We outline a multiprocessor architecture that uses modular arithmetic to implement numerical computation with 900 bits of intermediate precision. A proposed prototype, to be implemented with off-the-shelf parts, will perform high-precision arithmetic as fast as some workstations and mini- computers can perform IEEE double-precision arithmetic. We discuss how the structure of modular arithmetic conveniently maps into a simple, pipelined multiprocessor architecture. We present techniques we developed to overcome a few classical drawbacks of modular arithmetic. Our architecture is suitable to and essential for the study of chaotic dynamical systems.

The purpose of this case study was to understand whether or not the use of an invented number system, called Orpda, helped teachers develop a deeper understanding of place value in hopes that this will translate into their own teaching of place value concepts. Thirteen teachers enrolled in a graduate mathematics education course served as the participants for this study. Data were collected from teachers’ reflections on various activities related to Orpda, pre- and post-Orpda concept maps teachers created, online discussions between the teachers, teacher demographic sheets, and an interview with the instructor of the course. Analysis of the teachers' reflections revealed that Orpda increased teachers’ attention to three critical components necessary for developing a conceptual understanding of place value, namely unitizing, regrouping, and recognizing the meaning of different place values within a multi-digit number. In addition, Orpda encouraged teachers to reflect on their own teaching of place value. Comparing the structures of the teachers' pre-Orpda and post-Orpda concept maps showed changes in some cases but did not reveal clear patterns. Analysis of the categories teachers included in pre- and post-Orpda maps revealed that teachers were moving from a procedural to a more conceptual view of place value, as did the analysis of squared adjacency matrices created from each teacher's pre- and post-Orpda concept maps. Four conclusions can be drawn from this study: (a) Orpda increased teachers' attention to the importance of unitizing in place value, (b) Orpda encouraged teachers to reflect deeply on their thinking, (c) Concept maps show promise for revealing and documenting changes in conceptual understanding, and (d) Orpda increased teachers' attention to the importance of patterns in understanding place value. Further research is needed using Orpda with different groups and numbers of teachers, and in different settings, e.g., longer full semesters and teacher professional development meetings. Research exploring the use of follow-up interviews to accompany concept maps and enhance the assessment of conceptual understanding is also recommended. This study indicates two recommendations for practice in teacher education, the importance of a classroom environment that supports reflection, and the careful choosing of activities to provide appropriate challenge.

This doctoral thesis approaches the problem of designing versatile architectures for cryptographic hardware. By the term versatile we define hardware architectures capable of supporting a variety of arithmetic operations and algorithms useful in cryptography, with no need to reconfigure the internal interconnections of the integrated circuit. A versatile architecture could offer considerable benefits to the end-user. By embedding a variety of crucial operations in a common architecture, the user is able to switch seamlessly the underlying cryptographic protocols, which not only gives an added value in the design from flexibility but also from practicality point of view. The total cost of a cryptographic application can be also benefited; assuming a versatile integrated circuit which requires no additional circuitry for other vital operations (for example input–output