Despite the documented need to train and educate more cybersecurity professionals, we have little rigorous evidence to inform educators on effective ways to engage, educate, or retain cybersecurity students. To begin addressing this gap in our knowledge, we are conducting a series of think-aloud interviews with cybersecurity students to study how students reason about core cybersecurity concepts. We have recruited these students from three diverse institutions: University of Maryland, Baltimore County, Prince George's Community College, and Bowie State University. During these interviews, students grapple with security scenarios designed to probe student understanding of cybersecurity, especially adversarial thinking. We are analyzing student statements using a structured qualitative method, novice-led paired thematic analysis, to document student misconceptions and problematic reasonings. We intend to use these findings to develop Cybersecurity Assessment Tools that can help us assess the effectiveness of pedagogies. These findings can also inform the development of curricula, learning exercises, and other educational materials and policies.

We present and experimentally evaluate Spread Identity (SI) – a new dynamic network address remapping mechanism that provides anonymity and DDoS defense capabilities for Internet communications. For each session between a source and destination host, the trusted source gateway dynamically and randomly assigns an IP address for the source host from the pool of all routable IP addresses allocated to the source organization. Similarly, in response to a name resolution query from the source gateway, the trusted authoritative DNS server for the destination organization dynamically assigns an IP address for the destination host from the pool of all routable IP addresses allocated to the destination organization. These assignments depend upon the state of the server (including load, residual capacity, time of day) and policy. Different hosts can share the same IP address when communicating with distinct peers. Each gateway creates a NAT entry, valid for the communication session, based on the dynamic assignment by its organization. An eavesdropper listening to packets flowing through the Internet between the source and destination gateways learns only the source and destination domains; the eavesdropper cannot see the actual complete IP addresses of the source and destination hosts. In addition, SI enhances DDoS defense capabilities by enabling packet filtering based on destination addresses. With multiple IP addresses for the same destination, filtering based on destination addresses can block attackers without necessarily blocking legitimate users. Deploying SI requires changes to organizational gateways and, possibly, to the edge-routers that interface with organizational gateways; but network mechanisms farther upstream, including the core routers in the Internet, remain unchanged. Likewise, the installed base of operating systems running individual hosts in the internal network, together with the end-user application suites they support, remain untouched. SI mechanisms are backward compatible, incrementally deployable, and robustly scalable. A naïve implementation of SI can increase the DNS traffic; however, when SI is implemented at both the source and the destination ends, it is possible for SI to reduce DNS traffic. Ns-2 simulations and experiments on the DeterLab test bed corroborate the main hypotheses and demonstrate advantages of the SI paradigm. Ns-2 simulations demonstrate that file transfer success rates for our SI DDoS protection mechanism are similar to those of filter- and capability-based approaches, with lower file transfer times than for filter-based approaches. DeterLab trials demonstrate that SI consumes similar resources (connection establishment time, network address translation table size, packet forwarding rate and memory) to those of a typical single NAT system, though with higher name resolution times.

Maintaining the confidentiality of data sent to clouds is a vital issue that must be satisfactorily resolved as a pre-condition in order for the cloud computing paradigm to survive and thrive. In this paper we explore a completely new paradigm to approximately achieve the functionality implied by "Oblivious Data Processing(ODP)". Our strategy is to partition the data as well as the underlying Residue Number System (RNS) in the Residue Domain (RD) and then distribute and process the partitions independently. We have also outlined some new fundamental limitations to ODP in general and identified application scenarios where ODP can be achieved as well as those wherein ODP appears to be doomed to fail in the long run.

