In 2001, Boneh, Lynn, and Shacham presented a pairing-based signature scheme known as the BLS signature scheme. In 2003, Boneh, Gentry, Lynn, and Shacham presented the first aggregate signature scheme called the BGLS aggregate signature scheme. The BGLS scheme allows for N users with N signatures to combine their signatures into a single signature. The size of the resulting signature is independent of N. The BGLS signature scheme enjoys roughly the same level of security as the BLS scheme. In 2005, Waters presented a pairing-based signature scheme which does not assume the existence of random oracles. In 2007, Lu, Ostrovsky, Sahai, Shacham, and Waters presented the LOSSW aggregate signature scheme which does not assume the existence of random oracles. The BLS, BGLS, Waters, and LOSSW authors each chose to work with a restricted class of pairings. In each scheme, it is clear that the scheme extend to arbitrary pairings. We present the schemes in their full generality, explore variations of the schemes, and discuss optimizations that can be made when using specific pairings. Each of the schemes we discuss is secure assuming that the computational Diffie-Hellman (CDH) assumption holds. We improve on the security reduction for a variation of the BGLS signature scheme which allows for some restrictions of the BGLS signature scheme can be dropped and provides a stronger guarantee of security. We show that the BGLS scheme can be modified to reduce public-key size in presence of a certifying authority, when a certain type of pairing is used. We show that patient-free bit-compression can be applied to each of the scheme with a few modifications.

In 2001, Boneh, Lynn, and Shacham presented a pairing-based signature scheme known as the BLS signature scheme. In 2003, Boneh, Gentry, Lynn, and Shacham presented the first aggregate signature scheme called the BGLS aggregate signature scheme. The BGLS scheme allows for N users with N signatures to combine their signatures into a single signature. The size of the resulting signature is independent of N. The BGLS signature scheme enjoys roughly the same level of security as the BLS scheme. In 2005, Waters presented a pairing-based signature scheme which does not assume the existence of random oracles. In 2007, Lu, Ostrovsky, Sahai, Shacham, and Waters presented the LOSSW aggregate signature scheme which does not assume the existence of random oracles. The BLS, BGLS, Waters, and LOSSW authors each chose to work with a restricted class of pairings. In each scheme, it is clear that the scheme extend to arbitrary pairings. We present the schemes in their full generality, explore variations of the schemes, and discuss optimizations that can be made when using specific pairings. Each of the schemes we discuss is secure assuming that the computational Diffie-Hellman (CDH) assumption holds. We improve on the security reduction for a variation of the BGLS signature scheme which allows for some restrictions of the BGLS signature scheme can be dropped and provides a stronger guarantee of security. We show that the BGLS scheme can be modified to reduce public-key size in presence of a certifying authority, when a certain type of pairing is used. We show that patient-free bit-compression can be applied to each of the scheme with a few modifications.

