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Bommireddipalli, Nithesh Venkata Ramana Surya. "Tutorial on Elliptic Curve Arithmetic and Introduction to Elliptic Curve Cryptography (ECC)." University of Cincinnati / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1511866832906148.
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Kultinov, Kirill. "Software Implementations and Applications of Elliptic Curve Cryptography." Wright State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=wright1559232475298514.
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Lien, E.-Jen. "EFFICIENT IMPLEMENTATION OF ELLIPTIC CURVE CRYPTOGRAPHY IN RECONFIGURABLE HARDWARE." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1333761904.
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baktir, selcuk. "Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography." Digital WPI, 2008. https://digitalcommons.wpi.edu/etd-dissertations/272.
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Efficient implementation of the number theoretic transform(NTT), also known as the discrete Fourier transform(DFT) over a finite field, has been studied actively for decades and found many applications in digital signal processing. In 1971 Schonhage and Strassen proposed an NTT based asymptotically fast multiplication method with the asymptotic complexity O(m log m log log m) for multiplication of $m$-bit integers or (m-1)st degree polynomials. Schonhage and Strassen's algorithm was known to be the asymptotically fastest multiplication algorithm until Furer improved upon it in 2007. However, unfortunately, both algorithms bear significant overhead due to the conversions between the time and frequency domains which makes them impractical for small operands, e.g. less than 1000 bits in length as used in many applications. With this work we investigate for the first time the practical application of the NTT, which found applications in digital signal processing, to finite field multiplication with an emphasis on elliptic curve cryptography(ECC). We present efficient parameters for practical application of NTT based finite field multiplication to ECC which requires key and operand sizes as short as 160 bits in length. With this work, for the first time, the use of NTT based finite field arithmetic is proposed for ECC and shown to be efficient. We introduce an efficient algorithm, named DFT modular multiplication, for computing Montgomery products of polynomials in the frequency domain which facilitates efficient multiplication in GF(p^m). Our algorithm performs the entire modular multiplication, including modular reduction, in the frequency domain, and thus eliminates costly back and forth conversions between the frequency and time domains. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to ECC. This work presents the first hardware implementation of a frequency domain multiplier suitable for ECC and the first hardware implementation of ECC in the frequency domain. We introduce a novel area/time efficient ECC processor architecture which performs all finite field arithmetic operations in the frequency domain utilizing DFT modular multiplication over a class of Optimal Extension Fields(OEF). The proposed architecture achieves extension field modular multiplication in the frequency domain with only a linear number of base field GF(p) multiplications in addition to a quadratic number of simpler operations such as addition and bitwise rotation. With its low area and high speed, the proposed architecture is well suited for ECC in small device environments such as smart cards and wireless sensor networks nodes. Finally, we propose an adaptation of the Itoh-Tsujii algorithm to the frequency domain which can achieve efficient inversion in a class of OEFs relevant to ECC. This is the first time a frequency domain finite field inversion algorithm is proposed for ECC and we believe our algorithm will be well suited for efficient constrained hardware implementations of ECC in affine coordinates.
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Krisell, Martin. "Elliptic Curve Digital Signatures in RSA Hardware." Thesis, Linköpings universitet, Informationskodning, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-81084.
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A digital signature is the electronic counterpart to the hand written signature. It can prove the source and integrity of any digital data, and is a tool that is becoming increasingly important as more and more information is handled electronically. Digital signature schemes use a pair of keys. One key is secret and allows the owner to sign some data, and the other is public and allows anyone to verify the signature. Assuming that the keys are large enough, and that a secure scheme is used, it is impossible to find the private key given only the public key. Since a signature is valid for the signed message only, this also means that it is impossible to forge a digital signature. The most well-used scheme for constructing digital signatures today is RSA, which is based on the hard mathematical problem of integer factorization. There are, however, other mathematical problems that are considered even harder, which in practice means that the keys can be made shorter, resulting in a smaller memory footprint and faster computations. One such alternative approach is using elliptic curves. The underlying mathematical problem of elliptic curve cryptography is different to that of RSA, however some structure is shared. The purpose of this thesis was to evaluate the performance of elliptic curves compared to RSA, on a system designed to efficiently perform the operations associated with RSA. The discovered results are that the elliptic curve approach offers some great advantages, even when using RSA hardware, and that these advantages increase significantly if special hardware is used. Some usage cases of digital signatures may, for a few more years, still be in favor of the RSA approach when it comes to speed. For most cases, however, an elliptic curve system is the clear winner, and will likely be dominant within a near future.En digital signatur är den elektroniska motsvarigheten till en handskriven signatur. Den kan bevisa källa och integritet för valfri data, och är ett verktyg som blir allt viktigare i takt med att mer och mer information hanteras digitalt. Digitala signaturer använder sig av två nycklar. Den ena nyckeln är hemlig och tillåter ägaren att signera data, och den andra är offentlig och tillåter vem som helst att verifiera signaturen. Det är, under förutsättning att nycklarna är tillräck- ligt stora och att det valda systemet är säkert, omöjligt att hitta den hemliga nyckeln utifrån den offentliga. Eftersom en signatur endast är giltig för datan som signerades innebär detta också att det är omöjligt att förfalska en digital signatur. Den mest välanvända konstruktionen för att skapa digitala signaturer idag är RSA, som baseras på det svåra matematiska problemet att faktorisera heltal. Det finns dock andra matematiska problem som anses vara ännu svårare, vilket i praktiken innebär att nycklarna kan göras kortare, vilket i sin tur leder till att mindre minne behövs och att beräkningarna går snabbare. Ett sådant alternativ är att använda elliptiska kurvor. Det underliggande matematiska problemet för kryptering baserad på elliptiska kurvor skiljer sig från det som RSA bygger på, men de har en viss struktur gemensam. Syftet med detta examensarbete var att utvärdera hur elliptiska kurvor presterar jämfört med RSA, på ett system som är designat för att effektivt utföra RSA. De funna resultaten är att metoden med elliptiska kurvor ger stora fördelar, även om man nyttjar hårdvara avsedd för RSA, och att dessa fördelar ökar mångfaldigt om speciell hårdvara används. För några användarfall av digitala signaturer kan, under några år framöver, RSA fortfarande vara fördelaktigt om man bara tittar på hastigheten. För de flesta fall vinner dock elliptiska kurvor, och kommer troligen vara dominant inom kort.
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Hitchcock, Yvonne Roslyn. "Elliptic curve cryptography for lightweight applications." Thesis, Queensland University of Technology, 2003. https://eprints.qut.edu.au/15838/1/Yvonne_Hitchcock_Thesis.pdf.
