Efficient algorithms for pairing-based cryptosystems

We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics.We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairing-based cryptography

On Small Degree Extension Fields in Cryptology

This thesis studies the implications of using public key cryptographic primitives that are based in, or map to, the multiplicative group of finite fields with small extension degree. A central observation is that the multiplicative group of extension fields essentially decomposes as a product of algebraic tori, whose properties allow for improved communication efficiency. Part I of this thesis is concerned with the constructive implications of this idea. Firstly, algorithms are developed for the efficient implementation of torus-based cryptosystems and their performance compared with previous work. It is then shown how to apply these methods to operations required in low characteristic pairing-based cryptography. Finally, practical schemes for high-dimensional tori are discussed. Highly optimised implementations and benchmark timings are provided for each of these systems. Part II addresses the security of the schemes presented in Part I, i.e., the hardness of the discrete logarithm problem. Firstly, an heuristic analysis of the effectiveness of the Function Field Sieve in small characteristic is given. Next presented is an implementation of this algorithm for characteristic three fields used in pairing-based cryptography. Finally, a new index calculus algorithm for solving the discrete logarithm problem on algebraic tori is described and analysed

An integrated security Protocol communication scheme for Internet of Things using the Locator/ID Separation Protocol Network

Internet of Things communication is mainly based on a machine-to-machine pattern, where devices are globally addressed and identified. However, as the number of connected devices increase, the burdens on the network infrastructure increase as well. The major challenges are the size of the routing tables and the efficiency of the current routing protocols in the Internet backbone. To address these problems, an Internet Engineering Task Force (IETF) working group, along with the research group at Cisco, are still working on the Locator/ID Separation Protocol as a routing architecture that can provide new semantics for the IP addressing, to simplify routing operations and improve scalability in the future of the Internet such as the Internet of Things. Nonetheless, The Locator/ID Separation Protocol is still at an early stage of implementation and the security Protocol e.g. Internet Protocol Security (IPSec), in particular, is still in its infancy. Based on this, three scenarios were considered: Firstly, in the initial stage, each Locator/ID Separation Protocol-capable router needs to register with a Map-Server. This is known as the Registration Stage. Nevertheless, this stage is vulnerable to masquerading and content poisoning attacks. Secondly, the addresses resolving stage, in the Locator/ID Separation Protocol the Map Server (MS) accepts Map-Request from Ingress Tunnel Routers and Egress Tunnel Routers. These routers in trun look up the database and return the requested mapping to the endpoint user. However, this stage lacks data confidentiality and mutual authentication. Furthermore, the Locator/ID Separation Protocol limits the efficiency of the security protocol which works against redirecting the data or acting as fake routers. Thirdly, As a result of the vast increase in the different Internet of Things devices, the interconnected links between these devices increase vastly as well. Thus, the communication between the devices can be easily exposed to disclosures by attackers such as Man in the Middle Attacks (MitM) and Denial of Service Attack (DoS). This research provided a comprehensive study for Communication and Mobility in the Internet of Things as well as the taxonomy of different security protocols. It went on to investigate the security threats and vulnerabilities of Locator/ID Separation Protocol using X.805 framework standard. Then three Security protocols were provided to secure the exchanged transitions of communication in Locator/ID Separation Protocol. The first security protocol had been implemented to secure the Registration stage of Locator/ID separation using ID/Based cryptography method. The second security protocol was implemented to address the Resolving stage in the Locator/ID Separation Protocol between the Ingress Tunnel Router and Egress Tunnel Router using Challenge-Response authentication and Key Agreement technique. Where, the third security protocol had been proposed, analysed and evaluated for the Internet of Things communication devices. This protocol was based on the authentication and the group key agreement via using the El-Gamal concept. The developed protocols set an interface between each level of the phase to achieve security refinement architecture to Internet of Things based on Locator/ID Separation Protocol. These protocols were verified using Automated Validation Internet Security Protocol and Applications (AVISPA) which is a push button tool for the automated validation of security protocols and achieved results demonstrating that they do not have any security flaws. Finally, a performance analysis of security refinement protocol analysis and an evaluation were conducted using Contiki and Cooja simulation tool. The results of the performance analysis showed that the security refinement was highly scalable and the memory was quite efficient as it needed only 72 bytes of memory to store the keys in the Wireless Sensor Network (WSN) device