converters) it is easy to deduce that the total cost of development and fabrication of these extra components is eliminated, thus reducing the total production cost. We follow a systematic approach for developing and presenting the proposed versatile architectures. First, an in-depth analysis of the algorithms of interest is carried out, in order to identify new research areas and weaknesses of existing solutions. The proposed algorithms and architectures operate on Galois Fields GF of the form GF(p) for integers and GF(2^n) for polynomials. Alternative number representation systems such as Residue Number System (RNS) for integers and Polynomial Residue Number System (PRNS) for polynomials are employed. The mathematical validity of the proposed algorithms and the applicability of RNS and PRNS in the context of cryptographic algorithms is also presented. The derived algorithms are decomposed in a way that versatile structures can be formulated and the corresponding hardware is developed and evaluated. New cryptanalytic properties of the proposed algorithms against certain types of attacks are also highlighted. Furthermore, we try to approach a fundamental problem in Very Large Scale Integration (VLSI) design, that is the problem of evaluating and comparing architectures using models independent from the underlying fabrication technology. We also provide generic methods to evaluate the optimal operation parameters of the proposed architectures and methods to optimize the proposed architectures in terms of speed, area, and area x speed product, based on the needs of the underlying application. The proposed methodologies can be expanded to include applications other than cryptography. Finally, novel algorithms based on new mathematical and design problems for the crucial operation of modular multiplication are presented. The new algorithms preserve the versatile characteristics discussed previously and it is proved that, along with existing algorithms in the literature, they may forma large family of algorithms applicable in cryptography, unified under the common frame of the proposed versatile architectures. / Η παρούσα διατριβή άπτεται του θέματος της ανάπτυξης ευέλικτων αρχιτεκτονικών κρυπτογραφίας σε ολοκληρωμένα κυκλώματα υψηλής ολοκλήρωσης (VLSI). Με τον όρο ευέλικτες ορίζονται οι αρχιτεκτονικές που δύνανται να υλοποιούν πλήθος βασικών αριθμητικών πράξεων για την εκτέλεση κρυπτογραφικών αλγορίθμων, χωρίς την ανάγκη επαναπροσδιορισμού των εσωτερικών διατάξεων στο ολοκληρωμένο κύκλωμα. Η χρήση ευέλικτων αρχιτεκτονικών παρέχει πολλαπλά οφέλη στο χρήστη. Η ενσωμάτωση κρίσιμων πράξεων απαραίτητων στη κρυπτογραφία σε μια κοινή αρχιτεκτονική δίνει τη δυνατότητα στο χρήστη να εναλλάσσει το υποστηριζόμενο κρυπτογραφικό πρωτόκολλο, εισάγοντας έτσι χαρακτηριστικά ευελιξίας και πρακτικότητας, χωρίς επιπρόσθετη επιβάρυνση του συστήματος σε υλικό. Αξίζει να σημειωθεί πως οι εναλλαγές αυτές δεν απαιτούν τη παρέμβαση του χρήστη. Σημαντική είναι η συνεισφορά μιας ευέλικτης αρχιτεκτονικής και στο κόστος μιας εφαρμογής. Αναλογιζόμενοι ένα ολοκληρωμένο κύκλωμα που μπορεί να υλοποιεί αυτόνομα όλες τις απαραίτητες πράξεις ενός αλγόριθμου χωρίς την εξάρτηση από εξωτερικά υποσυστήματα (π.χ. μετατροπείς εισόδου–εξόδου), είναι εύκολο να αντιληφθούμε πως το τελικό κόστος της εκάστοτε εφαρμογής μειώνεται σημαντικά καθώς μειώνονται οι ανάγκες υλοποίησης και διασύνδεσης επιπρόσθετων υποσυστημάτων στο ολοκληρωμένο κύκλωμα. Η ανάπτυξη των προτεινόμενων αρχιτεκτονικών ακολουθεί μια δομημένη προσέγγιση. Διενεργείται εκτενής μελέτη για τον προσδιορισμό γόνιμων ερευνητικών περιοχών και εντοπίζονται προβλήματα και δυνατότητες βελτιστοποίησης υπαρχουσών κρυπτογραφικών λύσεων. Οι νέοι αλγόριθμοι που αναπτύσσονται αφορούν τα Galois πεδία GF(p) και GF(2^n) και χρησιμοποιούν εναλλακτικές αριθμητικές αναπαράστασης δεδομένων όπως το αριθμητικό σύστημα υπολοίπων (Residue Number System (RNS)) για ακέραιους αριθμούς και