Based　on　preimage　sampleable　functions　and　hard　problems　for　lattices,　we　propose　two　lattice-based　multi-signature　schemes,　which　include　the　trivial　broadcasting　multi-signature　scheme　and　the　sequential　multi-signature　scheme.　In　the　broadcasting　multi-signature　scheme,　the　length　of　the　multi-signature　varies　linearly　with　the　number　of　the　signers.　In　the　sequential　multi-signature　scheme,　the　length　of　the　multi-signature　is　fixed　and　independent　of　the　number　of　the　signers.　These　two　multi-signature　schemes　can　allow　multiple　signers　to　sign　the　same　message　cooperatively.