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∑-protocol has been proved to be a very powerful cryptographic tool and widely used in numerous important cryptographic applications. In this paper, the authors make use of ∑-protocol as a main tool to resolve the following difficult problems 1-3 and to construct three efficient cryptographic protocols 4-6:1) How to construct a protocol for proving a secret integer to be a Blum integer with form PQ, where P, Q are two different primes and both = 3(mod 4);2) How to construct a protocol for proving a secret polynomial with exact degree t - 1 in a (t, n)-threshold secret sharing scheme;3) How to construct witness indistinguishable and witness hiding protocol not from zero-knowledge proof;4) A publicly verifiable secret sharing scheme with information-theoretic security;5) A delegateable signature scheme under the existence of one-way permutations;6) Non-interactive universal designated verifier signature schemes.