το πολυωνυμικό αριθμητικό σύστημα υπολοίπων (Polynomial Residue Number System (PRNS)) για πολυώνυμα. Αποδεικνύεται η μαθηματική τους ορθότητα και βελτιστοποιούνται κατά τέτοιο τρόπο ώστε να σχηματίζουν ευέλικτες δομές. Αναπτύσσεται το κατάλληλο υλικό (hardware) και διενεργείται μελέτη χρήσιμων ιδιοτήτων των νέων αλγορίθμων, όπως για παράδειγμα νέες κρυπταναλυτικές ιδιότητες. Επιπρόσθετα, προσεγγίζουμε στα πλαίσια της διατριβής ένα βασικό πρόβλημα της επιστήμης σχεδιασμού ολοκληρωμένων συστημάτων μεγάλης κλίμακας (Very Large Scale Integration (VLSI)). Συγκεκριμένα, προτείνονται μέθοδοι σύγκρισης αρχιτεκτονικών ανεξαρτήτως τεχνολογίας καθώς και τρόποι εύρεσης των βέλτιστων συνθηκών λειτουργίας των προτεινόμενων αρχιτεκτονικών. Οι μέθοδοι αυτές επιτρέπουν στον σχεδιαστή να παραμετροποιήσει τις προτεινόμενες αρχιτεκτονικές με βάση τη ταχύτητα, επιφάνεια, ή το γινόμενο ταχύτητα x επιφάνεια. Οι προτεινόμενες μεθοδολογίες μπορούν εύκολα να επεκταθούν και σε άλλες εφαρμογές πέραν της κρυπτογραφίας. Τέλος, προτείνονται νέοι αλγόριθμοι για τη σημαντικότατη για την κρυπτογραφία πράξη του πολλαπλασιασμού με υπόλοιπα. Οι νέοι αλγόριθμοι ενσωματώνουν από τη μία τις ιδέες των ευέλικτων δομών, από την άλλη όμως βασίζονται σε νέες ιδέες και μαθηματικά προβλήματα τα οποία προσπαθούμε να προσεγγίσουμε και να επιλύσουμε. Αποδεικνύεται πως είναι δυνατή η ενοποίηση μιας μεγάλης οικογένειας αλγορίθμων για χρήση στην κρυπτογραφία, υπό τη στέγη των προτεινόμενων μεθοδολογιών για ευέλικτο σχεδιασμό.

Στα ασύρµατα δίκτυα η τεχνολογία MIMO-OFDM έχει ευρέως υιοθετηθεί µε στόχο την αύξηση του ρυθµού δεδοµένων σε υπηρεσίες υψηλής ποιότητας. Στο δέκτη ενός συστήµατος MIMO-OFDM ο υπολογισµός του µητρώου συντελεστών που χρειάζεται ο εξισωτής (equalizer) είναι ένα κρίσιµο σηµείο µε υψηλή υπολογιστική πολυπλοκότητα. Η καθυστέρηση που απαιτείται για την εκτέλεση του συγκεκριµένου υπολογισµού επηρεάζει άµεσα το ρυθµό περάτωσης (throughput) και την ποιότητα υπηρεσίας (QoS). Σε συστήµατα wi-fi (για παράδειγµα στο IEEE 802.11ac) στην αρχή κάθε πακέτου µεταδίδονται προσυµφωνηµένα σύµβολα, ώστε να εκπαιδεύσουν τον εξισωτή. Η συµβατική µέθοδος περιµένει να έρθουν όλα τα σύµβολα και στη συνέχεια υπολογίζει τοµητρώο του εξισωτή. Μια πιο πρόσφατη µέθοδος που αποσκοπεί στη µείωση της παραπάνω καθυστέρησης είναι ο προοδευτικός υπολογισµός του µητρώου του εξισωτή. Στη µέθοδο αυτή, γίνονται υπολογισµοί ανά σύµβολο εκπαίδευσης χωρίς να επηρεάζεται το τελικό αποτέλεσµα. Στα πλαίσια της διπλωµατικής υλοποιήθηκαν αρχιτεκτονικές υλικού για εξισωτές που αξιοποιούν την ανωτέρω µέθοδο και απεικονίστηκαν σε αναπτυξιακό σύστηµα µε FPGA. Επίσης, διερευνήθηκε ο ρόλος της αναπαράστασης των δεδοµένων στην πολυπλοκότητα του συστήµατος και βελτιστοποιήθηκαν οι σχετικές σχεδιαστικές παράµετροι, µε έµφαση στη χρήση της λογαριθµικής αριθµητικής. / In wireless networks, the MIMO-OFDM technology has been widely adopted in order to increase the data rate at high quality of service (QoS). In the receiver, the MIMO equalizer matrix calculation is an important part with high computational complexity. The delay of the matrix calculation affects directly the system throughput and QoS. In wi-fi systems (e.g. IEEE 802.11ac), training symbols are transmitted at the beginning of every frame, in order to train the equalizer. The conventional method waits all the training symbols to arrive before starting the calculations. A recent method proposes the progressive calculation of the equalizer matrix in order to decrease the processing delay. In this method, the equalizer matrix calculation starts upon receiving the first training symbol. The object of this thesis is the development of hardware architectures for equalizers that follow the progressive method and their implementation on FPGA. Furthermore, we analyzed the impact of the data representation on the system complexity and optimized the respective design parameters, using logarithmic number system.

