The motivation of this thesis is to understand how nodes can cooperate in a particular relay channel, say a relay channel with orthogonal link between relay and destination. We are especially interested in the scenario where relay cannot decode the message sent because the coding rate is too large vis-a-vis what it can handle, and try to investigate the optimality of compress-and-forward strategy for this scenario. Specifically, noting that relay's compression is based on the unconditional distribution of its observation, it is natural to ask if relay can do better if it utilizes the source's codebook information, such that the performance of the relay network is improved? To answer this key question, we need to find the posterior Conditional Distribution of channel Output given Codebook Information (namely CDOCI) for the channel between source and relay. Firstly, we model it as Binary Symmetric Channel (BSC) and show that under the now standard random coding framework, if the input distribution is uniform, then with high probability, the CDOCI is asymptotically uniform when coding rate is greater than the channel capacity and the block length is sufficiently large. Then it is shown that under the discrete memoryless channel (DMC), for those strongly typical output sequences, with high probability, the CDOCI is also asymptotically uniform and close to the unconditional distribution, for rate above capacity and sufficiently large block length. These results implicate that relay can hardly do better with codebook information used. To confirm this implication, we show that for sufficiently large block length, the rate needed for relay to forward its observation when the codebook information is utilized approaches the rate needed when the relay simply ignore the codebook information, if the coding rate at source is larger than channel capacity. Now the answer to the above key question is apparent: in the cases of BSC and DMC, even if the relay tries to utilize the information obtained by knowing the codebook used at the source, it can hardly do better than simply ignore the codebook information. Therefore, the compress-and-forward strategy is kind of optimal in this sense, under the random coding framework.

The motivation of this thesis is to understand how nodes can cooperate in a particular relay channel, say a relay channel with orthogonal link between relay and destination. We are especially interested in the scenario where relay cannot decode the message sent because the coding rate is too large vis-a-vis what it can handle, and try to investigate the optimality of compress-and-forward strategy for this scenario. Specifically, noting that relay's compression is based on the unconditional distribution of its observation, it is natural to ask if relay can do better if it utilizes the source's codebook information, such that the performance of the relay network is improved? To answer this key question, we need to find the posterior Conditional Distribution of channel Output given Codebook Information (namely CDOCI) for the channel between source and relay. Firstly, we model it as Binary Symmetric Channel (BSC) and show that under the now standard random coding framework, if the input distribution is uniform, then with high probability, the CDOCI is asymptotically uniform when coding rate is greater than the channel capacity and the block length is sufficiently large. Then it is shown that under the discrete memoryless channel (DMC), for those strongly typical output sequences, with high probability, the CDOCI is also asymptotically uniform and close to the unconditional distribution, for rate above capacity and sufficiently large block length. These results implicate that relay can hardly do better with codebook information used. To confirm this implication, we show that for sufficiently large block length, the rate needed for relay to forward its observation when the codebook information is utilized approaches the rate needed when the relay simply ignore the codebook information, if the coding rate at source is larger than channel capacity. Now the answer to the above key question is apparent: in the cases of BSC and DMC, even if the relay tries to utilize the information obtained by knowing the codebook used at the source, it can hardly do better than simply ignore the codebook information. Therefore, the compress-and-forward strategy is kind of optimal in this sense, under the random coding framework.

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<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>

M. Tech (Department of Chemistry, Faculty of Applied and Computer Sciences), Vaal University of Technology. / Activated carbons have been produced from various carbonaceous source materials including nutshells, peat, wood, coir, lignite, coal and petroleum pitch and the produced carbons have been used for adsorption of inorganic and organic compounds from numerous matrices. Activated carbons are characterized by large surface area and high degree of microporosity. The ability of activated carbon to adsorb gold from solutions, which is present in very low concentrations while loading to fairly high concentrations, has made it an attractive material for the concentration of gold from dilute solutions. Consequently, the use of activated carbons for the recovery of gold from cyanide-leached liquors has gained wide acceptance in the mining industry. However, the price of commercially prepared carbons, the time taken for them to arrive, the breakage of the carbon during transportation and the interest in utilization of various local wastes for the generation of adsorbents has led to a search of more cost effective and time friendly source for the activated carbons. The focus of this research was to conduct a study in which a comparison was conducted between untreated coconut shell derived activated carbon (CAC) and Macadamia nutshell derived activated carbon (MAC) for the adsorption of gold. These activated carbons were modified with HNO3 and H3PO4 to increase their surface adsorption properties. This was done in order to explore if these activated carbons prepared from Macadamia shells could be an attractive alternative or a complementary supplement to the coconut shell based carbons that are currently being used in the gold extraction industry. The modification of the commercially prepared Macadamia activated carbons was done with 3 different concentrations for both nitric acid and phosphoric acid. The modified activated carbons were labelled MACP20%, MACP40% and MACP60%, to signify the materials prepared from 20% (v/v) H3PO4, 40% (v/v) H3PO4 and 60% (v/v) H3PO4 , respectively . Same labelling was used for 20% (v/v) HNO3, 40% (v/v) HNO3 and 55% (v/v) HNO3 modifications to correspond to MACN20%, MACN40% and MACN55%, respectively. Also, untreated coconut shell derived activated carbon (CAC) and Macadamia nutshell derived activated carbon (MAC) were investigated for gold adsorption for comparison purposes. All the activated carbons prepared in the iii | P a g e study were characterized with Brunauer-Emmet-Teller (BET), scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), Thermo gravimetric analysis (TGA), elemental analysis (EL) and X-ray diffraction spectroscopy (XRD). The physical properties of the activated carbons were done by determining attrition, ash content, volatile matter, and moisture content of all the activated carbons. Various parameters that affect selective adsorption such as the effect of initial concentration, time, agitation speed, interfering species, and dose of the adsorbent were investigated. Optimal parameters for gold ion adsorption were as follows: solution pH, 10; contact time, 6 h; agitation speed 150 rpm; sorbent amount 4 g and 5.5 ppm for initial concentration of gold. The observed selectivity order was not the same for all the adsorbents but the adsorption of gold was found to be mostly influenced by the presence of nickel and least influenced by copper. The MACP60% was found to be the most effective from the three concentrations investigated for the phosphoric acid modified activated carbons yet proved to have lower adsorption capabilities compared to CAC. The MACN55% was found to be the most efficient and displayed similar adsorption capabilities to those of CAC.