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Elliptic curves were first proposed as a basis for public key cryptography in the mid 1980's. They provide public key cryptosystems based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP) , which is so called because of its similarity to the discrete logarithm problem (DLP) over the integers modulo a large prime. One benefit of elliptic curve cryptosystems (ECCs) is that they can use a much shorter key length than other public key cryptosystems to provide an equivalent level of security. For example, 160 bit ECCs are believed to provide about the same level of security as 1024 bit RSA. Also, the level of security provided by an ECC increases faster with key size than for integer based discrete logarithm (dl) or RSA cryptosystems. ECCs can also provide a faster implementation than RSA or dl systems, and use less bandwidth and power. These issues can be crucial in lightweight applications such as smart cards. In the last few years, ECCs have been included or proposed for inclusion in internationally recognized standards. Thus elliptic curve cryptography is set to become an integral part of lightweight applications in the immediate future. This thesis presents an analysis of several important issues for ECCs on lightweight devices. It begins with an introduction to elliptic curves and the algorithms required to implement an ECC. It then gives an analysis of the speed, code size and memory usage of various possible implementation options. Enough details are presented to enable an implementer to choose for implementation those algorithms which give the greatest speed whilst conforming to the code size and ram restrictions of a particular lightweight device. Recommendations are made for new functions to be included on coprocessors for lightweight devices to support ECC implementations Another issue of concern for implementers is the side-channel attacks that have recently been proposed. They obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time and the information is then used to break implementations that have not incorporated appropriate defences. A new method of defence to protect an implementation from the simple power analysis (spa) method of attack is presented in this thesis. It requires 44% fewer additions and 11% more doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm. The algorithm forms a contribution to the current range of possible spa defences which has a good speed but low memory usage. Another topic of paramount importance to ECCs for lightweight applications is whether the security of fixed curves is equivalent to that of random curves. Because of the inability of lightweight devices to generate secure random curves, fixed curves are used in such devices. These curves provide the additional advantage of requiring less bandwidth, code size and processing time. However, it is intuitively obvious that a large precomputation to aid in the breaking of the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve which would be unavailable for a random curve. Therefore, it would appear that fixed curves are less secure than random curves, but quantifying the loss of security is much more difficult. The thesis performs an examination of fixed curve security taking this observation into account, and includes a definition of equivalent security and an analysis of a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. A lower bound on the expected time to solve such ECDLPs using this method is presented, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. It is concluded that adding a total of 11 bits to the size of a fixed curve provides an equivalent level of security compared to random curves. The final part of the thesis deals with proofs of security of key exchange protocols in the Canetti-Krawczyk proof model. This model has been used since it offers the advantage of a modular proof with reusable components. Firstly a password-based authentication mechanism and its security proof are discussed, followed by an analysis of the use of the authentication mechanism in key exchange protocols. The Canetti-Krawczyk model is then used to examine secure tripartite (three party) key exchange protocols. Tripartite key exchange protocols are particularly suited to ECCs because of the availability of bilinear mappings on elliptic curves, which allow more efficient tripartite key exchange protocols.
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Hitchcock, Yvonne Roslyn. "Elliptic Curve Cryptography for Lightweight Applications." Queensland University of Technology, 2003. http://eprints.qut.edu.au/15838/.
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Elliptic curves were first proposed as a basis for public key cryptography in the mid 1980's. They provide public key cryptosystems based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP) , which is so called because of its similarity to the discrete logarithm problem (DLP) over the integers modulo a large prime. One benefit of elliptic curve cryptosystems (ECCs) is that they can use a much shorter key length than other public key cryptosystems to provide an equivalent level of security. For example, 160 bit ECCs are believed to provide about the same level of security as 1024 bit RSA. Also, the level of security provided by an ECC increases faster with key size than for integer based discrete logarithm (dl) or RSA cryptosystems. ECCs can also provide a faster implementation than RSA or dl systems, and use less bandwidth and power. These issues can be crucial in lightweight applications such as smart cards. In the last few years, ECCs have been included or proposed for inclusion in internationally recognized standards. Thus elliptic curve cryptography is set to become an integral part of lightweight applications in the immediate future. This thesis presents an analysis of several important issues for ECCs on lightweight devices. It begins with an introduction to elliptic curves and the algorithms required to implement an ECC. It then gives an analysis of the speed, code size and memory usage of various possible implementation options. Enough details are presented to enable an implementer to choose for implementation those algorithms which give the greatest speed whilst conforming to the code size and ram restrictions of a particular lightweight device. Recommendations are made for new functions to be included on coprocessors for lightweight devices to support ECC implementations Another issue of concern for implementers is the side-channel attacks that have recently been proposed. They obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time and the information is then used to break implementations that have not incorporated appropriate defences. A new method of defence to protect an implementation from the simple power analysis (spa) method of attack is presented in this thesis. It requires 44% fewer additions and 11% more doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm. The algorithm forms a contribution to the current range of possible spa defences which has a good speed but low memory usage. Another topic of paramount importance to ECCs for lightweight applications is whether the security of fixed curves is equivalent to that of random curves. Because of the inability of lightweight devices to generate secure random curves, fixed curves are used in such devices. These curves provide the additional advantage of requiring less bandwidth, code size and processing time. However, it is intuitively obvious that a large precomputation to aid in the breaking of the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve which would be unavailable for a random curve. Therefore, it would appear that fixed curves are less secure than random curves, but quantifying the loss of security is much more difficult. The thesis performs an examination of fixed curve security taking this observation into account, and includes a definition of equivalent security and an analysis of a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. A lower bound on the expected time to solve such ECDLPs using this method is presented, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. It is concluded that adding a total of 11 bits to the size of a fixed curve provides an equivalent level of security compared to random curves. The final part of the thesis deals with proofs of security of key exchange protocols in the Canetti-Krawczyk proof model. This model has been used since it offers the advantage of a modular proof with reusable components. Firstly a password-based authentication mechanism and its security proof are discussed, followed by an analysis of the use of the authentication mechanism in key exchange protocols. The Canetti-Krawczyk model is then used to examine secure tripartite (three party) key exchange protocols. Tripartite key exchange protocols are particularly suited to ECCs because of the availability of bilinear mappings on elliptic curves, which allow more efficient tripartite key exchange protocols.
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Murdica, Cédric. "Sécurité physique de la cryptographie sur courbes elliptiques." Thesis, Paris, ENST, 2014. http://www.theses.fr/2014ENST0008/document.
