Weil Pairing Research Articles

Mumford-Tate groups of 1-motives and Weil pairing

We show how the geometry of a 1-motive M (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group Galmot(M). Fixing periods matrices ΠM and ΠM⁎ associated respectively to a 1-motive M and to its Cartier dual M⁎, we describe the action of the Mumford-Tate group of M on these matrices. In the semi-elliptic case, according to the geometry of M we classify polynomial relations between the periods of M and we compute exhaustively the matrices representing the Mumford-Tate group of M. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives.

On 12-congruences of elliptic curves

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over [Formula: see text] with [Formula: see text]-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of [Formula: see text]-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the [Formula: see text], [Formula: see text] or [Formula: see text]-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the [Formula: see text]-invariants of the pair of elliptic curves.

Implementation of Boneh - Lynn - Shacham short digital signature scheme using Weil bilinear pairing based on supersingular elliptic curves

One option for a digital signature solution for devices with low memory and low bandwidth transmission over channels uses a short digital signature scheme based on Weil bilinear pairing aimed at short processing times, fast computation, and convenient deployment on applications. The computational technique of non-degenerate bilinear pairings uses supersingular elliptic curves over a finite field Fpl (where p is a sufficiently large prime number) and has the advantage of being able to avoid Weil-descent, Menezes-Okamoto-Vanstone (MOV) attacks, and attacks by the Number Field Sieve algorithm. Compared to Elliptic Curve Digital Signature Algorithm (ECDSA) digital signature schemes, generating a digital signature for a Boneh-Lynn-Shacham (BLS) scheme using Weil bilinear pairing on a supersingular elliptic curve is simple. In this study, the authors replace non-degenerate bilinear pairing calculations on a supersingular elliptic curve with a Weil pairing with PεE(Fp ), QεE(Fp1) and a higher security multiplier α=12 in the BLS short digital signature scheme. The execution time of the BLS short digital signature program showed improvement compared to the commercial ECDSA digital signature scheme.

Fast change of level and applications to isogenies

Let $$(A, {{\mathscr {L}}}, \Theta _n)$$ be a dimension g abelian variety together with a level n theta structure over a field k of odd characteristic. We thus denote by $$(\theta ^{\Theta _{{\mathscr {L}}}}_{i})_{(\mathbb {Z}/ n \mathbb {Z})^g} \in \Gamma (A, {{\mathscr {L}}})$$ the associated standard basis. For a positive integer $$\ell $$ relatively prime to n and the characteristic of k, we study change of level algorithms which allow one to compute level $$\ell n$$ theta functions $$(\theta _i^{\Theta _{{{\mathscr {L}}^\ell }}}(x))_{i \in (\mathbb {Z}/ \ell n \mathbb {Z})^g}$$ from the knowledge of level n theta functions $$(\theta ^{\Theta _{{\mathscr {L}}}}_i(x))_{(\mathbb {Z}/n \mathbb {Z})^g}$$ or vice versa. The classical duplication formulas are an example of change of level algorithm to go from level n to level 2n. The main result of this paper states that there exists an algorithm to go from level n to level $$\ell n$$ in $$O(n^g \ell ^{2g})$$ operations in k. We derive an algorithm to compute an isogeny $$f : A \rightarrow B$$ from the knowledge of $$(A, {{\mathscr {L}}}, \Theta _n)$$ and $$K \subset A[\ell ]$$ isotropic for the Weil pairing which computes f(x) for $$x \in A(k)$$ in $$O((n\ell )^g)$$ operations in k. We remark that this isogeny computation algorithm is of quasi-linear complexity in the size of K.

On the decisional Diffie–Hellman problem for class group actions on oriented elliptic curves

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order {mathcal {O}} in an unknown ideal class [{mathfrak {a}}] in {{,textrm{cl},}}({mathcal {O}}) that connects two given {mathcal {O}}-oriented elliptic curves (E, iota ) and (E', iota ') = [{mathfrak {a}}](E, iota ). When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sotáková and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie–Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski’s recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time.

On the symplectic type of isomorphisms of the 𝑝-torsion of elliptic curves

Let p ≥ 3 p \geq 3 be a prime. Let E / Q E/\mathbb {Q} and E ′ / Q E’/\mathbb {Q} be elliptic curves with isomorphic p p -torsion modules E [ p ] E[p] and E ′ [ p ] E’[p] . Assume further that either (i) every G Q G_\mathbb {Q} -modules isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] admits a multiple λ ⋅ ϕ \lambda \cdot \phi with λ ∈ F p × \lambda \in \mathbb {F}_p^\times preserving the Weil pairing; or (ii) no G Q G_\mathbb {Q} -isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime ℓ ≠ p \ell \neq p . Firstly, we determine the primes ℓ \ell for which the local curves E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell , to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from p p . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form y 2 = x p − ℓ y^2 = x^p - \ell and y 2 = x p − 2 ℓ y^2 = x^p - 2\ell where ℓ \ell is a prime; we also give incremental results on the Fermat equation x 2 + y 3 = z p x^2 + y^3 = z^p . As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the p p -torsion of two non-isogenous elliptic curves defined over Q \mathbb {Q} .