abstract: Redundant Binary (RBR) number representations have been extensively used in the past for high-throughput Digital Signal Processing (DSP) systems. Data-path components based on this number system have smaller critical path delay but larger area compared to conventional two's complement systems. This work explores the use of RBR number representation for implementing high-throughput DSP systems that are also energy-efficient. Data-path components such as adders and multipliers are evaluated with respect to critical path delay, energy and Energy-Delay Product (EDP). A new design for a RBR adder with very good EDP performance has been proposed. The corresponding RBR parallel adder has a much lower critical path delay and EDP compared to two's complement carry select and carry look-ahead adder implementations. Next, several RBR multiplier architectures are investigated and their performance compared to two's complement systems. These include two new multiplier architectures: a purely RBR multiplier where both the operands are in RBR form, and a hybrid multiplier where the multiplicand is in RBR form and the other operand is represented in conventional two's complement form. Both the RBR and hybrid designs are demonstrated to have better EDP performance compared to conventional two's complement multipliers. The hybrid multiplier is also shown to have a superior EDP performance compared to the RBR multiplier, with much lower implementation area. Analysis on the effect of bit-precision is also performed, and it is shown that the performance gain of RBR systems improves for higher bit precision. Next, in order to demonstrate the efficacy of the RBR representation at the system-level, the performance of RBR and hybrid implementations of some common DSP kernels such as Discrete Cosine Transform, edge detection using Sobel operator, complex multiplication, Lifting-based Discrete Wavelet Transform (9, 7) filter, and FIR filter, is compared with two's complement systems. It is shown that for relatively large computation modules, the RBR to two's complement conversion overhead gets amortized. In case of systems with high complexity, for iso-throughput, both the hybrid and RBR implementations are demonstrated to be superior with lower average energy consumption. For low complexity systems, the conversion overhead is significant, and overpowers the EDP performance gain obtained from the RBR computation operation. / Dissertation/Thesis / M.S. Electrical Engineering 2011

Nos últimos anos, os sistemas integrados em silício (SOCs - Systems-on-Chip) têm se tornado menos imunes a ruído, em decorrência dos ajustes necessários na tecnologia CMOS (Complementary Metal-Oxide-Silicon) para garantir o funcionamento dos transistores com dimensões nanométricas. Dentre tais ajustes, a redução da tensão de alimentação e da tensão de limiar (threshold) tornam os SOCs mais suscetíveis a falhas transientes, principalmente aquelas provocadas pela colisão de partículas energéticas que provêm do espaço e encontram-se presentes na atmosfera terrestre. Quando uma partícula energética de alta energia colide com o dreno de um transistor que está desligado, ela perde energia e produz pares elétron-lacuna livres, resultando em uma trilha de ionização. A ionização pode gerar um pulso transiente de tensão que pode ser interpretado como uma mudança no sinal lógico. Em um circuito combinacional, o pulso pode propagar-se até ser armazenado em um elemento de memória. Tal fenômeno é denominado Single-Event Transient (SET). Como a tendência é que as dimensões dos dispositivos fabricados com tecnologia CMOS continuem reduzindo por mais alguns anos, a ocorrência de SETs em SOCs operando na superfície terrestre tende a aumentar, exigindo a adoção de técnicas de tolerância a falhas no projeto de SOCs. O presente trabalho tem por objetivo avaliar circuitos somadores tolerantes a falhas transientes encontrados na literatura. Duas arquiteturas de somadores foram escolhidas: Ripple Carry