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La Cryptographie sur les Courbes Elliptiques (abréviée ECC de l'anglais Elliptic Curve Cryptography) est devenue très importante dans les cartes à puces car elle présente de meilleures performances en temps et en mémoire comparée à d'autres cryptosystèmes asymétriques comme RSA. ECC est présumé incassable dans le modèle dit « Boite Noire », où le cryptanalyste a uniquement accès aux entrées et aux sorties. Cependant, ce n'est pas suffisant si le cryptosystème est embarqué dans un appareil qui est physiquement accessible à de potentiels attaquants. En plus des entrés et des sorties, l'attaquant peut étudier le comportement physique de l'appareil. Ce nouveau type de cryptanalyse est appelé cryptanalyse physique. Cette thèse porte sur les attaques physiques sur ECC. La première partie fournit les pré-requis sur ECC. Du niveau le plus bas au plus élevé, ECC nécessite les outils suivants : l'arithmétique sur les corps finis, l'arithmétique sur courbes elliptiques, la multiplication scalaire sur courbes elliptiques et enfin les protocoles cryptographiques. La deuxième partie expose un état de l'art des différentes attaques physiques et contremesures sur ECC. Pour chaque attaque, nous donnons le contexte dans lequel elle est applicable. Pour chaque contremesure, nous estimons son coût en temps et en mémoire. Nous proposons de nouvelles attaques et de nouvelles contremesures. Ensuite, nous donnons une synthèse claire des attaques suivant le contexte. Cette synthèse est utile pendant la tâche du choix des contremesures. Enfin, une synthèse claire de l'efficacité de chaque contremesure sur les attaques est donnéeElliptic Curve Cryptography (ECC) has gained much importance in smart cards because of its higher speed and lower memory needs compared with other asymmetric cryptosystems such as RSA. ECC is believed to be unbreakable in the black box model, where the cryptanalyst has access to inputs and outputs only. However, it is not enough if the cryptosystem is embedded on a device that is physically accessible to potential attackers. In addition to inputs and outputs, the attacker can study the physical behaviour of the device. This new kind of cryptanalysis is called Physical Cryptanalysis. This thesis focuses on physical cryptanalysis of ECC. The first part gives the background on ECC. From the lowest to the highest level, ECC involves a hierarchy of tools: Finite Field Arithmetic, Elliptic Curve Arithmetic, Elliptic Curve Scalar Multiplication and Cryptographie Protocol. The second part exhibits a state-of-the-art of the different physical attacks and countermeasures on ECC.For each attack, the context on which it can be applied is given while, for each countermeasure, we estimate the lime and memory cost. We propose new attacks and new countermeasures. We then give a clear synthesis of the attacks depending on the context. This is useful during the task of selecting the countermeasures. Finally, we give a clear synthesis of the efficiency of each countermeasure against the attacks
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Abu-Mahfouz, Adnan Mohammed. "Elliptic curve cryptosystem over optimal extension fields for computationally constrained devices." Diss., University of Pretoria, 2004. http://hdl.handle.net/2263/25330.
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Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller.Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006.
Electrical, Electronic and Computer Engineering
unrestricted
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Felding, Eric. "Simuleringar av elliptiska kurvor för elliptisk kryptografi." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-158133.
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This thesis describes the theory behind elliptic-curve Diffie-Hellman key exchanges. All the way from the definition of a group until how the operator over an elliptic curve forms an abelian group. This is illustrated with clear examples. After that a smaller study is made to determine if there is a connection betweenthe size of the underlying field, the amount of points on the curve and the order of the points to determine how hard it is to find out the secret key in elliptic-curve Diffie-Hellman key exchanges. No clear connection is found. Since elliptic curves over extension fields have more computational heavy operations, it is concluded that these curves serve no practical use in elliptic-curve Diffie-Hellman key exchange.Denna rapport går igenom teorin bakom Diffie-Hellmans nyckelutbyte över elliptiska kurvor. Från definitionen av en grupp hela vägen till hur operatorn över en elliptisk kurva utgör en abelsk grupp gås igenom och görs tydligt med konstruktiva exempel. Sedan görs en mindre undersökning av sambandet mellan storleken av den underliggande kroppen, antal punkter på kurvan och ordning av punkterna på kurvan, det vill säga svårigheten att hitta den hemliga nyckeln framtagen med Diffie-Hellmans nyckelutbyte för elliptiska kurvor. Ingen tydlig koppling hittas. Då elliptiska kurvor över utvidgade kroppar har mer beräkningstunga operationer dras slutsatsen att dessa kurvor inte är praktiska inom Diffie-Hellman nyckelutbyte över elliptiska kurvor.
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Chabrier, Thomas. "Arithmetic recodings for ECC cryptoprocessors with protections against side-channel attacks." Phd thesis, Université Rennes 1, 2013. http://tel.archives-ouvertes.fr/tel-00910879.
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This PhD thesis focuses on the study, the hardware design, the theoretical and practical validation, and eventually the comparison of different arithmetic operators for cryptosystems based on elliptic curves (ECC). Provided solutions must be robust against some side-channel attacks, and efficient at a hardware level (execution speed and area). In the case of ECC, we want to protect the secret key, a large integer, used in the scalar multiplication. Our protection methods use representations of numbers, and behaviour of algorithms to make more difficult some attacks. For instance, we randomly change some representations of manipulated numbers while ensuring that computed values are correct. Redundant representations like signed-digit representation, the double- (DBNS) and multi-base number system (MBNS) have been studied. A proposed method provides an on-the-fly MBNS recoding which operates in parallel to curve-level operations and at very high speed. All recoding techniques have been theoretically validated, simulated extensively in software, and finally implemented in hardware (FPGA and ASIC). A side-channel attack called template attack is also carried out to evaluate the robustness of a cryptosystem using a redundant number representation. Eventually, a study is conducted at the hardware level to provide an ECC cryptosystem with a regular behaviour of computed operations during the scalar multiplication so as to protect against some side-channel attacks.
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Judge, Lyndon Virginia. "Design Methods for Cryptanalysis." Thesis, Virginia Tech, 2012. http://hdl.handle.net/10919/35980.
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Security of cryptographic algorithms relies on the computational difficulty of deriving the secret key using public information. Cryptanalysis, including logical and implementation attacks, plays an important role in allowing the security community to estimate their cost, based on the computational resources of an attacker. Practical implementations of cryptanalytic systems require complex designs that integrate multiple functional components with many parameters.
In this thesis, methodologies are proposed to improve the design process of cryptanalytic systems and reduce the cost of design space exploration required for optimization.
First, Bluespec, a rule-based HDL, is used to increase the abstraction level of hardware design and support efficient design space exploration. Bluespec is applied to implement a hardware-accelerated logical attack on ECC with optimized modular arithmetic components. The language features of Bluespec support exploration and this is demonstrated by applying Bluespec to investigate the speed area tradeoff resulting from various design parameters and demonstrating performance that is competitive with prior work. This work also proposes a testing environment for use in verifying the implementation attack resistance of secure systems. A modular design approach is used to provide separation between the device being tested and the test script, as well as portability, and openness. This yields an open-source solution that supports implementation attack testing independent of the system platform, implementation details, and type of attack under evaluation. The suitability of the proposed test environment for implementation attack vulnerability analysis is demonstrated by applying the environment to perform an implementation attack on AES.
The design of complex cryptanalytic hardware can greatly benefit from better design methodologies and the results presented in this thesis advocate the importance of this aspect.Master of Science
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Kirlar, Baris Bulent. "Elliptic Curve Pairing-based Cryptography." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612613/index.pdf.
