Constructing Permutation Rational Functions From Isogenies

A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994)

Efficient and Secure ECDSA Algorithm and its Applications: A Survey

Public-key cryptography algorithms, especially elliptic curve cryptography (ECC)and elliptic curve digital signature algorithm (ECDSA) have been attracting attention frommany researchers in different institutions because these algorithms provide security andhigh performance when being used in many areas such as electronic-healthcare, electronicbanking,electronic-commerce, electronic-vehicular, and electronic-governance. These algorithmsheighten security against various attacks and the same time improve performanceto obtain efficiencies (time, memory, reduced computation complexity, and energy saving)in an environment of constrained source and large systems. This paper presents detailedand a comprehensive survey of an update of the ECDSA algorithm in terms of performance,security, and applications

Failure of the Point Blinding Countermeasure Against Fault Attack in Pairing-Based Cryptography

Article published in the proceedings of the C2SI conference, May 2015.Pairings are mathematical tools that have been proven to be very useful in the construction of many cryptographic protocols. Some of these protocols are suitable for implementation on power constrained devices such as smart cards or smartphone which are subject to side channel attacks. In this paper, we analyse the efficiency of the point blinding countermeasure in pairing based cryptography against side channel attacks. In particular,we show that this countermeasure does not protect Miller's algorithm for pairing computation against fault attack. We then give recommendation for a secure implementation of a pairing based protocol using the Miller algorithm

Computing information on domain parameters from public keys selected uniformly at random

The security of many cryptographic schemes and protocols rests on the conjectured computational intractability of the discrete logarithm problem in some group of prime order. Such schemes and protocols require domain parameters that specify and a specific generator g. In this paper we consider the problem of computing information on the domain parameters from public keys selected uniformly at random from . We show that it is not possible to compute any information on the generator g regardless of the number of public keys observed. In the case of elliptic curves E(GF(p)) or E(GF(2^n)) on short Weierstrass form, or E(K) on Edwards form, twisted Edwards form or Montgomery form, where K is a non-binary field, we show how to compute the domain parameters excluding the generator from four keys on affine form. Hence, if the domain parameters excluding the generator are to be kept private, points may not be transmitted on affine form. It is an open question whether point compression is a sufficient requirement. Regardless of whether points are transmitted on affine or compressed form, it is in general possible to create a distinguisher for the domain parameters, excluding the generator, both in the case of the elliptic curve groups previously mentioned, and in the case of multiplicative subgroups of GF(p). We propose that a good method for preventing all of the above attacks may be to use blinding schemes, and suggest new applications for existing blinding schemes originally designed for steganographic applications

Proxy Blind Signature using Hyperelliptic Curve Cryptography

Blind signature is the concept to ensure anonymity of e-coins. Untracebility and unlinkability are two main properties of real coins and should also be mimicked electronically. A user has to fulll above two properties of blind signature for permission to spend an e-coin. During the last few years, asymmetric cryptosystems based on curve based cryptographiy have become very popular, especially for embedded applications. Elliptic curves(EC) are a special case of hyperelliptic curves (HEC). HEC operand size is only a fraction of the EC operand size. HEC cryptography needs a group order of size at least 2160. In particular, for a curve of genus two eld Fq with p 280 is needeed. Therefore, the eld arithmetic has to be performed using 80-bit long operands. Which is much better than the RSA using 1024 bit key length. The hyperelliptic curve is best suited for the resource constraint environments. It uses lesser key and provides more secure transmisstion of data

Cryptographically Strong Elliptic Curves of Prime Order

The purpose of this paper is to generate cryptographically strong elliptic curves over prime fields Fp, where p is a Mersenne prime, one of the special primes or a random prime. We search for elliptic curves which orders are also prime numbers. The cryptographically strong elliptic curves are those for which the discrete logarithm problem is computationally hard. The required mathematical conditions are formulated in terms of parameters characterizing the elliptic curves.We present an algorithm to generate such curves. Examples of elliptic curves of prime order are generated with Magma

