autocorrelation functions were presented.Then by constructing matrixes

some necessary and sufficient conditions were given for this class of functions to satisfying cryptological properties of balancedness

correlation immunity and so on.Especially

it presented a necessary and sufficient condition to fulfilling 1st order correlation immunity.At last

when n was odd number

the matrixes of n variables multi-output Plateaued rotation symmetric functions had special properties

so a method was introduced to find the odd number variables multi-output Plateated rotation symmetric functions.And this method only need calculate part of the generalized Walsh cycle spectrum

and the calculation could be reduced by almost half of the amount.