Elliptic Curve Cryptography (ECC), independently proposed by Miller [Mil86] and Koblitz [Kob87] in mid 80’s, is finding momentum to consolidate its status as the public-key system of choice in a wide range of applications and to further expand this position to settings traditionally occupied by RSA and DL-based systems. The non-existence of known subexponential attacks on this cryptosystem directly translates to shorter keylengths for a given security level and, consequently, has led to implementations with better bandwidth usage, reduced power and memory requirements, and higher speeds. Moreover, the dramatic entry of pairing-based cryptosystems defined on elliptic curves at the beginning of the new millennium has opened the possibility of a plethora of innovative applications, solving in some cases longstanding problems in cryptography. Nevertheless, public-key cryptography (PKC) is still relatively expensive in comparison with its symmetric-key counterpart and it remains an open challenge to reduce further the computing cost of the most time-consuming PKC primitives to guarantee their adoption for secure communication in commercial and Internet-based applications. The latter is especially true for pairing computations. Thus, it is of paramount importance to research methods which permit the efficient realization of Elliptic Curve and Pairing-based Cryptography on the several new platforms and applications. This thesis deals with efficient methods and explicit formulas for computing elliptic curve scalar multiplication and pairings over fields of large prime characteristic with the objective of enabling the realization of software implementations at very high speeds. To achieve this main goal in the case of elliptic curves, we accomplish the following tasks: identify the elliptic curve settings with the fastest arithmetic; accelerate the precomputation stage in the scalar multiplication; study number representations and scalar multiplication algorithms for speeding up the evaluation stage; identify most efficient field arithmetic algorithms and optimize them; analyze the architecture of the targeted platforms for maximizing the performance of ECC operations; identify most efficient coordinate systems and optimize explicit formulas; and realize implementations on x86-64 processors with an optimal algorithmic selection among all studied cases. In the case of pairings, the following tasks are accomplished: accelerate tower and curve arithmetic; identify most efficient tower and field arithmetic algorithms and optimize them; identify the curve setting with the fastest arithmetic and optimize it; identify state-of-the-art techniques for the Miller loop and final exponentiation; and realize an implementation on x86-64 processors with optimal algorithmic selection. The most outstanding contributions that have been achieved with the methodologies above in this thesis can be summarized as follows: • Two novel precomputation schemes are introduced and shown to achieve the lowest costs in the literature for different curve forms and scalar multiplication primitives. The detailed cost formulas of the schemes are derived for most relevant scenarios. • A new methodology based on the operation cost per bit to devise highly optimized and compact multibase algorithms is proposed. Derived multibase chains using bases {2,3} and {2,3,5} are shown to achieve the lowest theoretical costs for scalar multiplication on certain curve forms and for scenarios with and without precomputations. In addition, the zero and nonzero density formulas of the original (width-w) multibase NAF method are derived by using Markov chains. The application of “fractional” windows to the multibase method is described together with the derivation of the corresponding density formulas. • Incomplete reduction and branchless arithmetic techniques are optimally combined for devising high-performance field arithmetic. Efficient algorithms for “small” modular operations using suitably chosen pseudo-Mersenne primes are carefully analyzed and optimized for incomplete reduction. • Data dependencies between contiguous field operations are discovered to be a source of performance degradation on x86-64 processors. Three techniques for reducing the number of potential pipeline stalls due to these dependencies are proposed: field arithmetic scheduling, merging of point operations and merging of field operations. • Explicit formulas for two relevant cases, namely Weierstrass and Twisted Edwards curves over and , are carefully optimized employing incomplete reduction, minimal number of operations and reduced number of data dependencies between contiguous field operations. • Best algorithms for the field, point and scalar arithmetic, studied or proposed in this thesis, are brought together to realize four high-speed implementations on x86-64 processors at the 128-bit security level. Presented results set new speed records for elliptic curve scalar multiplication and introduce up to 34% of cost reduction in comparison with the best previous results in the literature. • A generalized lazy reduction technique that enables the elimination of up to 32% of modular reductions in the pairing computation is proposed. Further, a methodology that keeps intermediate results under Montgomery reduction boundaries maximizing operations without carry checks is introduced. Optimized formulas for the popular tower are explicitly stated and a detailed operation count that permits to determine the theoretical cost improvement attainable with the proposed method is carried out for the case of an optimal ate pairing on a Barreto-Naehrig (BN) curve at the 128-bit security level. • Best algorithms for the different stages of the pairing computation, including the proposed techniques and optimizations, are brought together to realize a high-speed implementation at the 128-bit security level. Presented results on x86-64 processors set new speed records for pairings, introducing up to 34% of cost reduction in comparison with the best published result. From a general viewpoint, the proposed methods and optimized formulas have a practical impact in the performance of cryptographic protocols based on elliptic curves and pairings in a wide range of applications. In particular, the introduced implementations represent a direct and significant improvement that may be exploited in performance-dominated applications such as high-demand Web servers in which millions of secure transactions need to be generated.