AN IDENTITY-BASED ENCRYPTION SCHEME USING ISOGENY OF ELLIPTIC CURVES

Identity-Based Encryption is a public key cryptosystem that uses the receiver identifier information such as email address, IP address, name and etc, to compute a public and a private key in a cryptosystem and encrypt a message. A message receiver can obtain the secret key corresponding with his privacy information from Private Key Generator and he can decrypt the ciphertext. In this paper, we review Boneh-Franklin’s scheme and use bilinear map and Weil pairing’s properties to propose an identity-based cryptography scheme based on isogeny of elliptic curves.

Novel Pseudonym Generation for Secured Privacy in Vehicular Adhoc Networks

Intelligent Transport Communication (ITC) has gained popularity in the present day scenario, where communication in vehicle is possible and it can be between Vehicle to Vehicle or Vehicle to Infrastructure. Security is one of the important issue in Vehicular Adhoc Networks (VANETS) and some of the main features of security are Privacy, Confidentiality, Integrity, Authentication and Non-repudiation. In the present scenario privacy along with security is considered to be one of the important factor, since the malicious nodes can attack the vehicles at any point of time during communication. Thus , Secured Privacy in Vehicular Adhoc Networks by Generating Efficient Pseudonym based on Weil Pair Algorithm (SPGPWP), is developed to address secured privacy in VANET’s . In the proposed algorithm, communication between vehicles are provided by using pseudonyms generated by road side units with the help of main authority. Our SPGPWP also includes that the pseudonyms received by vehicles from road side units are modified within the vehicle by using Weil pairing algorithm and provides secure communication between vehicle to vehicle without revealing its original identity, thus making the communication anonymous between all the vehicles except the intended vehicles. Simulation is performed to test the performance parameters such as latency, encryption, decryption time, and total time for communication. These performance parameters are compared with RSA (Rivest-Shamir-Adleman).

Klein coverings of genus 2 curves

We investigate the geometry of etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work \cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $\mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.

Beta Weil pairing revisited

In this paper, we extend the $$\beta $$ -Weil pairing, initially introduced in the setting of ordinary elliptic curves with even embedding degree by Aranha et al. [2], to ordinary elliptic curves of any embedding degree. We also propose a new optimal pairing which is the product of some rational functions with the same Miller loop and having a simple final exponentiation. The new pairing is appropriated for using the multi-pairing technique for an efficient implementation. We focus our computation at high security level. Exploiting the fact that the $$\beta $$ -Weil pairing is suitable for parallel execution, we first show that calculating the extended $$\beta $$ -Weil pairing over pairing-friendly elliptic curves with embedding degree 27 is more efficient than calculating the optimal ate pairing. Finally we show that calculating our new pairing over Barreto–Lynn–Scott curves with embedding degree 12 (BLS12) and pairing-friendly elliptic curves with embedding degree 15 is more efficient than calculating the optimal ate pairing.

Hyperelliptic curves on (1, 4)-polarised abelian surfaces

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We give a short proof for the known fact the only possibilities of genera of such curves are 2, 3, 4 and 5; then we focus on the genus five case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general (1, 4)-polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. Our proof of the existence of hyperelliptic curves on general (1, 4)-polarised abelian surfaces is different from that in the recent paper [10]. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every etale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of 2-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.

Pairing-Based Cryptography on Elliptic Curves

We give a brief overview of a recent branch of Public Key Cryptography, the so called Pairing-based Cryptography or Identity-based Cryptography. We describe the Weil pairing and its applications to cryptosystems and cryptographic protocols based on pairings as well as the elliptic curves suitable for the implementation of this kind of cryptography, the so called pairing-friendly curves. Some recent results of the authors are included.

Isogeny graphs of ordinary abelian varieties

Fix a prime number ell . Graphs of isogenies of degree a power of ell are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called mathfrak l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the “volcanoes” occurring as graphs of isogenies of elliptic curves. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as (ell , ell )-isogenies: those whose kernels are maximal isotropic subgroups of the ell -torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.

AN EFFICIENT AUTHENTICATION PROTOCOL USING ZERO KNOWLEDGE PROPERTY AND PAIRING ON ELLIPTIC CURVES

The systematic introduction to zero knowledge proof protocol has important theoretical guidance and practical significance on attracting more scholars involved in research as well as expanding application fields. Zero-knowledge proofs were first conceived in 1985 by Shafi Golwasser, Silvio Micalli and Charles Rackoff in a draft of the knowledge complexity of interactive proof systems. The goal of the present paper is to introduce a new identity based scheme which is a combination of zero-knowledge interactive proof and weil pairing on elliptic curves. The concept of weil pairing was first introduced by Andre Weil in 1940. It plays an important role in the theoretical study of the arithmetic of elliptic curves and Abelian varieties. It has also recently become extremely useful in cryptologic constructions related to these objects.

