In this work, we study some generalized multipartite access structures and linear secret sharing schemes for their realizations. Given a multipartite set of participants with m compartments (or levels) and m conditions to be satisfied by an authorized set, we firstly examine the intermediary access structures arousing from the natural case concerning that any c out of m of these conditions suffice, instead of requiring anyone or all of the m conditions simultaneously, yielding to generalizations for both the compartmented and hierarchical cases. These are realized essentially by employing a series of Lagrange interpolations and a simple frequently-used connective tool called access structure product, as well as some known constructions for existing ideal schemes. The resulting schemes are non-ideal but perfect. We also consider nested multipartite access structures, where we let a compartment to be defined within another, so that the access structure is composed of some multipartite substructures. We extend formerly employed bivariate interpolation techniques to multivariate interpolation, in order to realize such access structures. The generic scheme we consider is perfect with a high probability such as 1-O(1/q) on a finite field F_q. In particular, we propose a non-nested generalization for the conventional compartmented access structures, which depicts a stronger way of controlling the additional participants.

<p>Secure computation allows mutually distrusting parties to compute over private data. Such collaborations have widespread applications in social, scientific, commercial, and security domains. However, the overhead of achieving security is a major bottleneck to the adoption of such technologies. In this context, this thesis aims to design the most secure protocol within budgeted computational or network resources by mathematically formulating it as an optimization problem. </p> <p>With the rise in CPU power and cheap RAM, the offline-online model for secure computation has become the prominent model for real-world security systems. This thesis investigates the above-mentioned optimization problem in the information-theoretic offline-online model. In particular, this thesis presents the following selected sample of our research in greater detail. </p> <p>Round and Communication Complexity: Chor-Kushilevitz-Beaver characterized the round and communication complexity of secure two-party computation. Since then, the case of functions with randomized output remained unexplored. We proved the decidability of determining these complexities. Next, if such a protocol exists, we construct the optimal protocol; otherwise, we present an obstruction to achieving security. </p> <p>Rate and Capacity of secure computation: The efficiency of converting the offline samples into secure computation during the online phase is essential. However, investigating this ``production rate'' for general secure computations seems analytically intractable. Towards this objective, we introduce a new model of secure computation -- one without any communication -- that has several practical applications. We lay the mathematical foundations of formulating rate and capacity questions in this framework. Our research identifies the first tight rate and capacity results (a la Shannon) in secure computation. </p> <p>Reverse multiplication embedding: We identify a new problem in algebraic complexity theory that unifies several efficiency objectives in cryptography. Reverse multiplication embedding seeks to implement as many (base field) multiplications as possible using one extension field multiplication. We present optimal construction using algebraic function fields. This embedding has subsequently led to efficient improvement of secure computation, homomorphic encryption, proof systems, and leakage-resilient cryptography. </p> <p>Characterizing the robustness to side-channel attacks: Side-channel attacks present a significant threat to the offline phase. We introduce the cryptographic analog of common information to characterize the offline phase's robustness quantitatively. We build a framework for security and attack analysis. In the context of robust threshold cryptography, we present a state-of-the-art attack, threat assessment, and security fix for Shamir's secret-sharing. </p> <p><br></p>