125 research outputs found
Easy decision-Diffie-Hellman groups
The decision-Diffie-Hellman problem (DDH) is a central computational problem
in cryptography. It is known that the Weil and Tate pairings can be used to
solve many DDH problems on elliptic curves. Distortion maps are an important
tool for solving DDH problems using pairings and it is known that distortion
maps exist for all supersingular elliptic curves. We present an algorithm to
construct suitable distortion maps. The algorithm is efficient on the curves
usable in practice, and hence all DDH problems on these curves are easy. We
also discuss the issue of which DDH problems on ordinary curves are easy
Still Wrong Use of Pairings in Cryptography
Several pairing-based cryptographic protocols are recently proposed with a
wide variety of new novel applications including the ones in emerging
technologies like cloud computing, internet of things (IoT), e-health systems
and wearable technologies. There have been however a wide range of incorrect
use of these primitives. The paper of Galbraith, Paterson, and Smart (2006)
pointed out most of the issues related to the incorrect use of pairing-based
cryptography. However, we noticed that some recently proposed applications
still do not use these primitives correctly. This leads to unrealizable,
insecure or too inefficient designs of pairing-based protocols. We observed
that one reason is not being aware of the recent advancements on solving the
discrete logarithm problems in some groups. The main purpose of this article is
to give an understandable, informative, and the most up-to-date criteria for
the correct use of pairing-based cryptography. We thereby deliberately avoid
most of the technical details and rather give special emphasis on the
importance of the correct use of bilinear maps by realizing secure
cryptographic protocols. We list a collection of some recent papers having
wrong security assumptions or realizability/efficiency issues. Finally, we give
a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page
Pairings in Cryptology: efficiency, security and applications
Abstract
The study of pairings can be considered in so many di�erent ways that it
may not be useless to state in a few words the plan which has been adopted,
and the chief objects at which it has aimed. This is not an attempt to write
the whole history of the pairings in cryptology, or to detail every discovery,
but rather a general presentation motivated by the two main requirements
in cryptology; e�ciency and security.
Starting from the basic underlying mathematics, pairing maps are con-
structed and a major security issue related to the question of the minimal
embedding �eld [12]1 is resolved. This is followed by an exposition on how
to compute e�ciently the �nal exponentiation occurring in the calculation
of a pairing [124]2 and a thorough survey on the security of the discrete log-
arithm problem from both theoretical and implementational perspectives.
These two crucial cryptologic requirements being ful�lled an identity based
encryption scheme taking advantage of pairings [24]3 is introduced. Then,
perceiving the need to hash identities to points on a pairing-friendly elliptic
curve in the more general context of identity based cryptography, a new
technique to efficiently solve this practical issue is exhibited.
Unveiling pairings in cryptology involves a good understanding of both
mathematical and cryptologic principles. Therefore, although �rst pre-
sented from an abstract mathematical viewpoint, pairings are then studied
from a more practical perspective, slowly drifting away toward cryptologic
applications
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
Identity based cryptography from bilinear pairings
This report contains an overview of two related areas of research in cryptography
which have been prolific in significant advances in recent years. The first of
these areas is pairing based cryptography. Bilinear pairings over elliptic curves
were initially used as formal mathematical tools and later as cryptanalysis tools
that rendered supersingular curves insecure. In recent years, bilinear pairings
have been used to construct many cryptographic schemes. The second area
covered by this report is identity based cryptography. Digital certificates are
a fundamental part of public key cryptography, as one needs a secure way of
associating an agent’s identity with a random (meaningless) public key. In
identity based cryptography, public keys can be arbitrary bit strings, including
readable representations of one’s identity.Fundação para a Ci~Encia e Tecnologia - SFRH/BPD/20528/2004
Converting Pairing-Based Cryptosystems from Composite-Order Groups to Prime-Order Groups
We develop an abstract framework that encompasses the key properties of bilinear groups of composite order that are required to construct secure pairing-based cryptosystems, and we show how to use prime-order elliptic curve groups to construct bilinear groups with the same properties. In particular, we define a generalized version of the subgroup decision problem and give explicit constructions of bilinear groups in which the generalized subgroup decision assumption follows from the decision Diffie-Hellman assumption, the decision linear assumption, and/or related assumptions in prime-order groups.
We apply our framework and our prime-order group constructions to create more efficient versions of cryptosystems that originally required composite-order groups. Specifically, we consider the Boneh-Goh-Nissim encryption scheme, the Boneh-Sahai-Waters traitor tracing system, and the Katz-Sahai-Waters attribute-based encryption scheme. We give a security theorem for the prime-order group instantiation of each system, using assumptions of comparable complexity to those used in the composite-order setting. Our conversion of the last two systems to prime-order groups answers a problem posed by Groth and Sahai
Pairing computation on hyperelliptic curves of genus 2
Bilinear pairings have been recently used to construct cryptographic schemes with new and novel properties, the most celebrated example being the Identity Based Encryption scheme of Boneh and Franklin. As pairing computation is generally the most computationally intensive part of any painng-based cryptosystem, it is essential to investigate new ways in which to compute pairings efficiently.