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In this thesis, we explore the pairing-based cryptography on elliptic curves from the theoretical and implementation point of view. In this respect, we first study so-called pairing-friendly elliptic curves used in pairing-based cryptography. We classify these curves according to their construction methods and study them in details.
Inspired of the work of Koblitz and Menezes, we study the elliptic curves in the form $y^{2}=x^{3}-c$ over the prime field $F_{q}$ and compute explicitly the number of points $#E(mathbb{F}_{q})$. In particular, we show that the elliptic curve $y^{2}=x^{3}-1$ over $mathbb{F}_{q}$ for the primes $q$ of the form $27A^{2}+1$ has an embedding degree $k=1$ and belongs to Scott-Barreto families in our classification. Finally, we give examples of those primes $q$ for which the security level of the pairing-based cryptographic protocols on the curve $y^{2}=x^{3}-1$ over $mathbb{F}_{q}$ is equivalent to 128-, 192-, or 256-bit AES keys.
From the implementation point of view, it is well-known that one of the most important part of the pairing computation is final exponentiation. In this respect, we show explicitly how the final exponentiation is related to the linear recurrence relations. In particular, this correspondence gives that finding an algoritm to compute final exponentiation is equivalent to finding an algorithm to compute the $m$-th term of the associated linear recurrence relation. Furthermore, we list all those work studied in the literature so far and point out how the associated linear recurrence computed efficiently.
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Hutchinson, Aaron. "Algorithms in Elliptic Curve Cryptography." Thesis, Florida Atlantic University, 2019. http://pqdtopen.proquest.com/#viewpdf?dispub=10980188.
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Elliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Diffie-Hellman (ECDH) key exchange algorithm are widely used in practice today for their efficiency and small key sizes. More recently, the Supersingular Isogeny-based Diffie-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the post-quantum setting. For ECDSA and ECDH, efficient and secure algorithms for scalar multiplication of points are necessary for modern use of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny from a given finite subgroup of an elliptic curve in a fast and secure fashion.
We therefore find strong motivation to study and improve the algorithms used in elliptic curve cryptography, and to develop new algorithms to be deployed within these protocols. In this thesis we design and develop d-MUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1-dimensional Montgomery ladder addition chain and the 2-dimensional addition chain due to Dan J. Bernstein. We analyze the construction and derive many optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the final chapter of the thesis we analyze the operations carried out in the construction of an isogeny from a given subgroup, as performed in SIDH. We detail how to efficiently make use of parallel processing when constructing this isogeny.
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Ozturk, Erdinc. "Low Power Elliptic Curve Cryptography." Digital WPI, 2005. https://digitalcommons.wpi.edu/etd-theses/691.
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This M.S. thesis introduces new modulus scaling techniques for transforming a class of primes into special forms which enable efficient arithmetic. The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of a scaled modulus. Our inversion algorithm exhibits superior performance to the Euclidean algorithm and lends itself to efficient hardware implementation due to its simplicity. Using the scaled modulus technique and our specialized inversion algorithm we develop an elliptic curve processor architecture. The resulting architecture successfully utilizes redundant representation of elements in GF(p) and provides a low-power, high speed, and small footprint specialized elliptic curve implementation. We also introduce a unified Montgomery multiplier architecture working on the extension fields GF(p), GF(2) and GF(3). With the increasing research activity for identity based encryption schemes, there has been an increasing need for arithmetic operations in field GF(3). Since we based our research on low-power and small footprint applications, we designed a unified architecture rather than having a seperate hardware for GF{3}. To the best of our knowledge, this is the first time a unified architecture was built working on three different extension fields.
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Ozturk, Erdinc. "Low Power Elliptic Curve Cryptography." Link to electronic thesis, 2004. http://www.wpi.edu/Pubs/ETD/Available/etd-050405-143155/.
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Thesis (M.S.) -- Worcester Polytechnic Institute.Keywords: low power; montgomery multiplication; elliptic curve crytography; modulus scaling; unified architecture; inversion; redundant signed digit. Includes bibliographical references (p.55-59).
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Ozcan, Ayca Bahar. "Performance Analysis Of Elliptic Curve Multiplication Algorithms For Elliptic Curve Cryptography." Master's thesis, METU, 2006. http://etd.lib.metu.edu.tr/upload/12607698/index.pdf.
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Elliptic curve cryptography (ECC) has been introduced as a public-key cryptosystem, which offers smaller key sizes than the other known public-key systems at equivalent security level. The key size advantage of ECC provides faster computations, less memory consumption, less processing power and efficient bandwidth usage. These properties make ECC attractive especially for the next generation public-key cryptosystems. The implementation of ECC involves so many arithmetic operationsone of them is the elliptic curve point multiplication operation, which has a great influence on the performance of ECC protocols. In this thesis work, we have studied on elliptic curve point multiplication methods which are proposed by many researchers. The software implementations of these methods are developed in C programming language on Pentium 4 at 3 GHz. We have used NIST-recommended elliptic curves over prime and binary fields, by using efficient finite field arithmetic. We have then applied our elliptic curve point multiplication implementations to Elliptic Curve Digital Signature Algorithm (ECDSA), and compared different methods. The timing results are presented and comparisons with recent studies have been done.
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Kumar, Sandeep S. "Elliptic curve cryptography for constrained devices." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=982216998.
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Hedges, Mary. "Elliptic curve cryptography and identity-based encryption." Connect to online resource, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:1442908.
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Enos, Graham. "Binary Edwards curves in elliptic curve cryptography." Thesis, The University of North Carolina at Charlotte, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3563153.
Full textAbstract:
Edwards curves are a new normal form for elliptic curves that exhibit some cryptographically desirable properties and advantages over the typical Weierstrass form. Because the group law on an Edwards curve (normal, twisted, or binary) is complete and unified, implementations can be safer from side channel or exceptional procedure attacks. The different types of Edwards provide a better platform for cryptographic primitives, since they have more security built into them from the mathematic foundation up.
Of the three types of Edwards curves—original, twisted, and binary—there hasn't been as much work done on binary curves. We provide the necessary motivation and background, and then delve into the theory of binary Edwards curves. Next, we examine practical considerations that separate binary Edwards curves from other recently proposed normal forms. After that, we provide some of the theory for binary curves that has been worked on for other types already: pairing computations. We next explore some applications of elliptic curve and pairing-based cryptography wherein the added security of binary Edwards curves may come in handy. Finally, we finish with a discussion of e2c2, a modern C++11 library we've developed for Edwards Elliptic Curve Cryptography.
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Ng, Chiu-wa, and 吳潮華. "Elliptic curve cryptography: a study and FPGAimplementation." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B29706336.
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Ontiveros, Mercedes Beatriz. "Elliptic curve cryptography mapped with channel coding." Thesis, University of Newcastle Upon Tyne, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.423718.