Implementation of elliptic and hyperelliptic curve cryptographic algorithms has been the focus of a great deal of recent research directed at increasing efficiency. Elliptic curve cryptography (ECC) was introduced independently by Koblitz and Miller in the 1980s. Hyperelliptic curve cryptography (HECC), a generalization of the elliptic curve case, allows a decreasing field size as the genus increases. The work presented in this thesis examines the problems created by limited area, power, and computation time when elliptic and hyperelliptic curves are integrated into constrained devices such as wireless sensor network (WSN) and smart cards. The lack of a battery in wireless sensor network limits the processing power of these devices, but they still require security. It was widely believed that devices with such constrained resources cannot incorporate a strong HECC processor for performing cryptographic operations such as elliptic curve scalar multiplication (ECSM) or hyperelliptic curve divisor multiplication (HCDM). However, the work presented in this thesis has demonstrated the feasibility of integrating an HECC processor into such devices through the use of the proposed architecture synthesis and optimization techniques for several inversion-free algorithms. The goal of this work is to develop a hardware implementation of binary elliptic and hyperelliptic curves. The focus is on the modeling of three factors: register allocation, operation scheduling, and storage binding. These factors were then integrated into architecture synthesis and optimization techniques in order to determine the best overall implementation suitable for constrained devices. The main purpose of the optimization is to reduce the area and power. Through analysis of the architecture optimization techniques for both datapath and control unit synthesis, the number of registers was reduced by an average of 30%. The use of the proposed efficient explicit formula for the different algorithms also enabled a reduction in the number of read/write operations from/to the register file, which reduces the processing power consumption. As a result, an overall HECC processor requires from 1843 to 3595 slices for a Xilinix XC4VLX200 and the total computation time is limited to between 10.08 ms to 15.82 ms at a maximum frequency of 50 MHz for a varity of inversion-free coordinate systems in hyperelliptic curves. The value of the new model has been demonstrated with respect to its implementation in elliptic and hyperelliptic curve crypogrpahic algorithms, through both synthesis and simulations. In summary, a framework has been provided for consideration of interactions with synthesis and optimization through architecture modeling for constrained enviroments. Insights have also been presented with respect to improving the design process for cryptogrpahic algorithms through datapath and control unit analysis.

Three quantum cryptographic protocols of multiuser quantum networks with embedded authentication, allowing quantum key distribution or quantum direct communication, are discussed in this work. The security of the protocols against different types of attacks is analysed with a focus on various impersonation attacks and the man-in-the-middle attack. On the basis of the security analyses several improvements are suggested and implemented in order to adjust the investigated vulnerabilities. Furthermore, the impact of the eavesdropping test procedure on impersonation attacks is outlined. The framework of a general eavesdropping test is proposed to provide additional protection against security risks in impersonation attacks. / In der Diplomarbeit werden drei verschiedene quantenkryptographische Protokolle mit dem Schwerpunkt auf authentifizierten Quantennetzwerken analysiert. Die Sicherheit der Protokolle gegenüber verschiedenen Angriffen wird untersucht, wobei der Fokus auf kompletten Personifikationsattacken („impersonation attacks“) liegt. Auf Basis der Sicherheitsanalyse und den Netzwerkanforderungen werden entsprechende Verbesserungen vorgeschlagen. Um die Gefahr von Personifikationen realistisch abschätzen zu können, wird außerdem der Einfluss des Testablaufs analysiert. Um zusätzlichen Schutz gegen Personifikationsattacken zu gewährleisten, werden die Rahmenbedingungen für eine allgemeine Testspezifikation festgelegt.

Dans cette thèse, nous nous intéressons à la cryptanalyse de certaines primitives de cryptographie symétrique qui utilisent les concepts de construction du schéma de chiffrement AES. Nous commençons par une analyse de l'AES lui-même dans trois modèles de sécurité différents: le modèle standard, le modèle à clefs reliées et le modèle ouvert. Dans le modèle standard, où l'adversaire cherche à récupérer la clef secrète, nous décrivons les meilleures attaques différentielles existantes sur cet algorithme de chiffrement, en améliorant les attaques différentielles précédemment publiées. Ensuite, nous procédons à une analyse structurelle de l'AES dans le modèle à clefs reliées. Nous montrons des résultats d'impossibilité, indiquant que l'on ne peut pas prouver la sécurité de la structure de l'AES contre les attaques différentielles dans ce modèle. Enfin, dans le modèle ouvert, nous proposons le premier distingueur pour neuf tours d'AES-128, ce qui résout un problème ouvert depuis plusieurs années dans la communauté symétrique. Dans une deuxième partie, nous analysons en détail l'application de l'attaque par rebond sur les primitives basées sur l'AES. Nous montrons qu'il est possible de considérer un tour de plus dans la première des deux phases de cette stratégie, ce qui améliore les meilleurs résultats connus sur les permutations à base d'AES. Ceci résout le problème ouvert consistant à augmenter le nombre total de tours attaqués grâce à cette technique. Nous montrons également qu'il est possible de relâcher certaines contraintes pour augmenter la probabilité de succès de la deuxième étape. Ceci conduit à une diminution des complexités de toutes les attaques publiées. Nous appliquons ces améliorations à la fonction de hachage Grostl, obtenant les meilleures attaques sur la permutation interne. Finalement, nous nous intéressons à la fonction de hachage ECHO pour montrer qu'il est possible d'appliquer plusieurs fois l'attaque par rebond et ainsi attaquer plus de tours de la permutation interne.