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

A note on the construction of pairing-friendly elliptic curves for composite order protocols

In pairing-based cryptography, the security of protocols using composite order groups relies on the difficulty of factoring a composite number $N$. Boneh~\etal~proposed the Cocks-Pinch method to construct ordinary pairing-friendly elliptic curves having a subgroup of composite order $N$. Displaying such a curve as a public parameter implies revealing a square root $s$ of the complex multiplication discriminant $-D$ modulo $N$. We exploit this information leak and the structure of the endomorphism ring of the curve to factor the RSA modulus, under certain conditions. Our conclusion is that the values of $s$ modulo each prime in the factorization of $N$ should be chosen as high entropy input parameters when running the Cocks-Pinch algorithm

On the Efficiency and Security of Cryptographic Pairings

Pairing-based cryptography has been employed to obtain several advantageous cryptographic protocols. In particular, there exist several identity-based variants of common cryptographic schemes. The computation of a single pairing is a comparatively expensive operation, since it often requires many operations in the underlying elliptic curve. In this thesis, we explore the efficient computation of pairings. Computation of the Tate pairing is done in two steps. First, a Miller function is computed, followed by the final exponentiation. We discuss the state-of-the-art optimizations for Miller function computation under various conditions. We are able to shave off a fixed number of operations in the final exponentiation. We consider methods to effectively parallelize the computation of pairings in a multi-core setting and discover that the Weil pairing may provide some advantage under certain conditions. This work is extended to the 192-bit security level and some unlikely candidate curves for such a setting are discovered. Electronic Toll Pricing (ETP) aims to improve road tolling by collecting toll fares electronically and without the need to slow down vehicles. In most ETP schemes, drivers are charged periodically based on the locations, times, distances or durations travelled. Many ETP schemes are currently deployed and although these systems are efficient, they require a great deal of knowledge regarding driving habits in order to operate correctly. We present an ETP scheme where pairing-based BLS signatures play an important role. Finally, we discuss the security of pairings in the presence of an efficient algorithm to invert the pairing. We generalize previous results to the setting of asymmetric pairings as well as give a simplified proof in the symmetric setting

Practical backward unlinkable revocation in FIDO, German e-ID, Idemix and U-Prove

FIDO, German e-ID, Idemix and U-Prove constitute privacy-enhanced public-key infrastructures allowing users to authenticate in an anonymous way. This however hampers timely revocation in a privacy friendly way. From a legal perspective, revocation typically should be effective within 24 hours after user reporting. It should also be backward unlinkable, i.e. user anonymity cannot be removed after revocation. We describe a new, generic revocation mechanism based on pairing based encryption and apply it to supplement the systems mentioned. This allows for both flexible and privacy friendly revocation. Protocol execution takes less than a quarter of a second on modern smartcards. An additional property is that usage after revocation is linkable, allowing users to identify fraudulent usage after revocation. Our technique is the first Verifier Local Revocation scheme with backwards unlinkable revocation for the systems mentioned. This also allows for a setup resembling the well-known Online Certificate Status Protocol (OCSP). Here the service provider sends a pseudonym to a revocation provider that returns its status. As the information required for this is not secret the status service can be distributed over many cloud services. In addition to the status service our technique also supports the publication of a central revocation list

Efficient algorithms for supersingular isogeny Diffie-Hellman

We propose a new suite of algorithms that significantly improve the performance of supersingular isogeny Diffie-Hellman (SIDH) key exchange. Subsequently, we present a full-fledged implementation of SIDH that is geared towards the 128-bit quantum and 192-bit classical security levels. Our library is the first constant-time SIDH implementation and is up to 2.9 times faster than the previous best (non-constant-time) SIDH software. The high speeds in this paper are driven by compact, inversion-free point and isogeny arithmetic and fast SIDH-tailored field arithmetic: on an Intel Haswell processor, generating ephemeral public keys takes 46 million cycles for Alice and 54 million cycles for Bob, while computing the shared secret takes 44 million and 52 million cycles, respectively. The size of public keys is only 564 bytes, which is significantly smaller than most of the popular post-quantum key exchange alternatives. Ultimately, the size and speed of our software illustrates the strong potential of SIDH as a post-quantum key exchange candidate and we hope that these results encourage a wider cryptanalytic effort