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Morozov, Sergey Victorovich. "Elliptic Curve Cryptography on Heterogeneous Multicore Platform." Thesis, Virginia Tech, 2010. http://hdl.handle.net/10919/34872.
Full textAbstract:
Elliptic curve cryptography (ECC) is becoming the algorithm of choice for digital signature generation and authentication in embedded context. However, performance of ECC and the underlying modular arithmetic on embedded processors remains a concern. At the same time, more complex system-on-chip platforms with multiple heterogeneous cores are commonly available in mobile phones and other embedded devices. In this work we investigate the design space for ECC on TI's OMAP 3530 platform, with a focus of utilizing the on-chip DSP core to improve the performance and efficiency of ECC point multiplication on the target platform. We examine multiple aspects of ECC and heterogeneous design such as algorithm-level choices for elliptic curve operations and the effect of interprocessor communication overhead on the design partitioning. We observe how the limitations of the platform constrict the design space of ECC. However, by closely studying the platform and efficiently partitioning the design between the general purpose ARM core and the DSP, we demonstrate a significant speed-up of the resulting ECC implementation. Our system focused approach allows us to accurately measure the performance and power profiles of the resulting implementation. We conclude that heterogeneous multiprocessor design can significantly improve the performance and power consumption of ECC operations, but that the integration cost and the overhead of interprocessor communication cannot be ignored in any actual system.Master of Science
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Guajardo, Jorge. "Efficient Algorithms for Elliptic Curve Cryptosystems." Digital WPI, 2000. https://digitalcommons.wpi.edu/etd-theses/185.
Full textAbstract:
Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF((2n)m). The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide ablolute performance measures for the field operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation.
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Rosner, Martin Christopher. "Elliptic Curve Cryptosystems on Reconfigurable Hardware." Digital WPI, 1999. https://digitalcommons.wpi.edu/etd-theses/883.
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"Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other public-key schemes based on the discrete logarithm in finite fields and the integer factorisation problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structure are composite Galois fields GF((2n)m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplication architectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves or real-world size can be implemented on commercially available FPGAs."
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Hassan, Mohamed Nabil. "Low resource scalable elliptic curve cryptography on FPGA." Thesis, University of Sheffield, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.522417.
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Benits, Junior Waldyr Dias. "Applications of Frobenius expansions in elliptic curve cryptography." Thesis, Royal Holloway, University of London, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.499834.
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Erbsen, Andres. "Crafting certified elliptic curve cryptography implementations in Coq." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/112843.
Full textAbstract:
Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 103-106).
Elliptic curve cryptography has become a de-facto standard for protecting the privacy and integrity of internet communications. To minimize the operational cost and enable near-universal adoption, increasingly sophisticated implementation techniques have been developed. While the complete specification of an elliptic curve cryptosystem (in terms of middle school mathematics) fits on the back of a napkin, the fast implementations span thousands of lines of low-level code and are only intelligible to a small group of experts. However, the complexity of the code makes it prone to bugs, which have rendered well-designed security systems completely ineffective. I describe a principled approach for writing crypto code simultaneously with machine-checkable functional correctness proofs that compose into an end-to-end certificate tying highly optimized C code to the simplest specification used for verification so far. Despite using template-based synthesis for creating low-level code, this workflow offers good control over performance: I was able to match the fastest C implementation of X25519 to within 1% of arithmetic instructions per inner loop and 7% of overall execution time. While the development method itself relies heavily on a proof assistant such as Coq and most techniques are explained through code snippets, every Coq feature is introduced and motivated when it is first used to accommodate a non-Coq-savvy reader.
by Andres Erbsen.
M. Eng.
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Piñol, Piñol Oriol. "Implementation and Evaluation of BSD Elliptic Curve Cryptography." Thesis, KTH, Skolan för informations- och kommunikationsteknik (ICT), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-176839.
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Security is recently arising as an important issue for the Internet of Things (IoT). Efficient ways to provide secure communication between devices and sensors is crucial for the IoT devices, which are becoming more and more used and spread in a variety of fields. In this context, Elliptic Curve Cryptography (ECC) is considered as a strong candidate to provide security while being able to be functional in an environment with strong requirements and limitations such as wireless sensor networks (WSN). The solutions used need to be efficient for devices that have some important restrictions on memory availability and battery life. In this master thesis we present a lightweight BSD-based implementation of the Elliptic Curve Cryptography (ECC) for the Contiki OS and its evaluation. We show the feasibility of the implementation and use of this cryptography in the IoT by a thorough evaluation of the solution by analyzing the performance using different implementations and optimizations of the used algorithms, and also by evaluating it in a real hardware environment. The evaluation of ECC shows that it can adapt to the upcoming challenges, thanks to the level of security that it provides with a smaller size of keys when compared to other legacy cryptography schemes.
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Ho, Sun Wah. "A cryptosystem based on chaotic and elliptic curve cryptography /." access full-text access abstract and table of contents, 2005. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?mphil-it-b19886238a.pdf.
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Thesis (M.Phil.)--City University of Hong Kong, 2005."Submitted to Department of Computer Engineering and Information Technology in partial fulfillment of the requirements for the degree of Master of Philosophy" Includes bibliographical references (leaves 109-111)
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Baktir, Selcuk. "Frequency domain finite field arithmetic for elliptic curve cryptography." Worcester, Mass. : Worcester Polytechnic Institute, 2008. http://www.wpi.edu/Pubs/ETD/Available/etd-050508-142044/.
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Thesis (Ph.D.)--Worcester Polytechnic Institute.Keywords: discrete fourier transform; ECC; elliptic curve cryptography; inversion; finite fields; multiplication; DFT; number theoretic transform; NTT. Includes bibliographical references (leaves 78-85).
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Wilcox, Nicholas. "A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography." Oberlin College Honors Theses / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1528649455201473.
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CANAVESE, DANIELE. "Automatic generation of high speed elliptic curve cryptography code." Doctoral thesis, Politecnico di Torino, 2016. http://hdl.handle.net/11583/2652694.
Full textAbstract:
Apparently, trust is a rare commodity when power, money or life itself are at stake. History is full of examples. Julius Caesar did not trust his generals, so that: ``If he had anything confidential to say, he wrote it in cipher, that is, by so changing the order of the letters of the alphabet, that not a word could be made out. If anyone wishes to decipher these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for A, and so with the others.''
And so the history of cryptography began moving its first steps. Nowadays, encryption has decayed from being an emperor's prerogative and became a daily life operation. Cryptography is pervasive, ubiquitous and, the best of all, completely transparent to the unaware user. Each time we buy something on the Internet we use it. Each time we search something on Google we use it. Everything without (almost) realizing that it silently protects our privacy and our secrets.