Privacy-preserving Payments for Transportation Systems

The operation of our society heavily relies on high mobility of people. Not only our social life but also our economy and trade are built upon a system where people need to be able to move around easily. The costs for building and maintaining a suitable transportation infrastructure to satisfy those needs are high, and to charge users is thus a central requirement. This calls for well functioning payment systems satisfying the multitude of requirements that transportation systems impose on them. Electronic payment systems have many benefits over traditional cash payments as they are easy to maintain, can be more secure, reduce revenue collection costs, and can reduce the execution time of a payment. However, as a drawback, currently employed electronic payment systems usually reveal a payer’s identity during a payment which greatly infringes customer privacy. In the transportation domain this allows to generate fine grain patterns of customers’ locations. Cryptographic payment protocols called e-cash have been proposed which allow to preserve a customer’s privacy. E-cash provides provable guarantees for both security and user privacy, as it allows secure, unlinkable payments which do not reveal the identity of the payer during a payment. From a security and privacy perspective these protocols present a good solution. However, even though e-cash protocols have been proposed three decades ago, there are relatively few actual implementations. One reason for this is their high computational complexity which makes an implementation on potential mobile payment devices rather difficult. While customers usually value their privacy they often do not accept to sacrifice convenience. A fast execution of payments is thus a hard constraint, which conflicts with the computational complexity of e-cash schemes. This dissertation analyzes how e-cash can be used to solve the issue of privacy in the domain of transportation payments while satisfying the unique requirements of transportation payment systems and achieving high security and ease of use. Highlyefficient implementations of the underlying cryptographic primitives of e-cash schemes on constrained devices as they might be used in the transportation setting are presented. Based on the efficient implementations of these primitives, e-cash schemes are analyzed with regards to speed and hardware requirements. The results show that e-cash presents a good solution for privacy-preserving payments in the domain of public transport, if the number of coins that have to be spent can be limited. It is further practically shown that this limitation can be alleviated relying on the e-cash based privacy-preserving pre-payments with refunds scheme (P4R). Moreover, it is demonstrated that the promising feature of supporting the encoding of user attributes into electronic coins can be implemented at only moderate extra cost. Finally, an ecash based e-mobility payment scheme is presented which highlights the flexibility and unique advantages of e-cash based transportation payment schemes