We present a method to visualize and authenticate a cryptographically negotiated key for a secure phone call. That is, each caller is presented with a graphical representation of the key and through verbal interaction (i.e., side-channel authentication) they describe what they see. If they agree, the key is authenticated and the secure media session continues. The strength of the approach lies in the vocal recognition of the callers, and their ability to confirm the image displayed by their system. The necessary degree of visual recognition is achieved by using basic shapes, color and count. People, regardless of language or age, can easily identifying these images. Our experience shows that they can communicate what they see with little effort and terminate the call when they differ. We believe that this approach reverses the current trend in security to divest users from the underlying cryptographic principles supporting secure systems by abstracting these principles to a comprehensible and visual form. This paper demonstrates that visualization and the human factor can play a pivotal role in establishing a secure communication channel. This short paper discusses how a key is visualized and provides some initial user feedback. We have named this approach the Short Authentication SymbolS VisuallY (SASSY.)

The core contribution of this paper is a three-fold improvement of the Haar discrete wavelet transform (DWT). It is modified to efficiently transform a multiclass- (rather than numerical-) valued function over a multidimensional (rather than low dimensional) domain, or transform a multiclass-valued decision tree into another useful representation. We prove that this multidimensional, multiclass DWT uses dynamic programming to minimize (within its framework) the number of nontrivial wavelet coefficients needed to summarize a training set or decision tree. It is a spatially localized algorithm that takes linear time in the number of training samples, after a sort. Convergence of the DWT to benchmark training sets seems to degrade with rising dimension in this test of high dimensional wavelets, which have been seen as difficult to implement. This multiclass multidimensional DWT has tightly coupled applications from learning "dyadic" decision trees directly from training data, rebalancing or converting preexisting decision trees to fixed depth boolean or threshold neural networks (in effect parallelizing the evaluation of the trees), or learning rule/exception sets represented as a new form of tree called an "E-tree", which could greatly help interpretation/visualization of a dataset.

Modular reduction is a fundamental operation in cryptographic systems. Most well known modular reduction methods including Barrett's and Montgomery's algorithms leverage some-pre computations to avoid divisions so that the main complexity of these methods lies in a sequence of two long multiplications. For large wordlengths a multiplication which is tantamount to a linear convolution is performed via the fast Fourier transform (FFT) or other transform-based techniques as in the Schonhage-Strassen multiplication algorithm. We show a fundamental property (the separation principle): in a modular reduction based on long multiplications, the linear convolution required by one of the two long multiplications can be replaced by a cyclic convolution, and the halves can be separated using other information available due to the intrinsic redundancy of the operations. This reduces the number of operations by about 25%. We demonstrate that both Barrett's and Montgomery's methods can be sped up by using the aforementioned fundamental principle. It is shown that a direct application of this algorithm to modular exponentiation (either using Barrett's or Montgomery's methods) can be expected to yield about about 17% speedup.

The explosive growth in wireless (and wired) networking technologies and services indicates that multiple means of network connectivity will become available in the near future. For example, stationary and mobile hosts currently support Internet access via wired LANs, Wireless LANs/PANs (e.g., 802.11x, 802.15) or wide area wireless cellular phone and data networks (like GSM). In essence, heterogeneous multi-homing is now a necessity for all hosts (mobile or non-mobile). In order to tap the full potential of such heterogeneous multi-homing, we introduce the novel “Spread Identity (SI)” communications paradigm. Therein, the concept of multi-homing is extended to allow each interface to simultaneously assume multiple addresses and dynamically acquire and release them as needed which is tantamount to “Spreading Identity” at the network( IP) level and has fundamental implications for security. In this paper we show how the spread Identity mechanisms can effectively (1) Mitigate DDOS attacks by rate-limiting the number of name-resolution responses. (2) Quickly detect and neutralize resource-overload type DDOS attacks that cannot be prevented by rate-limiting (3) Enable surviving the remaining types of DDOS attacks by quenching destination addresses they target (in essence by changing the Identity) (4) and preventing future attack flows by returning NULL addresses, and re-directing the attackers against one-another. We demonstrate that Spread Identity mechanisms can also be leveraged to bolster the security of single sourceto- destination flows. SI mechanisms can attain the same level of security as that of a single link with Strong Security Infrastructure (SSI) at a lower cost (in terms of the infrastructure required and the encryption effort needed). The fundamental concept of Spreading-Identity revealed herein is more general and potentially applicable to other scenarios beyond Internet/Electronic communications.

