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In this paper, we propose a novel encryption scheme based on the concepts of the commutative law of the  k -th order linear recurrences over the finite field   q   for  k  &gt; 2. The proposed encryption scheme is an ephemeral-static, which is useful in situations like email where the recipient may not be online. The security of the proposed encryption scheme depends on the difficulty of solving some Linear Feedback Shift Register (LFSR) problems. It has also the property of semantic security. For  k  = 2, we propose another encryption scheme by using the commutative law of Lucas and Fibonacci sequences having also the same security as proposed for  k  &gt; 2.

On the k -th order lfsr sequence with public key cryptosystems