Adder (RCA) e Binary Signed Digit Adder (BSDA). O RCA foi escolhido por ser o tipo de somador de menor custo e por isso, amplamente utilizado em SOCs. Já o BSDA foi escolhido porque utiliza o sistema numérico de dígito binário com sinal (Binary Signed Digit – BSD). Por ser um sistema de representação redundante, o uso de BSD facilita a aplicação de técnicas de tolerância a falhas baseadas em redundância de informação. Os somadores protegidos avaliados foram projetados com as seguintes técnicas: Redundância Modular Tripla (Triple Modular Redundancy - TMR) e Recomputação com Entradas e Saídas Invertidas (RESI), no caso do RCA, e codificação 1 de 3 e verificação de paridade, no caso do BSDA. As 9 arquiteturas de somadores foram simuladas no nível elétrico usando o Modelo Tecnológico Preditivo (Predictive Technology Model - PTM) de 45nm e considerando quatro comprimentos de operandos: 4, 8, 16 e 32 bits. Os resultados obtidos permitiram quantificar o número de transistores, o atraso crítico e a potência média consumida por cada arquitetura protegida. Também foram realizadas campanhas de injeção de falhas, por meio de simulações no nível elétrico, para estimar o grau de proteção de cada arquitetura. Os resultados obtidos servem para guiar os projetistas de SOCs na escolha da arquitetura de somador tolerante a falhas mais adequada aos requisitos de cada projeto. / In the past recent years, integrated systems on a chip (Systems-on-chip - SOCs) became less immune to noise due to the adjusts in CMOS technology needed to assure the operation of nanometric transistors. Among such adjusts, the reductions in supply voltage and threshold voltage make SOSs more susceptible to transient faults, mainly those provoked by the collision of charged particles coming from the outer space that are present in the atmosphere. When a heavily energy charged particle hits the drain region of a transistor that is at the off state it produces free electron-hole pairs, resulting in an ionizing track. The ionization may generate a transient voltage pulse that can be interpreted as a change in the logic signal. In a combinational circuit, the pulse may propagate up to the primary outputs and may be captured by the output storage element. Such phenomenon is referred to as Single-Event Transient (SET). Since it is expected that transistor dimensions will continue to reduce in the next technological nodes, the occurrence of SETs at Earth surface will increase and therefore, fault tolerance techniques will become a must in the design of SOSs. The present work targets the evaluation of transient fault-tolerant adders found in the literature. Two adder architectures were chosen: the Ripple-Carry Adder (RCA) and the Binary Signed Digit Adder (BSDA). The RCA was chosen because it is the least expensive and therefore, the most used architecture for SOS design. The BSDA, in turn, was chosen because it uses the Binary Signed Digit (BSD) system. As a redundant number system, the BSD paves the way to the implementation of fault-tolerant adders using information redundancy. The evaluated fault-tolerant adders were implemented by using the following techniques: Triple Module Redundancy (TMR) and Recomputing with Inverted Inputs and Outputs (RESI), in the case of the RCA, and 1 out of 3 coding and parity verification, in the case of the BSDA. A total of 9 adder architectures were simulated at the electric-level using the Predictive Technology Model (PTM) for 45nm in four different bitwidths: 4, 8, 16 and 32. The obtained results allowed for quantifying the number of transistors, critical delay and average power consumption for each fault-tolerant architecture. Fault injection campaigns were also accomplished by means of electric-level simulations to estimate the degree of protection of each architecture. The results obtained in the present work may be used to guide SOS designers in the choice of the fault-tolerant adder architecture that is most likely to satisfy the design requirements.