Encryption is a very interesting instrument in the "toolbox of security" because it has very few side effects, at least on the user side. A particularly important one is the intrinsic slow down that its use imposes in the communications. High speed cryptography is very important for the Internet, where busy servers proliferate. Being faster is a double advantage: more throughput and less server overhead. In this context, however, the public key algorithms starts with a big handicap. They have very bad performances if compared to their symmetric counterparts. Due to this reason their use is often reduced to the essential operations, most notably key exchanges and digital signatures. The high speed public key cryptography challenge is a very practical topic with serious repercussions in our technocentric world. Using weak algorithms with a reduced key length to increase the performances of a system can lead to catastrophic results.
In 1985, Miller and Koblitz independently proposed to use the group of rational points of an elliptic curve over a finite field to create an asymmetric algorithm. Elliptic Curve Cryptography (ECC) is based on a problem known as the ECDLP (Elliptic Curve Discrete Logarithm Problem) and offers several advantages with respect to other more traditional encryption systems such as RSA and DSA. The main benefit is that it requires smaller keys to provide the same security level since breaking the ECDLP is much harder. In addition, a good ECC implementation can be very efficient both in time and memory consumption, thus being a good candidate for performing high speed public key cryptography. Moreover, some elliptic curve based techniques are known to be extremely resilient to quantum computing attacks, such as the SIDH (Supersingular Isogeny Diffie-Hellman).
Traditional elliptic curve cryptography implementations are optimized by hand taking into account the mathematical properties of the underlying algebraic structures, the target machine architecture and the compiler facilities. This process is time consuming, requires a high degree of expertise and, ultimately, error prone. This dissertation' ultimate goal is to automatize the whole optimization process of cryptographic code, with a special focus on ECC. The framework presented in this thesis is able to produce high speed cryptographic code by automatically choosing the best algorithms and applying a number of code-improving techniques inspired by the compiler theory. Its central component is a flexible and powerful compiler able to translate an algorithm written in a high level language and produce a highly optimized C code for a particular algebraic structure and hardware platform. The system is generic enough to accommodate a wide array of number theory related algorithms, however this document focuses only on optimizing primitives based on elliptic curves defined over binary fields.
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Woodbury, Adam D. "Efficient algorithms for elliptic curve cryptosystems on embedded systems." Link to electronic version, 2001. http://www.wpi.edu/Pubs/ETD/Available/etd-1001101-195321/.
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Bathgate, Jonathan. "Elliptic Curves and their Applications to Cryptography." Thesis, Boston College, 2007. http://hdl.handle.net/2345/389.
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Thesis advisor: Benjamin HowardIn the last twenty years, Elliptic Curve Cryptography has become a standard for the transmission of secure data. The purpose of my thesis is to develop the necessary theory for the implementation of elliptic curve cryptosystems, using elementary number theory, abstract algebra, and geometry. This theory is based on developing formulas for adding rational points on an elliptic curve. The set of rational points on an elliptic curve form a group over the addition law as it is defined. Using the group law, my study continues into computing the torsion subgroup of an elliptic curve and considering elliptic curves over finite fields. With a brief introduction to cryptography and the theory developed in the early chapters, my thesis culminates in the explanation and implementation of three elliptic curve cryptosystems in the Java programming language
Thesis (BA) — Boston College, 2007
Submitted to: Boston College. College of Arts and Sciences
Discipline: Mathematics
Discipline: College Honors Program
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Mus, Koksal. "An Alternative Normal Form For Elliptic Curve Cryptography: Edwards Curves." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12611065/index.pdf.
Full textAbstract:
A new normal form x2 + y2 = c2(1 + x2y2) of elliptic curves was introduced by M. Harold
Edwards in 2007 over the field k having characteristic different than 2. This new form has
very special and important properties such that addition operation is strongly unified and
complete for properly chosen parameter c . In other words, doubling can be done by using
the addition formula and any two points on the curve can be added by the addition formula
without exception. D. Bernstein and T. Lange added one more parameter d to the normal
form to cover a large class of elliptic curves, x2 + y2 = c2(1 + dx2y2) over the same field.
In this thesis, an expository overview of the literature on Edwards curves is given. First, the
types of Edwards curves over the nonbinary field k are introduced, addition and doubling over
the curves are derived and efficient algorithms for addition and doubling are stated with their
costs. Finally, known elliptic curves and Edwards curves are compared according to their
cryptographic applications. The way to choose the Edwards curve which is most appropriate
for cryptographic applications is also explained.
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Lester, Jeremy W. "The Elliptic Curve Group Over Finite Fields: Applications in Cryptography." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1348847698.
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Buop, George Onyango. "Data storage security for cloud computing using elliptic curve cryptography." Master's thesis, University of Cape Town, 2020. http://hdl.handle.net/11427/32489.
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Institutions and enterprises are moving towards more service availability, managed risk and at the same time, aim at reducing cost. Cloud Computing is a growing technology, thriving in the fields of information communication and data storage. With the proliferation of online activity, more and more information is saved as data every day. This means that more data is being stored in the cloud than ever before. Data that is stored online often holds private information – such as addresses, payment details and medical documentation. These become the target of cyber criminals. There is therefore growing need to protect these data from threats and issues such as data breach and leakage, data loss, account takeover or hijackings, among others. Cryptography refers to securing the information and communication techniques based on mathematical concepts and algorithms which transform messages in ways that are hard to decipher. Cryptography is one of the techniques we could protect data stored in the cloud as it enables security properties of data confidentiality and integrity. This research investigates the security issues that affect storage of data in the cloud. This thesis also discusses the previous research work and the currently available technology and techniques that are used for securing data in the cloud. This thesis then presents a novel scheme for security of data stored in Cloud Computing by using Elliptic Curve Integrated Encryption Scheme (ECIES) that provides for confidentiality and integrity. This scheme also uses Identity Based Cryptography (IBC) for more efficient key management. The proposed scheme combines the security of Identity- Based Cryptography (IBC), Trusted cloud (TC), and Elliptic Curve Cryptography (ECC) to reduce system complexity and provide more security for cloud computing applications. The research shows that it is possible to securely store confidential user data on a Public Cloud such as Amazon S3 or Windows Azure Storage without the need to trust the Cloud Provider and with minimal overhead in processing time. The results of implementing the proposed scheme shows faster and more efficient communication operation when it comes to key generation as well as encryption and decryption. The difference in the time taken for these operations is as a result of the use of ECC algorithm which has a small key size and hence highly efficient compared with other types of asymmetric cryptography. The results obtained show the scheme is more efficient, when compared with other classification techniques in the literature.
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Khan, Zia. "Efficient design and implementation of elliptic curve cryptography on FPGA." Thesis, University of Sheffield, 2016. http://etheses.whiterose.ac.uk/13516/.