Algebraic Curves and Cryptographic Protocols for the e-society

Amb l'augment permanent de l'adopció de sistemes intel·ligents de tot tipus en la societat actual apareixen nous reptes. Avui en dia quasi tothom en la societat moderna porta a sobre almenys un telèfon intel·ligent, si no és que porta encara més dispositius capaços d'obtenir dades personals, com podria ser un smartwatch per exemple. De manera similar, pràcticament totes les cases tindran un comptador intel·ligent en el futur pròxim per a fer un seguiment del consum d'energia. També s'espera que molts més dispositius del Internet de les Coses siguin instal·lats de manera ubiqua, recol·lectant informació dels seus voltants i/o realitzant accions, com per exemple en sistemes d'automatització de la llar, estacions meteorològiques o dispositius per la ciutat intel·ligent en general. Tots aquests dispositius i sistemes necessiten enviar dades de manera segura i confidencial, les quals poden contindre informació sensible o de caire privat. A més a més, donat el seu ràpid creixement, amb més de nou mil milions de dispositius en tot el món actualment, s'ha de tenir en compte la quantitat de dades que cal transmetre. En aquesta tesi mostrem la utilitat de les corbes algebraiques sobre cossos finits en criptosistemes de clau pública, en particular la de les corbes de gènere 2, ja que ofereixen la mida de clau més petita per a un nivell de seguretat donat i això redueix de manera significativa el cost total de comunicacions d'un sistema, a la vegada que manté un rendiment raonable. Analitzem com la valoració 2-àdica del cardinal de la Jacobiana augmenta en successives extensions quadràtiques, considerant corbes de gènere 2 en cossos de característica senar, incloent les supersingulars. A més, millorem els algoritmes actuals per a computar la meitat d'un divisor d'una corba de gènere 2 sobre un cos binari, cosa que pot ser útil en la multiplicació escalar, que és l'operació principal en criptografia de clau pública amb corbes. Pel que fa a la privacitat, presentem un sistema de pagament d'aparcament per mòbil que permet als conductors pagar per aparcar mantenint la seva privacitat, i per tant impedint que el proveïdor del servei o un atacant obtinguin un perfil de conducta d'aparcament. Finalment, oferim protocols de smart metering millorats, especialment pel que fa a la privacitat i evitant l'ús de terceres parts de confiança.Con el aumento permanente de la adopción de sistemas inteligentes de todo tipo en la sociedad actual aparecen nuevos retos. Hoy en día prácticamente todos en la sociedad moderna llevamos encima al menos un teléfono inteligente, si no es que llevamos más dispositivos capaces de obtener datos personales, como podría ser un smartwatch por ejemplo. De manera similar, en el futuro cercano la mayoría de las casas tendrán un contador inteligente para hacer un seguimiento del consumo de energía. También se espera que muchos más dispositivos del Internet de las Cosas sean instalados de manera ubicua, recolectando información de sus alrededores y/o realizando acciones, como por ejemplo en sistemas de automatización del hogar, estaciones meteorológicas o dispositivos para la ciudad inteligente en general. Todos estos dispositivos y sistemas necesitan enviar datos de manera segura y confidencial, los cuales pueden contener información sensible o de ámbito personal. Además, dado su rápido crecimiento, con más de nueve mil millones de dispositivos en todo el mundo actualmente, hay que tener en cuenta la cantidad de datos a transmitir. En esta tesis mostreamos la utilidad de las curvas algebraicas sobre cuerpos finitos en criptosistemas de clave pública, en particular la de las curvas de género 2, ya que ofrecen el tamaño de clave más pequeño para un nivel de seguridad dado y esto disminuye de manera significativa el coste total de comunicaciones del sistema, a la vez que mantiene un rendimiento razonable. Analizamos como la valoración 2-ádica del cardinal de la Jacobiana aumenta en sucesivas extensiones cuadráticas, considerando curvas de género 2 en cuerpos de característica importa, incluyendo las supersingulares. Además, mejoramos los algoritmos actuales para computar la mitad de un divisor de una curva de género 2 sobre un cuerpo binario, lo cual puede ser útil en la multiplicación escalar, que es la operación principal en criptografía de clave pública con curvas. Respecto a la privacidad, presentamos un sistema de pago de aparcamiento por móvil que permite a los conductores pagar para aparcar manteniendo su privacidad, y por lo tanto impidiendo que el proveedor del servicio o un atacante obtengan un perfil de conducta de aparcamiento. Finalmente, ofrecemos protocolos de smart metering mejorados, especialmente en lo relativo a la privacidad y evitando el uso de terceras partes de confianza.With the ever increasing adoption of smart systems of every kind throughout society, new challenges arise. Nowadays, almost everyone in modern societies carries a smartphone at least, if not even more devices than can also gather personal data, like a smartwatch or a fitness wristband for example. Similarly, practically all homes will have a smart meter in the near future for billing and energy consumption monitoring, and many other Internet of Things devices are expected to be installed ubiquitously, obtaining information of their surroundings and/or performing some action, like for example, home automation systems, weather detection stations or devices for the smart city in general. All these devices and systems need to securely and privately transmit some data, which can be sensitive and personal information. Moreover, with a rapid increase of their number, with already more than nine billion devices worldwide, the amount of data to be transmitted has to be considered. In this thesis we show the utility of algebraic curves over finite fields in public key cryptosystems, specially genus 2 curves, since they offer the minimum key size for a given security level and that significantly reduces the total communication costs of a system, while maintaining a reasonable performance. We analyze how the 2-adic valuation of the cardinality of the Jacobian increases in successive quadratic extensions, considering genus 2 curves with odd characteristic fields, including supersingular curves. In addition, we improve the current algorithms for computing the halving of a divisor of a genus 2 curve over binary fields, which can be useful in scalar multiplication, the main operation in public key cryptography using curves. As regards to privacy, we present a pay-by-phone parking system which enables drivers to pay for public parking while preserving their privacy, and thus impeding the service provider or an attacker to obtain a profile of parking behaviors. Finally, we offer better protocols for smart metering, especially regarding privacy and the avoidance of trusted third parties

Privacy-Preserving Shortest Path Computation

Navigation is one of the most popular cloud computing services. But in virtually all cloud-based navigation systems, the client must reveal her location and destination to the cloud service provider in order to learn the fastest route. In this work, we present a cryptographic protocol for navigation on city streets that provides privacy for both the client's location and the service provider's routing data. Our key ingredient is a novel method for compressing the next-hop routing matrices in networks such as city street maps. Applying our compression method to the map of Los Angeles, for example, we achieve over tenfold reduction in the representation size. In conjunction with other cryptographic techniques, this compressed representation results in an efficient protocol suitable for fully-private real-time navigation on city streets. We demonstrate the practicality of our protocol by benchmarking it on real street map data for major cities such as San Francisco and Washington, D.C.Comment: Extended version of NDSS 2016 pape