Elliptic Curve Cryptography (ECC), independently proposed by Miller [Mil86] and Koblitz [Kob87] in mid 80’s, is finding momentum to consolidate its status as the public-key system of choice in a wide range of applications and to further expand this position to settings traditionally occupied by RSA and DL-based systems. The non-existence of known subexponential attacks on this cryptosystem directly translates to shorter keylengths for a given security level and, consequently, has led to implementations with better bandwidth usage, reduced power and memory requirements, and higher speeds. Moreover, the dramatic entry of pairing-based cryptosystems defined on elliptic curves at the beginning of the new millennium has opened the possibility of a plethora of innovative applications, solving in some cases longstanding problems in cryptography. Nevertheless, public-key cryptography (PKC) is still relatively expensive in comparison with its symmetric-key counterpart and it remains an open challenge to reduce further the computing cost of the most time-consuming PKC primitives to guarantee their adoption for secure communication in commercial and Internet-based applications. The latter is especially true for pairing computations. Thus, it is of paramount importance to research methods which permit the efficient realization of Elliptic Curve and Pairing-based Cryptography on the several new platforms and applications. This thesis deals with efficient methods and explicit formulas for computing elliptic curve scalar multiplication and pairings over fields of large prime characteristic with the objective of enabling the realization of software implementations at very high speeds. To achieve this main goal in the case of elliptic curves, we accomplish the following tasks: identify the elliptic curve settings with the fastest arithmetic; accelerate the precomputation stage in the scalar multiplication; study number representations and scalar multiplication algorithms for speeding up the evaluation stage; identify most efficient field arithmetic algorithms and optimize them; analyze the architecture of the targeted platforms for maximizing the performance of ECC operations; identify most efficient coordinate systems and optimize explicit formulas; and realize implementations on x86-64 processors with an optimal algorithmic selection among all studied cases. In the case of pairings, the following tasks are accomplished: accelerate tower and curve arithmetic; identify most efficient tower and field arithmetic algorithms and optimize them; identify the curve setting with the fastest arithmetic and optimize it; identify state-of-the-art techniques for the Miller loop and final exponentiation; and realize an implementation on x86-64 processors with optimal algorithmic selection. The most outstanding contributions that have been achieved with the methodologies above in this thesis can be summarized as follows: • Two novel precomputation schemes are introduced and shown to achieve the lowest costs in the literature for different curve forms and scalar multiplication primitives. The detailed cost formulas of the schemes are derived for most relevant scenarios. • A new methodology based on the operation cost per bit to devise highly optimized and compact multibase algorithms is proposed. Derived multibase chains using bases {2,3} and {2,3,5} are shown to achieve the lowest theoretical costs for scalar multiplication on certain curve forms and for scenarios with and without precomputations. In addition, the zero and nonzero density formulas of the original (width-w) multibase NAF method are derived by using Markov chains. The application of “fractional” windows to the multibase method is described together with the derivation of the corresponding density formulas. • Incomplete reduction and branchless arithmetic techniques are optimally combined for devising high-performance field arithmetic. Efficient algorithms for “small” modular operations using suitably chosen pseudo-Mersenne primes are carefully analyzed and optimized for incomplete reduction. • Data dependencies between contiguous field operations are discovered to be a source of performance degradation on x86-64 processors. Three techniques for reducing the number of potential pipeline stalls due to these dependencies are proposed: field arithmetic scheduling, merging of point operations and merging of field operations. • Explicit formulas for two relevant cases, namely Weierstrass and Twisted Edwards curves over and , are carefully optimized employing incomplete reduction, minimal number of operations and reduced number of data dependencies between contiguous field operations. • Best algorithms for the field, point and scalar arithmetic, studied or proposed in this thesis, are brought together to realize four high-speed implementations on x86-64 processors at the 128-bit security level. Presented results set new speed records for elliptic curve scalar multiplication and introduce up to 34% of cost reduction in comparison with the best previous results in the literature. • A generalized lazy reduction technique that enables the elimination of up to 32% of modular reductions in the pairing computation is proposed. Further, a methodology that keeps intermediate results under Montgomery reduction boundaries maximizing operations without carry checks is introduced. Optimized formulas for the popular tower are explicitly stated and a detailed operation count that permits to determine the theoretical cost improvement attainable with the proposed method is carried out for the case of an optimal ate pairing on a Barreto-Naehrig (BN) curve at the 128-bit security level. • Best algorithms for the different stages of the pairing computation, including the proposed techniques and optimizations, are brought together to realize a high-speed implementation at the 128-bit security level. Presented results on x86-64 processors set new speed records for pairings, introducing up to 34% of cost reduction in comparison with the best published result. From a general viewpoint, the proposed methods and optimized formulas have a practical impact in the performance of cryptographic protocols based on elliptic curves and pairings in a wide range of applications. In particular, the introduced implementations represent a direct and significant improvement that may be exploited in performance-dominated applications such as high-demand Web servers in which millions of secure transactions need to be generated.

