New signature schemes from the Hadamard-type <inline-formula><tex-math id="M1">$ (k,t)- $</tex-math></inline-formula>generalized Jacobsthal-Lehmer sequences on the <inline-formula><tex-math id="M2">$ 2- $</tex-m

In this paper, we introduce sequences from the $ t- $generalized Lehmer and Jacobsthal sequences called Hadamard-type $ (k,t)- $generalized Jacobsthal-Lehmer sequences. Then, we prove that these sequences are simply periodic. We define the Hadamard-type $ (k,t)- $generalized Jacobsthal-Lehmer sequences in finite groups and study the $ 2- $generator $ p- $group of nilpotency class $ 2 $. As an interesting application of group theory in cryptography, these sequences are used to develop digital signature algorithms.