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This thesis is concerned with challenging the design space of Elliptic Curve Cryptography (ECC) over binary Galois Field, GF(2m) in hardware on field-programmable gate array (FPGA) in terms of area, speed and latency. Novel contributions have been made at the algorithmic, architectural and implementation levels that produced leading performance figures in terms of key hardware implementation metrics on FPGA. This demonstrated performance will enable ECC to be deployed across a range of application requiring public key security using FPGA technology. The proposed low area ECC implementation outperforms relevant state of the art in both area-time and area2-time metrics. The proposed high throughput ECC implementation adopts a new digit serial multiplier over GF(2m) incorporating a novel pipelining technique along with algorithmic and architectural level modification to support parallel operations in the arithmetic level. The resulting throughput/area performance outperforms state of the art designs on FPGA to date. The proposed high-speed only implementation utilises a new full-precision multiplier and smart point multiplication scheduling to reduce the latency. The resulting high speed ECC design with three multipliers achieves the lowest reported latency figure to date with high speed (450 clock cycles to get 2.83 μs on Virtex7). Finally, the proposed low resources scalable ECC implementation is based on very low latency multiprecision multiplication and low latency multiprecision squaring. The scalable ECC point multiplication design over all NIST curves consumes very low latency and shows the best area-time performance on FPGA to date.
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Baktir, Selcuk. "Efficient algorithms for finite fields, with applications in elliptic curve cryptography." Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0501103-132249.
Full textAbstract:
Thesis (M.S.)--Worcester Polytechnic Institute.Keywords: multiplication; OTF; optimal extension fields; finite fields; optimal tower fields; cryptography; OEF; inversion; finite field arithmetic; elliptic curve cryptography. Includes bibliographical references (p. 50-52).
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Lutz, Jonathan. "High Performance Elliptic Curve Cryptographic Co-processor." Thesis, University of Waterloo, 2003. http://hdl.handle.net/10012/855.
Full textAbstract:
In FIPS 186-2, NIST recommends several finite fields to be used in the elliptic curve digital signature algorithm (ECDSA). Of the ten recommended finite fields, five are binary extension fields with degrees ranging from 163 to 571. The fundamental building block of the ECDSA, like any ECC based protocol, is elliptic curve scalar multiplication. This operation is also the most computationally intensive. In many situations it may be desirable to accelerate the elliptic curve scalar multiplication with specialized hardware.
In this thesis a high performance elliptic curve processor is developed which is optimized for the NIST binary fields. The architecture is built from the bottom up starting with the field arithmetic units. The architecture uses a field multiplier capable of performing a field multiplication over the extension field with degree 163 in 0. 060 microseconds. Architectures for squaring and inversion are also presented. The co-processor uses Lopez and Dahab's projective coordinate system and is optimized specifically for Koblitz curves. A prototype of the processor has been implemented for the binary extension field with degree 163 on a Xilinx XCV2000E FPGA. The prototype runs at 66 MHz and performs an elliptic curve scalar multiplication in 0. 233 msec on a generic curve and 0. 075 msec on a Koblitz curve.
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Sen, Nilanjan. "A Performance and Security Analysis of Elliptic Curve Cryptography Based Real-Time Media Encryption." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1609099/.
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This dissertation emphasizes the security aspects of real-time media. The problems of existing real-time media protections are identified in this research, and viable solutions are proposed. First, the security of real-time media depends on the Secure Real-time Transport Protocol (SRTP) mechanism. We identified drawbacks of the existing SRTP Systems, which use symmetric key encryption schemes, which can be exploited by attackers. Elliptic Curve Cryptography (ECC), an asymmetric key cryptography scheme, is proposed to resolve these problems. Second, the ECC encryption scheme is based on elliptic curves. This dissertation explores the weaknesses of a widely used elliptic curve in terms of security and describes a more secure elliptic curve suitable for real-time media protection. Eighteen elliptic curves had been tested in a real-time video transmission system, and fifteen elliptic curves had been tested in a real-time audio transmission system. Based on the performance, X9.62 standard 256-bit prime curve, NIST-recommended 256-bit prime curves, and Brainpool 256-bit prime curves were found to be suitable for real-time audio encryption. Again, X9.62 standard 256-bit prime and 272-bit binary curves, and NIST-recommended 256-bit prime curves were found to be suitable for real-time video encryption.The weaknesses of NIST-recommended elliptic curves are discussed and a more secure new elliptic curve is proposed which can be used for real-time media encryption. The proposed curve has fulfilled all relevant security criteria, but the corresponding NIST curve could not fulfill two such criteria. The research is applicable to strengthen the security of the Internet of Things (IoT) devices, especially VoIP cameras. IoT devices have resource constraints and thus need lightweight encryption schemes for security. ECC could be a better option for these devices. VoIP cameras use a similar methodology to traditional real-time video transmission, so this research could be useful to find a better security solution for these devices.
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McGee, John J. "René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field." Thesis, Virginia Tech, 2006. http://hdl.handle.net/10919/31911.
Full textAbstract:
Elliptic curves have a rich mathematical history dating back to Diophantus (c. 250 C.E.), who used a form of these cubic equations to find right triangles of integer area with rational sides. In more recent times the deep mathematics of elliptic curves was used by Andrew Wiles et. al., to construct a proof of Fermat's last theorem, a problem which challenged mathematicians for more than 300 years. In addition, elliptic curves over finite fields find practical application in the areas of cryptography and coding theory. For such problems, knowing the order of the group of points satisfying the elliptic curve equation is important to the security of these applications. In 1985 René Schoof published a paper [5] describing a polynomial time algorithm for solving this problem. In this thesis we explain some of the key mathematical principles that provide the basis for Schoof's method. We also present an implementation of Schoof's algorithm as a collection of Mathematica functions. The operation of each algorithm is illustrated by way of numerical examples.Master of Science
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Man, Kwan Pok. "Security enhancement on the cryptosystem based on chaotic and elliptic curve cryptography /." access abstract and table of contents access full-text, 2006. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?mphil-ee-b21471526a.pdf.
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Thesis (M.Phil.)--City University of Hong Kong, 2006."Submitted to Department of Electronic Engineering in partial fulfillment of the requirements for the degree of Master of Philosophy" Includes bibliographical references (leaves 93-97)
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Huang, Jian Li Hao. "FPGA implementations of elliptic curve cryptography and Tate pairing over binary field." [Denton, Tex.] : University of North Texas, 2007. http://digital.library.unt.edu/permalink/meta-dc-3963.
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Huang, Jian. "FPGA Implementations of Elliptic Curve Cryptography and Tate Pairing over Binary Field." Thesis, University of North Texas, 2007. https://digital.library.unt.edu/ark:/67531/metadc3963/.