Particular levels of partial fault tolerance (PFT) in feedforward artificial neural networks of a given size can be obtained by redundancy (replicating a smaller normally trained network), by design (training specifically to increase PFT), and by a combination of the two (replicating a smaller PFT-trained network). This letter investigates the method of achieving the highest PFT per network size (total number of units and connections) for classification problems. It concludes that for non-toy problems, there exists a normally trained network of optimal size that produces the smallest fully fault-tolerant network when replicated. In addition, it shows that for particular network sizes, the best level of PFT is achieved by training a network of that size for fault tolerance. The results and discussion demonstrate how the outcome depends on the levels of saturation of the network nodes when classifying data points. With simple training tasks, where the complexity of the problem and the size of the network are well within the ability of the training method, the hidden-layer nodes operate close to their saturation points, and classification is clean. Under such circumstances, replicating the smallest normally trained correct network yields the highest PFT for any given network size. For hard training tasks (difficult classification problems or network sizes close to the minimum), normal training obtains networks that do not operate close to their saturation points, and outputs are not as close to their targets. In this case, training a larger network for fault tolerance yields better PFT than replicating a smaller, normally trained network. However, since fault-tolerant training on its own produces networks that operate closer to their linear areas than normal training, replicating normally trained networks ultimately leads to better PFT than replicating fault-tolerant networks of the same initial size.

It was shown in Phatak, Choi, and Koren (1993) that the neural network implementation of the n-k-n encoder-decoder has a minimal size of k = 2, and it was shown how to construct the network. A proof was given in Phatak and Koren (1995) that in order to achieve perfect fault tolerance by replicating the hidden layer, the required number of replications is at least three. In this note, exact lower bounds are derived for the number of replications as a function of n, for the n-2-n network and for the n- log2n-n network.

Data striping across multiple network interfaces and its applications to mobile environments was recently investigated (Phatak and Goff, 2002). Therein, a network-layer Internet protocol (IP)-in-IP encapsulation mechanism was proposed to aggregate the bandwidth of multiple network paths by striping a single transport layer [transmission control protocol (TCP)] connection across them. The analysis and experimental results from that study demonstrated fundamental limitations to TCPs performance in such scenarios. In this paper, we therefore propose a method of overcoming these limitations by striping data at the transport layer. For a proof-of-concept demonstration we use a substantially enhanced version of the stream control transmission protocol (SCTP). Our analytical results and experimental data illustrate that there are significant advantages to using a transport protocol with native support for the simultaneous use of multiple network interfaces, as opposed to stretching TCP to a point where it is no longer effective. This work naturally leads to another fundamental issue of end-to-end support for host mobility at the transport layer. Our analysis and results demonstrate that transport layer support for multiple network (IP) interfaces, together with the capability to dynamically add or delete IP addresses can yield the following advantages: higher bandwidth, load balancing, and increased fairness, enhanced reliability, and end-to-end support for host mobility. This is independent of the underlying network-layer and, hence, is applicable to static/wired, as well as wireless/ad hoc environments. The proposed protocol offers a unified solution to both data striping across multiple networks interfaces, as well as end-to-end mobility support.

We compare implementations of Duprat and Muller's branching CORDIC and Phatak's double step branching (DSB)-CORDIC algorithms for sine and cosine evaluation. For reference we also report on classical CORDIC implementations for the same wordlengths. We have also implemented double stepping in the classical algorithm and report on the performance of this method. CORDIC evaluation of sine and cosine includes two parts, the zeroer and the rotator. We discuss implementation issues related to the minimization of the delay of each iteration of the algorithm (including delays for both the zeroer as well the rotator). We then examine hybrid methods that select the components from different algorithms (such as a DSB zeroer together with a classical rotator or vice versa).