Identity‐based schemes for a secured big data and cloud ICT framework in smart grid system

AbstractSmart grid is an intelligent cyber physical system (CPS). The CPS generates a massive amount of data for efficient grid operation. In this paper, a big data‐driven, cloud‐based information and communication technology (ICT) framework for smart grid CPS is proposed. The proposed ICT framework deploys hybrid cloud servers to enhance scalability and reliability of smart grid communication infrastructure. Because the data in the ICT framework contains much privacy of customers and important data for automated controlling, the security of data transmission must be ensured. In order to secure the communications over the Internet in the system, identity‐based schemes are proposed especially because of their advantage in key management. Specifically, an identity‐based signcryption (IBSC) scheme is proposed to provide confidentiality, non‐repudiation, and data integrity. For practical purposes, an identity‐based signature scheme is relaxed from the proposed IBSC to provide non‐repudiation only. Moreover, identity‐based schemes are also proposed to achieve signature delegation within the ICT framework. Security of the proposed IBSC scheme is rigorously analyzed in this work. Efficiency of the proposed IBSC scheme is demonstrated with an implementation using modified Weil pairing over an elliptic curve. Copyright © 2016 John Wiley & Sons, Ltd.

A note on Gekeler’s h-function

We give a brief introduction to Drinfeld modular forms, concentrating on the many equivalent constructions of the form h of weight q + 1 and type 1, to which we contribute some new characterizations involving Moore determinants, and an application to the Weil pairing on Drinfeld modules. We also define Drinfeld modular functions of non-zero type and provide a moduli interpretation of these.

Review on Forward Secure Identity based Ring Signature for Data Sharing in the Cloud

The key feature of cloud computing is one can access information any place, anywhere, at any time. So basically cloud computing is subscription based service where one can obtain network storage space and computer resources for data storage as well as data sharing. Due to high fame of cloud for data storage and sharing, large number of participants gets attracted to it but it leads to issue related to efficiency, Data integrity, privacy and authentication. To overcome these issues, concept of ring signature has been introduced for data sharing amongst large number of users. Ring signatures are used to provide user’s anonymity and signer’s privacy. Use of ID-based ring signature, removes the need of certificate verification which was done using public key infrastructure, hence reduce cost as well as introduction of forward security, further strengthen this system more. Use of weil pairing, keeps even shorter keys secure and it also requires less processing power. So the motivation of this paper is to propose a secure data reading and sharing scheme using above mentioned scheme.

Speeding up Pairing Computation Using Non-adjacent Form and ELM Method

The bilinear pairings such as Weil pairing and Tate pairing on elliptic curves have recently found many applications in cryptography. The first efficient algorithm for computing pairing was originally proposed by Miller and much subsequent research has been directed at many different aspects in order to improve efficiency. In 2003, Eisentrager, Lauter and Montgomery proposed a new point-double-addition method to speed up elliptic curve arithmetic computation and obtained a 7.8% performance improvement of the Miller algorithm of a general elliptic curve. In 2006, Blake et al. proposed a new concept based on the conjugate of a line to reduce the total number of lines in the Miller algorithm. In this paper we propose an enhancement of Eisentrager et al.'s algorithm for computing pairings. Our enhancement can further speed up the pairing computation by 5.9%.

Remote Login Authentication Scheme based on Bilinear Pairing and Fingerprint

The bilinear pairing, also known as Weil pairing or Tate pairing, is widely used in cryptography and its properties help to construct cryptographic schemes for different applications in which the security of the transmitted data is a major concern. In remote login authentication schemes, there are two major requirements: i) proving the identity of a user and the server for legitimacy without exposing their private keys and ii) freedom for a user to choose and change his password (private key) efficiently. Most of the existing methods based on the bilinear property have some security breaches due to the lack of features and the design issues. In this paper, we develop a new scheme using the bilinear property of an elliptic point and the biometric characteristics. Our method provides many features along with three major goals. a) Checking the correctness of the password before sending the authentication message, which prevents the wastage of communication cost; b) Efficient password change phase in which the user is asked to give a new password after checking the correctness of the current password without involving the server; c) User anonymity - enforcing the suitability of our scheme for applications in which a user does not want to disclose his identity. We use BAN logic to ensure the mutual authentication and session key agreement properties. The paper provides informal security analysis to illustrate that our scheme resists all the security attacks. Furthermore, we use the AVISPA tool for formal security verification of our scheme.

Further refinements of Miller’s algorithm on Edwards curves

Recently, Edwards curves have received a lot of attention in the cryptographic community due to their fast scalar multiplication algorithms. Then, many works on the application of these curves to pairing-based cryptography have been introduced. In this paper, we investigate refinements to Miller's algorithm that play a central role in paring computation. We first introduce a variant of Miller function that leads to a more efficient variant of Miller's algorithm on Edwards curves. Then, based on the new Miller function, we present a refinement to Miller's algorithm that significantly improves the performance in comparison with the original Miller's algorithm. Our analyses also show that the proposed refinement is approximately 25 % faster than Xu---Lin's refinements (CT-RSA, 2010). Last but not least, our approach is generic, hence the proposed algorithms allow to compute both Weil and Tate pairings on pairing-friendly Edwards curves of any embedding degree.