Pairings have found a range of applications in many areas of cryptography. As such, to utilize the enormous potential of pairing-based protocols one needs to efficiently compute pairings across various computing platforms. In this thesis, we give an introduction to pairing-based cryptography and describe the Tate pairing and its variants. We then describe some recent work to realize efficient computation of pairings. We further extend these optimizations and implement the O-Ate pairing on BN-curves on ARM and x86-64 platforms. Specifically, we extend the idea of lazy reduction to field inversion, optimize curve arithmetic, and construct efficient tower extensions to optimize field arithmetic. We also analyze the use of affine coordinates for pairing computation leading us to the conclusion that they are a competitive choice for fast pairing computation on ARM processors, especially at high security level. Our resulting implementation is more than three times faster than any previously reported implementation on ARM processors.

Implementation of elliptic and hyperelliptic curve cryptographic algorithms has been the focus of a great deal of recent research directed at increasing efficiency. Elliptic curve cryptography (ECC) was introduced independently by Koblitz and Miller in the 1980s. Hyperelliptic curve cryptography (HECC), a generalization of the elliptic curve case, allows a decreasing field size as the genus increases. The work presented in this thesis examines the problems created by limited area, power, and computation time when elliptic and hyperelliptic curves are integrated into constrained devices such as wireless sensor network (WSN) and smart cards. The lack of a battery in wireless sensor network limits the processing power of these devices, but they still require security. It was widely believed that devices with such constrained resources cannot incorporate a strong HECC processor for performing cryptographic operations such as elliptic curve scalar multiplication (ECSM) or hyperelliptic curve divisor multiplication (HCDM). However, the work presented in this thesis has demonstrated the feasibility of integrating an HECC processor into such devices through the use of the proposed architecture synthesis and optimization techniques for several inversion-free algorithms. The goal of this work is to develop a hardware implementation of binary elliptic and hyperelliptic curves. The focus is on the modeling of three factors: register allocation, operation scheduling, and storage binding. These factors were then integrated into architecture synthesis and optimization techniques in order to determine the best overall implementation suitable for constrained devices. The main purpose of the optimization is to reduce the area and power. Through analysis of the architecture optimization techniques for both datapath and control unit synthesis, the number of registers was reduced by an average of 30%. The use of the proposed efficient explicit formula for the different algorithms also enabled a reduction in the number of read/write operations from/to the register file, which reduces the processing power consumption. As a result, an overall HECC processor requires from 1843 to 3595 slices for a Xilinix XC4VLX200 and the total computation time is limited to between 10.08 ms to 15.82 ms at a maximum frequency of 50 MHz for a varity of inversion-free coordinate systems in hyperelliptic curves. The value of the new model has been demonstrated with respect to its implementation in elliptic and hyperelliptic curve crypogrpahic algorithms, through both synthesis and simulations. In summary, a framework has been provided for consideration of interactions with synthesis and optimization through architecture modeling for constrained enviroments. Insights have also been presented with respect to improving the design process for cryptogrpahic algorithms through datapath and control unit analysis.

Thesis (Ph.D.)--University of Wollongong, 2005. / Typescript. Includes bibliographical references: leaf 149-157.