Este trabalho mostra os diversos sistemas de representação numérica, incluindo o sistema numérico normalmente utilizado em circuitos e alguns sistemas alternativos. Uma maior ênfase é dada ao sistema numérico por resíduos. Este último apresenta características muito interessantes para o desenvolvimento de circuitos aritméticos nos dias atuais, como por exemplo, a alta paralelização. São estudadas também as principais arquiteturas de somadores e multiplicadores. Várias descrições de circuitos aritméticos são feitas e sintetizadas. A arquitetura de circuitos aritméticos utilizando o sistema numérico por resíduos também é estudada e implementada. Os dados da síntese destes circuitos são comparados com os dados dos circuitos aritméticos tradicionais. Com isto, é possível avaliar as potenciais vantagens de se utilizar o sistema numérico por resíduos no desenvolvimento de circuitos aritméticos. / This work shows various numerical representation systems, including the system normally used in current circuits and some alternative systems. A great emphasis is given to the residue number system. This last one presents very interesting characteristics for the development of arithmetic circuits nowadays, as for example, the high parallelization. The main architectures of adders and multipliers are also studied. Some descriptions of arithmetic circuits are made and synthesized. The architecture of arithmetic circuits using the residue number system is also studied and implemented. The synthesis data of these circuits are compared with the traditional arithmetic circuits results. Then it is possible to evaluate the potential advantages of using the residue number system in arithmetic circuits development.

Este trabalho mostra os diversos sistemas de representação numérica, incluindo o sistema numérico normalmente utilizado em circuitos e alguns sistemas alternativos. Uma maior ênfase é dada ao sistema numérico por resíduos. Este último apresenta características muito interessantes para o desenvolvimento de circuitos aritméticos nos dias atuais, como por exemplo, a alta paralelização. São estudadas também as principais arquiteturas de somadores e multiplicadores. Várias descrições de circuitos aritméticos são feitas e sintetizadas. A arquitetura de circuitos aritméticos utilizando o sistema numérico por resíduos também é estudada e implementada. Os dados da síntese destes circuitos são comparados com os dados dos circuitos aritméticos tradicionais. Com isto, é possível avaliar as potenciais vantagens de se utilizar o sistema numérico por resíduos no desenvolvimento de circuitos aritméticos. / This work shows various numerical representation systems, including the system normally used in current circuits and some alternative systems. A great emphasis is given to the residue number system. This last one presents very interesting characteristics for the development of arithmetic circuits nowadays, as for example, the high parallelization. The main architectures of adders and multipliers are also studied. Some descriptions of arithmetic circuits are made and synthesized. The architecture of arithmetic circuits using the residue number system is also studied and implemented. The synthesis data of these circuits are compared with the traditional arithmetic circuits results. Then it is possible to evaluate the potential advantages of using the residue number system in arithmetic circuits development.

Nos últimos anos, os sistemas integrados em silício (SOCs - Systems-on-Chip) têm se tornado menos imunes a ruído, em decorrência dos ajustes necessários na tecnologia CMOS (Complementary Metal-Oxide-Silicon) para garantir o funcionamento dos transistores com dimensões nanométricas. Dentre tais ajustes, a redução da tensão de alimentação e da tensão de limiar (threshold) tornam os SOCs mais suscetíveis a falhas transientes, principalmente aquelas provocadas pela colisão de partículas energéticas que provêm do espaço e encontram-se presentes na atmosfera terrestre. Quando uma partícula energética de alta energia colide com o dreno de um transistor que está desligado, ela perde energia e produz pares elétron-lacuna livres, resultando em uma trilha de ionização. A ionização pode gerar um pulso transiente de tensão que pode ser interpretado como uma mudança no sinal lógico. Em um circuito combinacional, o pulso pode propagar-se até ser armazenado em um elemento de memória. Tal fenômeno é denominado Single-Event Transient (SET). Como a tendência é que as dimensões dos dispositivos fabricados com tecnologia CMOS continuem reduzindo por mais alguns anos, a ocorrência de SETs em SOCs operando na superfície terrestre tende a aumentar, exigindo a adoção de técnicas de tolerância a falhas no projeto de SOCs. O presente trabalho tem por objetivo avaliar circuitos somadores tolerantes a falhas transientes encontrados na literatura. Duas arquiteturas de somadores foram escolhidas: Ripple Carry Adder (RCA) e Binary Signed Digit Adder (BSDA). O RCA foi escolhido por ser o tipo de somador de menor custo e por isso, amplamente utilizado em SOCs. Já o BSDA foi escolhido porque utiliza o sistema numérico de dígito binário com sinal (Binary Signed Digit – BSD). Por ser um sistema de representação redundante, o uso de BSD facilita a aplicação de técnicas de tolerância a falhas baseadas em redundância de informação. Os somadores protegidos avaliados foram projetados com as seguintes técnicas: Redundância Modular Tripla (Triple Modular Redundancy - TMR) e Recomputação com Entradas e Saídas Invertidas (RESI), no caso do RCA, e codificação 1 de 3 e verificação de paridade, no caso do BSDA. As 9 arquiteturas de somadores foram simuladas no nível elétrico usando o Modelo Tecnológico Preditivo (Predictive Technology Model - PTM) de 45nm e considerando quatro comprimentos de operandos: 4, 8, 16 e 32 bits. Os resultados obtidos permitiram quantificar o número de transistores, o atraso crítico e a potência média consumida por cada arquitetura protegida. Também foram realizadas campanhas de injeção de falhas, por meio de simulações no nível elétrico, para estimar o grau de proteção de cada arquitetura. Os resultados obtidos servem para guiar os projetistas de SOCs na escolha da arquitetura de somador tolerante a falhas mais adequada aos requisitos de cada projeto. / In the past recent years, integrated systems on a chip (Systems-on-chip - SOCs) became less immune to noise due to the adjusts in CMOS technology needed to assure the operation of nanometric transistors. Among such adjusts, the reductions in supply voltage and threshold voltage make SOSs more susceptible to transient faults, mainly those provoked by the collision of charged particles coming from the outer space that are present in the atmosphere. When a heavily energy charged particle hits the drain region of a transistor that is at the off state it produces free electron-hole pairs, resulting in an ionizing track. The ionization may generate a transient voltage pulse that can be interpreted as a change in the logic signal. In a combinational circuit, the pulse may propagate up to the primary outputs and may be captured by the output storage element. Such phenomenon is referred to as Single-Event Transient (SET). Since it is expected that transistor dimensions will continue to reduce in the next technological nodes, the occurrence of SETs at Earth surface will increase and therefore, fault tolerance techniques will become a must in the design of SOSs. The present work targets the evaluation of transient fault-tolerant adders found in the literature. Two adder architectures were chosen: the Ripple-Carry Adder (RCA) and the Binary Signed Digit Adder (BSDA). The RCA was chosen because it is the least expensive and therefore, the most used architecture for SOS design. The BSDA, in turn, was chosen because it uses the Binary Signed Digit (BSD) system. As a redundant number system, the BSD paves the way to the implementation of fault-tolerant adders using information redundancy. The evaluated fault-tolerant adders were implemented by using the following techniques: Triple Module Redundancy (TMR) and Recomputing with Inverted Inputs and Outputs (RESI), in the case of the RCA, and 1 out of 3 coding and parity verification, in the case of the BSDA. A total of 9 adder architectures were simulated at the electric-level using the Predictive Technology Model (PTM) for 45nm in four different bitwidths: 4, 8, 16 and 32. The obtained results allowed for quantifying the number of transistors, critical delay and average power consumption for each fault-tolerant architecture. Fault injection campaigns were also accomplished by means of electric-level simulations to estimate the degree of protection of each architecture. The results obtained in the present work may be used to guide SOS designers in the choice of the fault-tolerant adder architecture that is most likely to satisfy the design requirements.