The Haar discrete wavelet transform is used as inspiration for a new simple and fast algorithm to train multiclass dyadic decision trees. A localized ordering of the training data changes a multidimensional Haar transform into the one dimensional case, and avoids cache misses. The resulting tree has very simple structure convenient for direct implementation as a fixed-depth threshold network and very fast evaluation of the classification function. The simple structure of the tree is also conducive to good compression of the function. We implement and test the learning algorithm and threshold network but without incorporating sampled interpolated-value points for improved generalization. Compression appears good in low dimensions.

Web access from mobile devices presents its own unique challenges because of severe resource constraints on the mobile devices (power, form factor, bandwidth, etc.). Hence, instead of reacting to a user's requests, it would be better to try and predict a user's actions. This would allow time for the server (on the fixed-wired side) to pre-fetch data and pre-process it into a wireless-friendly format (such as the PQA format required by Palm Pilots). Information that flows on the wireless link should be tailored to match what the user wants (rather than making the user wander through Web, which wastes bandwidth and increases latency experienced by the user). Adaptive user clustering and profiling is essential to be able to accurately predict user actions. In this paper we present results of our clustering and personalization project. We compare several distance measures used in clustering. We introduce a new measure to assess the quality of clustering independent of the distance measure used in the clustering-algorithm. We also compare different strategies.

With ubiquitous computing and network access, multiple network conduits are becoming available to mobile as well as static hosts. Selection of the preferred mode of data transfer is a dynamic optimization problem depending on the type of application, its bandwidth/latency/jitter requirements, current network status, cost, power consumption, battery life, and so on. Furthermore, since wireless bandwidth is likely to remain a scarce resource, we foresee scenarios wherein mobile hosts require simultaneous data transfer across multiple IP interfaces to obtain higher overall bandwidth. We present a brief overview of related work identifying schemes that might be applicable to the problem, along with their feasibility, and pros and cons. We then propose a new mechanism to aggregate the bandwidth of multiple IP links by splitting a data flow across multiple network interfaces at the IP level. Our method is transparent to transport (TCP/UDP) and higher layers. We have analyzed the performance characteristics of the aggregation scheme and demonstrated significant gain when the links being aggregated have similar bandwidth and latency. The use of multiple interfaces also enhances reliability. Our analysis identifies the conditions under which the proposed scheme, or any other scheme that stripes a single TCP connection across multiple IP links, can be used to enhance throughput. Several interesting directions for future work have also been identified.

In mobile ad hoc networks, the topology of the network is constantly changing as nodes move in and out of each other's range, breaking and establishing links. TCP performs poorly in such networks because packets that are lost due to path disconnections trigger TCP's congestion avoidance mechanisms. We investigate the effect of preemptive routing protocols, where an alternative path is found before an actual disconnection occurs, on the performance of TCP. Preemptive routing should perform well for TCP traffic because it reduces the delays caused by TCP's unnecessary use of congestion avoidance when paths break. We observe this behavior under some, but not all scenarios. Specifically, it appears that when the network is saturated, the additional traffic introduced by preemptive routing causes small degradation in performance. In the analysis process, we encountered an unfairness problem resulting from interaction between the routing protocol and the MAC layer under multiple continuous transmission cases. Similar unfairness problems were encountered by other studies-however the observations of those studies related those problems to the number of hops, and not the routing effects as we observed. This motivates the study of fairer wireless MAC protocols for multi-hop and ad hoc networks.

