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One of the key research directions aligning with the current and emerging consumer technology is cybersecurity. A variety of technologies including Internet-of-Things (IoT), IoT-Edge/Fog Computing, and embedded systems are gaining more importance in designing smart applications (e.g. smart cities, smart villages, and smart healthcare) with minimal human interventions. All the devices and datacentres in these applications are connected to the Internet for easy and smooth data transmission. Considering the Internet and communication medium properties, attackers get an easy chance to become part of the system and participate in the data communication process. This article presents thoughts on paradigm shift next generation cryptosystems to overcome the vulnerabilities of the omnipresent conventional cryptosystems.

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Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. Let G i n be the family of restrained dominating sets of a graph G of order n with cardinality i, and let dr(Gn, i) = |G i n |. The restrained domination polynomial (RDP) of Gn, Dr(Gn, x) is defined as Dr(Gn, x) = n i=γr (Gn ) dr(Gn, i)x i . In this paper, we focus on the RDP of cycles and have, thus, introduced several novel ways to compute dr(Cn, i), where Cn is a cycle of order n. In the first approach, we use a recursive formula for dr(Cn, i); while in the other approach, we construct a generating function to compute dr(Cn, i). We also develop an algorithm, based on integer partitioning and circular permutation, to compute dr(Cn, i). This gives us an upper bound on the number of restrained dominating sets of a fixed size for Cn.