Efficient Doubling on Genus Two Curves over Binary Fields

In most algorithms involving elliptic and hyperelliptic curves, the costliest part consists in computing multiples of ideal classes. This paper investigates how to compute faster doubling over fields of characteristic two. We derive explicit doubling formulae making strong use of the defining equation of the curve. We analyze how many field operations are needed depending on the curve making clear how much generality one loses by the respective choices. Note, that none of the proposed types is known to be weak – one only could be suspicious because of the more special types. Our results allow to choose curves from a large enough variety which have extremely fast doubling needing only half the time of an addition. Combined with a sliding window method this leads to fast computation of scalar multiples. We also speed up the general case

A census of zeta functions of quartic K3 surfaces over F_2

We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.Comment: 11 pages; final version, minor changes; to appear in ANTS XI

Efficient Doubling on Genus Two Curves over Binary Fields

In most algorithms involving elliptic and hyperelliptic curves, the costliest part consists in computing multiples of ideal classes. This paper investigates how to compute faster doubling over fields of characteristic two. We derive explicit doubling formulae making strong use of the defining equation of the curve. We analyze how many field operations are needed depending on the curve making clear how much generality one loses by the respective choices. Note, that none of the proposed types is known to be weak – one only could be suspicious because of the more special types. Our results allow to choose curves from a large enough variety which have extremely fast doubling needing only half the time of an addition. Combined with a sliding window method this leads to fast computation of scalar multiples. We also speed up the general case

The Point Decomposition Problem over Hyperelliptic Curves: toward efficient computations of Discrete Logarithms in even characteristic

International audienceComputing discrete logarithms is generically a difficult problem. For divisor class groups of curves defined over extension fields, a variant of the Index-Calculus called Decomposition attack is used, and it can be faster than generic approaches. In this situation, collecting the relations is done by solving multiple instances of the Point m-Decomposition Problem (PDP$_m$). An instance of this problem can be modelled as a zero-dimensional polynomial system. Solving is done with Gröbner bases algorithms, where the number of solutions of the system is a good indicator for the time complexity of the solving process. For systems arising from a PDP$_m$ context, this number grows exponentially fast with the extension degree. To achieve an efficient harvesting, this number must be reduced as much as as possible. Extending the elliptic case, we introduce a notion of Summation Ideals to describe PDP m instances over higher genus curves, and compare to Nagao's general approach to PDP$_m$ solving. In even characteristic we obtain reductions of the number of solutions for both approaches, depending on the curve's equation. In the best cases, for a hyperelliptic curve of genus $g$, we can divide the number of solutions by $2^{(n−1)(g+1)}$. For instance, for a type II genus 2 curve defined over $\mathbb{F}_{2^{93}}$ whose divisor class group has cardinality a near-prime 184 bits integer, the number of solutions is reduced from 4096 to 64. This is enough to build the matrix of relations in around 7 days with 8000 cores using a dedicated implementation

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

Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation

We present a complete set of efficient explicit formulas for arithmetic in the degree 0 divisor class group of a genus two real hyperelliptic curve given in affine coordinates. In addition to formulas suitable for curves defined over an arbitrary finite field, we give simplified versions for both the odd and the even characteristic cases. Formulas for baby steps, inverse baby steps, divisor addition, doubling, and special cases such as adding a degenerate divisor are provided, with variations for divisors given in reduced and adapted basis. We describe the improvements and the correctness together with a comprehensive analysis of the number of field operations for each operation. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model

The modular approach to Diophantine equations over totally real fields

Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine equations over totally real number fields. We give an extensive overview of these obstacles as well as providing a survey of existing methods and results in this area

Quantum algorithms for algebraic problems

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic

Cryptographic Pairings: Efficiency and DLP security

This thesis studies two important aspects of the use of pairings in cryptography, efficient algorithms and security. Pairings are very useful tools in cryptography, originally used for the cryptanalysis of elliptic curve cryptography, they are now used in key exchange protocols, signature schemes and Identity-based cryptography. This thesis comprises of two parts: Security and Efficient Algorithms. In Part I: Security, the security of pairing-based protocols is considered, with a thorough examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the relationship between the two instances of the DLP will be presented along with a discussion about the appropriate selection of parameters to ensure particular security level. In Part II: Efficient Algorithms, some of the computational issues which arise when using pairings in cryptography are addressed. Pairings can be computationally expensive, so the Pairing-Based Cryptography (PBC) research community is constantly striving to find computational improvements for all aspects of protocols using pairings. The improvements given in this section contribute towards more efficient methods for the computation of pairings, and increase the efficiency of operations necessary in some pairing-based protocol