Existing on-demand ad-hoc routing algorithms initiate route discovery only after a path breaks, incurring a significant cost in detecting the disconnection and establishing a new route. In this work, we investigate adding proactive route selection and maintenance to on-demand ad-hoc routing algorithms. More specifically, when a path is likely to be broken, a warning is sent to the source indicating the likelihood of a disconnection. The source can then initiate path discovery early, potentially avoiding the disconnection altogether. A path is considered likely to break when the received packet power becomes close to the minimum detectable power (other approaches are possible). Care must be taken to avoid initiating false route warnings due to fluctuations in received power caused by fading, multipath effects and similar random transient phenomena. Experiments demonstrate that adding proactive route selection and maintenance to DSR and AODV (on-demand ad hoc routing protocols) significantly reduces the number of broken paths, with a small increase in protocol overhead. Packet latency and jitter also goes down in most cases. We also show some experimental results obtained by running TCP on top of the proactive routing schemes proposed. Several improvements and extensions are also discussed. Pro-active route selection and maintenance is general and can be used with other routing algorithms and optimizations to them.

It is well-known that constant-time addition, in which the execution delay is independent of operand lengths, is feasible only if the output is expressed in a redundant representation. There are many ways of introducing redundancy and the specifics of the redundant format employed can have a major impact on the performance of constant-time addition and digit set conversion. This paper presents a comprehensive analysis of constant-time addition and simultaneous format conversion. We consider full as well as partially redundant representations, where not all digit positions are redundant. The number of redundant digits and their positions can be arbitrary, yielding many possible redundant representations. Format conversion refers to changing the number and/or position of redundant digits in a representation. It is shown that such a format conversion is feasible during (ie, simultaneous with) constant time addition, even if all three operands (the two inputs and single output) are represented in distinct redundant formats. We exploit "equal-weight grouping" (EWG), wherein bits having the same weight are grouped together to achieve the constant-time addition and possible simultaneous format conversion. The analysis and data show that EWG leads to efficient implementations. We compare VLSI implementations of various constant-time addition cells and demonstrate that the conventional 4:2 compressor is the most efficient way to execute constant time-addition. We show interesting connections to prior results and indicate possible directions for further extensions.

In this paper we use a "uniformity" property of Riemann integration to obtain a single-hidden-layer neural network of fixed translates of a (not necessarily radial) basis function with a fixed "width" that approximates a (possibly infinite) set of target functions arbitrarily well in the supremum norm over a compact set. The conditions on the set of target functions are simple and intuitive: uniform boundedness and equicontinuity (so this result reduces to the "classical" theorems for a single target function). The uniformity property mentioned above refers to the existence of a single Riemann partition that achieves a prescribed accuracy of approximation of the Riemann integrals for a set of functions. A noteworthy feature of this simultaneous approximation scheme is that the nonlinear problem of finding the translates (also known as the "centers") needs to be solved only once. The only parameters that need to be adapted for a particular target function are the weights from hidden-to-output layer (which is a linear problem).

Optimizing TCP (transport layer) for mobility has been researched extensively. We present a brief summary of existing results which indicates that most schemes require intermediaries (such as base stations) to monitor the TCP traffic and actively participate in flow control in order to enhance performance. Although these methods simulate end-to-end semantics, they do not comprise true end-to-end signaling. As a result, these techniques are not applicable when the IP payload is encrypted. For instance IPSEC, which is expected to be standard under IPv6, encrypts the entire IP payload making it impossible for intermediaries to monitor TCP traffic unless those entities are part of the security association. In addition, these schemes require changes (in the TCP/IP code) at intermediate nodes making it difficult for the mobile clients to inter-operate with the existing infrastructure. In this paper we explore the "freeze-TCP" mechanism which is a true end-to-end scheme and does not require the involvement of any intermediaries (such as base stations) for flow control. Furthermore, this scheme does not require any changes on the "sender side" or intermediate routers; changes in TCP code are restricted to the mobile client side, making it possible to fully inter-operate with the existing infrastructure. We then outline a method which integrates the best attributes of freeze-TCP and some existing solutions. Performance results highlight the importance of pro-active action/signaling by the mobile-host. The data indicate that in most cases, simply reacting to disconnections tends to yield lower performance than pro-active mechanisms such as freeze-TCP.

