Directed signature is a type of function-based signature with the property that the signature only can be verified by a designated verifier and at certain times, the verifier should be able to convince anyone about the validity of the signature without revealing any secret information about the signature to the public. Taking into consideration the involvement of group decision making, some threshold directed signature schemes based on single number theoretic problems, such as integer factorization, discrete logarithm problem, and elliptic curve discrete logarithm problem, have been developed by cryptographers. Although the single-problem-based schemes are still invincible because there is still no cryptanalyst to find the solution to the problems, in the future, if the enemy or attacker manages to get the polynomial algorithm to solve the single problems, the schemes will no longer be practiced and applied. For such reason, in this paper, we propose a new threshold-directed signature scheme based on integer factorization and discrete logarithm problems. The advantage of our scheme is based on the assumption that it is very unlikely to solve two hard number theoretic problems simultaneously. We also show that our scheme is secured against some cryptographic attacks and also significantly efficient compared with threshold signature scheme based on single problem.
Time-constrained strong multi-designated verifier signature suitable for Internet of things–based collaborative fog computing systems