Full textAbstract:
Elliptic curve cryptography (ECC) is an alternative to traditional techniques for public key cryptography. It offers smaller key size without sacrificing security level. Tate pairing is a bilinear map used in identity based cryptography schemes. In a typical elliptic curve cryptosystem, elliptic curve point multiplication is the most computationally expensive component. Similarly, Tate pairing is also quite computationally expensive. Therefore, it is more attractive to implement the ECC and Tate pairing using hardware than using software. The bases of both ECC and Tate pairing are Galois field arithmetic units. In this thesis, I propose the FPGA implementations of the elliptic curve point multiplication in GF (2283) as well as Tate pairing computation on supersingular elliptic curve in GF (2283). I have designed and synthesized the elliptic curve point multiplication and Tate pairing module using Xilinx's FPGA, as well as synthesized all the Galois arithmetic units used in the designs. Experimental results demonstrate that the FPGA implementation can speedup the elliptic curve point multiplication by 31.6 times compared to software based implementation. The results also demonstrate that the FPGA implementation can speedup the Tate pairing computation by 152 times compared to software based implementation.
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Cai, Zhi, and 蔡植. "A study on parameters generation of elliptic curve cryptosystem over finite fields." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B31225639.
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Higuchi, Akira, 直史 高木, and Naofumi Takagi. "A fast addition algorithm for elliptic curve arithmetic in GF(2n) using projective coordinates." Elsevier, 2000. http://hdl.handle.net/2237/2754.
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Houssain, Hilal. "Elliptic curve cryptography algorithms resistant against power analysis attacks on resource constrained devices." Thesis, Clermont-Ferrand 2, 2012. http://www.theses.fr/2012CLF22286/document.
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Les systèmes de cryptographie à base de courbe elliptique (ECC) ont été adoptés comme des systèmes standardisés de cryptographie à clé publique (PKC) par l'IEEE, ANSI, NIST, SEC et WTLS. En comparaison avec la PKC traditionnelle, comme RSA et ElGamal, l'ECC offre le même niveau de sécurité avec des clés de plus petites tailles. Cela signifie des calculs plus rapides et une consommation d'énergie plus faible ainsi que des économies de mémoire et de bande passante. Par conséquent, ECC est devenue une technologie indispensable, plus populaire et considérée comme particulièrement adaptée à l’implémentation sur les dispositifs à ressources restreintes tels que les réseaux de capteurs sans fil (WSN). Le problème majeur avec les noeuds de capteurs chez les WSN, dès qu'il s'agit d’opérations cryptographiques, est les limitations de leurs ressources en termes de puissance, d'espace et de temps de réponse, ce qui limite la capacité du capteur à gérer les calculs supplémentaires nécessaires aux opérations cryptographiques. En outre, les mises en oeuvre actuelles de l’ECC sur WSN sont particulièrement vulnérables aux attaques par canaux auxiliaires (SCA), en particulier aux attaques par analyse de consommation (PAA), en raison de l'absence de la sécurité physique par blindage, leur déploiement dans les régions éloignées et le fait qu’elles soient laissées sans surveillance. Ainsi, les concepteurs de crypto-processeurs ECC sur WSN s'efforcent d'introduire des algorithmes et des architectures qui ne sont pas seulement résistants PAA, mais également efficaces sans aucun supplément en termes de temps, puissance et espace. Cette thèse présente plusieurs contributions dans le domaine des cryptoprocesseurs ECC conscientisés aux PAA, pour les dispositifs à ressources limitées comme le WSN. Premièrement, nous proposons deux architectures robustes et efficaces pour les ECC conscientisées au PAA. Ces architectures sont basées sur des algorithmes innovants qui assurent le fonctionnement de base des ECC et qui prévoient une sécurisation de l’ECC contre les PAA simples (SPA) sur les dispositifs à ressources limitées tels que les WSN. Deuxièmement, nous proposons deux architectures additionnelles qui prévoient une sécurisation des ECC contre les PAA différentiels (DPA). Troisièmement, un total de huit architectures qui incluent, en plus des quatre architectures citées ci-dessus pour SPA et DPA, deux autres architectures dérivées de l’architecture DPA conscientisée, ainsi que deux architectures PAA conscientisées. Les huit architectures proposées sont synthétisées en utilisant la technologie des réseaux de portes programmables in situ (FPGA). Quatrièmement, les huit architectures sont analysées et évaluées, et leurs performances comparées. En plus, une comparaison plus avancée effectuée sur le niveau de la complexité du coût (temps, puissance, et espace), fournit un cadre pour les concepteurs d'architecture pour sélectionner la conception la plus appropriée. Nos résultats montrent un avantage significatif de nos architectures proposées par rapport à la complexité du coût, en comparaison à d'autres solutions proposées récemment dans le domaine de la rechercheElliptic Curve Cryptosystems (ECC) have been adopted as a standardized Public Key Cryptosystems (PKC) by IEEE, ANSI, NIST, SEC and WTLS. In comparison to traditional PKC like RSA and ElGamal, ECC offer equivalent security with smaller key sizes, in less computation time, with lower power consumption, as well as memory and bandwidth savings. Therefore, ECC have become a vital technology, more popular and considered to be particularly suitable for implementation on resource constrained devices such as the Wireless Sensor Networks (WSN). Major problem with the sensor nodes in WSN as soon as it comes to cryptographic operations is their extreme constrained resources in terms of power, space, and time delay, which limit the sensor capability to handle the additional computations required by cryptographic operations. Moreover, the current ECC implementations in WSN are particularly vulnerable to Side Channel Analysis (SCA) attacks; in particularly to the Power Analysis Attacks (PAA), due to the lack of secure physical shielding, their deployment in remote regions and it is left unattended. Thus designers of ECC cryptoprocessors on WSN strive to introduce algorithms and architectures that are not only PAA resistant, but also efficient with no any extra cost in terms of power, time delay, and area. The contributions of this thesis to the domain of PAA aware elliptic curve cryptoprocessor for resource constrained devices are numerous. Firstly, we propose two robust and high efficient PAA aware elliptic curve cryptoprocessors architectures based on innovative algorithms for ECC core operation and envisioned at securing the elliptic curve cryptoprocessors against Simple Power Analysis (SPA) attacks on resource constrained devices such as the WSN. Secondly, we propose two additional architectures that are envisioned at securing the elliptic curve cryptoprocessors against Differential Power Analysis (DPA) attacks. Thirdly, a total of eight architectures which includes, in addition to the two SPA aware with the other two DPA awareproposed architectures, two more architectures derived from our DPA aware proposed once, along with two other similar PAA aware architectures. The eight proposed architectures are synthesized using Field Programmable Gate Array (FPGA) technology. Fourthly, the eight proposed architectures are analyzed and evaluated by comparing their performance results. In addition, a more advanced comparison, which is done on the cost complexity level (Area, Delay, and Power), provides a framework for the architecture designers to select the appropriate design. Our results show a significant advantage of our proposed architectures for cost complexity in comparison to the other latest proposed in the research field
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Chelton, William N. "Galois Field GF (2'') Arithmetic Circuits and Their Application in Elliptic Curve Cryptography." Thesis, University of Sheffield, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.490334.
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