Intermediate Signed Digit (SD) representation can facilitate fast and compact VLSI implementations of partial product accumulation trees. It achieves a reduction ratio of 2:1 at every level and also leads to more regular layouts. Its disadvantage is that the number of bit lines that need to be routed can be high. This can lead to a significant area overhead especially at smaller feature sizes where the wire/interconnect area and delay can be dominant. A Hybrid Signed Digit (HSD) representation lets some of the digits be unsigned bits, thereby reducing the number of bit lines. By arbitrarily varying the positions of and distances between consecutive signed digits, this representation can trade off latency for area and offers a continuum of choices between the two's complement representation on the one hand and fully Signed Digit (FSD or simply SD) representation on the other. We illustrate an A/spl middot/T (area/spl times/delay) efficient multiplier based on the HSD-1 representation which is one of the many possible HSD formats, wherein every alternate digit is signed and the rest are unsigned (ordinary) bits. It is seen that multipliers based on HSD-1 format require more transistors than those based on FSD format. However, they require fewer bit lines to be routed, which substantially reduces the interconnect area; thereby leading to a reduction in the total VLSI area and a lower A/spl middot/T product. The design reaffirms that the interconnect area can be significant, especially at small feature sizes.

In two operand addition, bit-wise intermediate variables such as the "propagate" and "generate" terms are defined/evaluated first. Basic carry propagation recursion is then expressed in terms of these variables and is "unrolled" to obtain a tree structure for fast execution. In CMOS VLSI technology, multiplexors are fast and efficient to implement. Hence, we investigate in this paper all possible two-bit encodings for the intermediate variables and identify the ones that enable multiplexor-based implementations. Some of these encodings enable further simplification of the multiplexor-based realizations. Our analysis also shows that adopting an intermediate signed-digit representation simply amounts to selecting one of the possible encodings. Thus, there is no inherent advantage to the use of intermediate signed-digit representations in a two operand addition. Finally, we extend our analysis to the generalized look-ahead-recursions proposed by R.W. Doran (1988).

It is demonstrated that fault tolerance, generalization and the Vapnik-Chertonenkis (VC) dimension are inter-related attributes. It is well known that the generalization error if plotted as a function of the VC dimension h, exhibits a well defined minimum corresponding to an optimal value of h, say h/sub opt/. We show that if the VC dimension h of an ANN satisfies h/spl les/h/sub opt/ (i.e., there is no excess capacity or redundancy), then fault tolerance and generalization are mutually conflicting attributes. On the other hand, if h>h/sub opt/ (i.e., there is excess capacity or redundancy), then fault tolerance and generalization are mutually synergistic attributes. In other words, training methods geared towards improving the fault tolerance can also lead to better generalization and vice versa, only when there is excess capacity or redundancy. This is consistent with our previous results indicating that complete fault tolerance in ANNs requires a significant amount of redundancy.

Duprat and Muller (1993) introduced the ingenious "Branching CORDIC" algorithm. It enables a fast implementation of CORDIC algorithm using signed digits and requires a constant normalization factor. The speedup is achieved by performing two basic CORDIC rotations in parallel in two separate modules. In their method, both modules perform identical computation except when the algorithm is in a "branching" [1]. We have improved the algorithm and show that it is possible to perform two circular mode rotations in a single step, with little additional hardware. In our method, both modules perform distinct computations at each step which leads to a better utilization of the hardware and the possibility of further speedup over the original method. Architectures for VLSI implementation of our algorithm are discussed.

Duprat and Muller (IEEE trans. Computers, vo1.42. no. 2, p. 168-178, Feb. 1993) introduced the ingenious "Branching CORDIC" algorithm. It enables a fast implementation of CORDIC algorithm using signed digits and requires a constant normalization factor. This correspondence corrects some errors in the original paper.