The vast majority of the literature on pairing computation focuscs solely on using elliptic curves. In this thesis we investigate pairing computation on supersingular hyperelliptic curves of genus 2 Our aim is to provide a practical alternative to using elliptic curves for pairing based cryptography. Specifically, we illustrate how to implement pairings efficiently using genus 2 curves, and how to attain performance comparable to using elliptic curves.
We show that pairing computation on genus 2 curves over F2m can outperform elliptic curves by using a new variant of the Tate pairing, called the r¡j pairing, to compute the fastest pairing implementation in the literature to date We also show for the first time how the final exponentiation required to compute the Tate pairing can be avoided for certain hyperelliptic curves.
We investigate pairing computation using genus 2 curves over large prime fields, and detail various techniques that lead to an efficient implementation, thus showing that these curves are a viable candidate for practical use
Security evaluation of a key management scheme based on bilinear maps on elliptic curves
In recent years, many applications of elliptic curves to cryptography have
been developed. Cryptosystems based on groups of rational points on elliptic curves allow more efficient alternatives to finite field cryptography, which usually requires groups with larger cardinality and lower efficiency. The existence of non-degenerate, bilinear maps on elliptic curves, called pairings, allow the construction of many efficient cryptosystems; however, their security must be carefully studied. We will study the security of a key menagement scheme introduced by Boneh, Gentry and Waters in 2005, which is based on the decisional version of the l-BDHE problem. This is a variant of the classical Diffie-Hellman problem, specifically constructed for pairing-based cryptography. Its hardness, is still a research topic and only some theoretical evidence exists. The aim of this work is to investigate the security of this broadcast encryption system, taking in account a model that proves the hardness of the l-BDHE problem, under strong assumptions. Drawbacks of this approach will be discussed: its main weakness is the system's behaviour during attack simulations, which is far from real. The main result of this thesis is a lower bound on the running time of an adversary solving the above problem.
Moreover, also the elliptic curve choice, when implementing an encryption
scheme, could affect its security. We will review the main criteria for this choice and we will investigate the existence of elliptic curves suitable for the system of our interest
Cryptographic Key Distribution In Wireless Sensor Networks Using Bilinear Pairings
It is envisaged that the use of cheap and tiny wireless sensors will soon bring a third wave of evolution in computing systems. Billions of wireless senor nodes will provide a bridge between information systems and the physical world. Wireless nodes deployed around the globe will monitor the surrounding environment as well as gather information about the people therein. It is clear that this revolution will put security solutions to a great test.
Wireless Sensor Networks (WSNs) are a challenging environment for applying security services. They differ in many aspects from traditional fixed networks, and standard cryptographic solutions cannot be used in this application space. Despite many research efforts, key distribution in WSNs still remains an open problem. Many of the proposed schemes suffer from high communication overhead and storage costs, low scalability and poor resilience against different types of attacks. The exclusive usage of simple and energy efficient symmetric cryptography primitives does not solve the security problem. On the other hand a full public key infrastructure which uses asymmetric techniques, digital signatures and certificate authorities seems to be far too complex for a constrained WSN environment. This thesis investigates a new approach to WSN security which addresses
many of the shortcomings of existing mechanisms. It presents a detailed description on how to provide practical Public Key Cryptography solutions for wireless sensor networks. The contributions to the state-of-the-art are added on all levels of development beginning with the basic arithmetic operations and finishing with complete security protocols. This work includes a survey of different key distribution protocols that have been developed for WSNs, with an evaluation of their limitations. It also proposes Identity- Based Cryptography (IBC) as an ideal technique for key distribution in sensor networks. It presents the first in-depth study of the application and implementation of Pairing- Based Cryptography (PBC) to WSNs. This is followed by a presentation of the state of the art on the software implementation of Elliptic Curve Cryptography (ECC) on typical WSNplatforms. New optimized algorithms for performing multiprecision multiplication on a broad range of low-end CPUs are introduced as well. Three novel protocols for key distribution are proposed in this thesis. Two of these are intended for non-interactive key exchange in flat and clustered networks respectively. A third key distribution protocol uses Identity-Based Encryption (IBE) to secure communication within a heterogeneous sensor network. This thesis includes also a comprehensive security evaluation that shows that proposed schemes are resistant to various attacks that are specific to WSNs. This work shows that by using the newest achievements in cryptography like pairings and IBC it is possible to deliver affordable public-key cryptographic solutions and to apply a sufficient level of security for the most demanding WSN applications
- …