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We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $$(\epsilon,\delta)$$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $$(\epsilon,\delta)$$-DP, improving upon previous results that only apply to at most three colors and for $$\epsilon$$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism. 

Due to the globalization of semiconductor manufacturing and test processes, the system-on-a-chip (SoC) designers no longer design the complete SoC and manufacture chips on their own. This outsourcing of the design and manufacturing of Integrated Circuits (ICs) has resulted in several threats, such as overproduction of ICs, sale of out-of-specification/rejected ICs, and piracy of Intellectual Properties (IPs). Logic locking has emerged as a promising defense strategy against these threats. However, various attacks about the extraction of secret keys have undermined the security of logic locking techniques. Over the years, researchers have proposed different techniques to prevent existing attacks. In this article, we propose a novel attack that can break any logic locking techniques that rely on the stored secret key. This proposed TAAL attack is based on implanting a hardware Trojan in the netlist, which leaks the secret key to an adversary once activated. As an untrusted foundry can extract the netlist of a design from the layout/mask information, it is feasible to implement such a hardware Trojan. All three proposed types of TAAL attacks can be used for extracting secret keys. We have introduced the models for both the combinational and sequential hardware Trojans that evade manufacturing tests. An adversary only needs to choose one hardware Trojan out of a large set of all possible Trojans to launch the TAAL attack.