A method is proposed to estimate the fault tolerance (FT) of feedforward artificial neural nets (ANNs) and synthesize robust nets. The fault model abstracts a variety of failure modes for permanent stuck-at type faults. A procedure is developed to build FT ANNs by replicating the hidden units. It exploits the intrinsic weighted summation operation performed by the processing units to overcome faults. Metrics are devised to quantify the FT as a function of redundancy. A lower bound on the redundancy required to tolerate all possible single faults is analytically derived. Less than triple modular redundancy (TMR) cannot provide complete FT for all possible single faults. The actual redundancy needed to synthesize a completely FT net is specific to the problem at hand and is usually much higher than that dictated by the general lower bound. The conventional TMR scheme of triplication and majority voting is the best way to achieve complete FT in most ANNs. Although the redundancy needed for complete FT is substantial, the ANNs exhibit good partial FT to begin with and degrade gracefully. The first replication yields maximum enhancement in partial FT compared with later successive replications. For large nets, exhaustive testing of all possible single faults is prohibitive, so the strategy of randomly testing a small fraction of the total number of links is adopted. It yields partial FT estimates that are very close to those obtained by exhaustive testing. When the fraction of links tested is held fixed, the accuracy of the estimate generated by random testing is seen to improve as the net size grows.

A novel hybrid number representation is proposed. It includes the two's complement representation and the signed-digit representation as special cases. The hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addition to any desired value between 1 and the entire word length. The framework reveals a continuum of number representations between the two extremes of two's complement and signed-digit number systems and allows a unified performance analysis of the entire spectrum of implementations of adders, multipliers and alike. We present several static CMOS implementations of a two-operand adder which employ the proposed representations. We then derive quantitative estimates of area (in terms of the required number of transistors) and the maximum carry propagation delay for such an adder. The analysis clearly illustrates the trade-offs between area and execution time associated with each of the possible representations. We also discuss adder trees for parallel multipliers and show that the proposed representations lead to compact adder trees with fast execution times. In practice, the area available to a designer is often limited. In such cases, the designer can select the particular hybrid representation that yields the most suitable implementation (fastest, lowest power consumption, etc.) while satisfying the area constraint. Similarly, if the worst case delay is predetermined, the designer can select a hybrid representation that minimizes area or power under the delay constraint.>

The cascade correlation is a very flexible, efficient and fast algorithm for supervised learning. It incrementally builds the network by adding hidden units one at a time, until the desired input/output mapping is achieved. It connects all the previously installed units to the new unit being added. Consequently, each new unit in effect adds a new layer and the fan-in of the hidden and output units keeps on increasing as more units get added. The resulting structure could be hard to implement in VLSI, because the connections are irregular and the fan-in is unbounded. Moreover, the depth or the propagation delay through the resulting network is directly proportional to the number of units and can be excessive. We have modified the algorithm to generate networks with restricted fan-in and small depth (propagation delay) by controlling the connectivity. Our results reveal that there is a tradeoff between connectivity and other performance attributes like depth, total number of independent parameters, and learning time.

A novel, hybrid number representation is proposed. It includes the two's complement representation and the signed-digit representation as special cases. Such a hybrid representation is capable of bounding the maximum length of carry propagation chains during addition. Thus, one can select a specific hybrid number representation that limits the maximum length of all the carry propagation chains to any desired value between one and the entire word length. The framework reveals a continuum of number representations between the two extremes of two's complement and signed-digit number systems and allows a unified performance analysis of the entire spectrum of adders based on the representation selected.>

The encoding problem (Rumelhart and McClelland 1986) is an important canonical problem. It has been widely used as a benchmark. Here, we have analytically derived minimal-sized nets necessary and sufficient to solve encoding problems of arbitrary size. The proofs are constructive: we construct n-2-n encoders and show that two hidden units are also necessary for n > 2. Moreover, the geometric approach employed is general and has much wider applications. For example, this method has also helped us derive lower bounds on the redundancy necessary for achieving complete fault tolerance (Phatak and Koren 1992a,b).