Nos últimos anos, os sistemas integrados em silício (SOCs - Systems-on-Chip) têm se tornado menos imunes a ruído, em decorrência dos ajustes necessários na tecnologia CMOS (Complementary Metal-Oxide-Silicon) para garantir o funcionamento dos transistores com dimensões nanométricas. Dentre tais ajustes, a redução da tensão de alimentação e da tensão de limiar (threshold) tornam os SOCs mais suscetíveis a falhas transientes, principalmente aquelas provocadas pela colisão de partículas energéticas que provêm do espaço e encontram-se presentes na atmosfera terrestre. Quando uma partícula energética de alta energia colide com o dreno de um transistor que está desligado, ela perde energia e produz pares elétron-lacuna livres, resultando em uma trilha de ionização. A ionização pode gerar um pulso transiente de tensão que pode ser interpretado como uma mudança no sinal lógico. Em um circuito combinacional, o pulso pode propagar-se até ser armazenado em um elemento de memória. Tal fenômeno é denominado Single-Event Transient (SET). Como a tendência é que as dimensões dos dispositivos fabricados com tecnologia CMOS continuem reduzindo por mais alguns anos, a ocorrência de SETs em SOCs operando na superfície terrestre tende a aumentar, exigindo a adoção de técnicas de tolerância a falhas no projeto de SOCs. O presente trabalho tem por objetivo avaliar circuitos somadores tolerantes a falhas transientes encontrados na literatura. Duas arquiteturas de somadores foram escolhidas: Ripple Carry Adder (RCA) e Binary Signed Digit Adder (BSDA). O RCA foi escolhido por ser o tipo de somador de menor custo e por isso, amplamente utilizado em SOCs. Já o BSDA foi escolhido porque utiliza o sistema numérico de dígito binário com sinal (Binary Signed Digit – BSD). Por ser um sistema de representação redundante, o uso de BSD facilita a aplicação de técnicas de tolerância a falhas baseadas em redundância de informação. Os somadores protegidos avaliados foram projetados com as seguintes técnicas: Redundância Modular Tripla (Triple Modular Redundancy - TMR) e Recomputação com Entradas e Saídas Invertidas (RESI), no caso do RCA, e codificação 1 de 3 e verificação de paridade, no caso do BSDA. As 9 arquiteturas de somadores foram simuladas no nível elétrico usando o Modelo Tecnológico Preditivo (Predictive Technology Model - PTM) de 45nm e considerando quatro comprimentos de operandos: 4, 8, 16 e 32 bits. Os resultados obtidos permitiram quantificar o número de transistores, o atraso crítico e a potência média consumida por cada arquitetura protegida. Também foram realizadas campanhas de injeção de falhas, por meio de simulações no nível elétrico, para estimar o grau de proteção de cada arquitetura. Os resultados obtidos servem para guiar os projetistas de SOCs na escolha da arquitetura de somador tolerante a falhas mais adequada aos requisitos de cada projeto. / In the past recent years, integrated systems on a chip (Systems-on-chip - SOCs) became less immune to noise due to the adjusts in CMOS technology needed to assure the operation of nanometric transistors. Among such adjusts, the reductions in supply voltage and threshold voltage make SOSs more susceptible to transient faults, mainly those provoked by the collision of charged particles coming from the outer space that are present in the atmosphere. When a heavily energy charged particle hits the drain region of a transistor that is at the off state it produces free electron-hole pairs, resulting in an ionizing track. The ionization may generate a transient voltage pulse that can be interpreted as a change in the logic signal. In a combinational circuit, the pulse may propagate up to the primary outputs and may be captured by the output storage element. Such phenomenon is referred to as Single-Event Transient (SET). Since it is expected that transistor dimensions will continue to reduce in the next technological nodes, the occurrence of SETs at Earth surface will increase and therefore, fault tolerance techniques will become a must in the design of SOSs. The present work targets the evaluation of transient fault-tolerant adders found in the literature. Two adder architectures were chosen: the Ripple-Carry Adder (RCA) and the Binary Signed Digit Adder (BSDA). The RCA was chosen because it is the least expensive and therefore, the most used architecture for SOS design. The BSDA, in turn, was chosen because it uses the Binary Signed Digit (BSD) system. As a redundant number system, the BSD paves the way to the implementation of fault-tolerant adders using information redundancy. The evaluated fault-tolerant adders were implemented by using the following techniques: Triple Module Redundancy (TMR) and Recomputing with Inverted Inputs and Outputs (RESI), in the case of the RCA, and 1 out of 3 coding and parity verification, in the case of the BSDA. A total of 9 adder architectures were simulated at the electric-level using the Predictive Technology Model (PTM) for 45nm in four different bitwidths: 4, 8, 16 and 32. The obtained results allowed for quantifying the number of transistors, critical delay and average power consumption for each fault-tolerant architecture. Fault injection campaigns were also accomplished by means of electric-level simulations to estimate the degree of protection of each architecture. The results obtained in the present work may be used to guide SOS designers in the choice of the fault-tolerant adder architecture that is most likely to satisfy the design requirements.