A crosslinked polymer with high ionic conductivity, an electrolyte using the crosslinked polymer and a process for producing the electrolyte, and a nonaqueous secondary battery using the electrolyte. Crosslinked polymers obtained by a crosslinking reaction between a compound having at least two substituents, in total, of at least one kind selected from the group consisting of .alpha.,.beta.-unsaturated sulfonyl, .alpha., .beta.-unsaturated nitryl and .alpha.,.beta.-unsaturated carbonyl groups in its molecule and a compound having at least two nucleophilic groups in its molecule.