A Summary of Recent and Old Results on the Security of the Diffie-Hellman Key Exchange Protocol in Finite Groups

A Summary of Recent and Old Results on the Security of the Diffie-Hellman Key Exchange Protocol in Finite Groups

Ionut Florescu (Stevens Institute of Technology, USA)
Copyright: © 2009 |Pages: 20
DOI: 10.4018/978-1-60566-262-6.ch010
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Abstract

Regarding fundamental protocols in cryptography, the Diffie-Hellman (Diffie and Hellman, 1976) public key exchange protocol is one of the oldest and most widely used in today’s applications. Consequently, many specific cryptographic implementations depend on its security. Typically, an underlying (finite dimensional) group is selected to provide candidates for the key. The study of the security of the exchange as depending on the structure of the underlying group is even today poorly understood, with the most common approaches relying on the security of the Discrete Logarithm problem or on the size of the group. Recent developments bring to attention that the relationship is not necessarily valid and that more research is needed that will relate the underlying structure of the group and the security of the Diffie- Hellman exchange. In this chapter, we describe the problem in detail, we present the relationship with the previously studied Discrete Logarithm and Computational Diffie-Hellman problems, we expose the various concepts of security, and we introduce a new statistical concept specifically designed to serve the assessment of the security of the exchange.
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Introduction

A key exchange protocol, is any algorithm through which two parties A and B agree on a common key . Once the key is established, any further information shared between the parties is encoded, transmitted and decoded using the key. The protocol is secure if any third party C finds it extremely hard (impossible in practice) to identify the key.

In a public key exchange protocol the two parties agree on a common key pooled from a set S while communicating over an insecure channel. The difference is that all the information exchanged over the insecure channel as well as the set of possible keys S is known by the perpetrator C. If C cannot tell apart from any other value in the set S, given the information observed guarantees that it is computationally unfeasible to gain “any” partial information on the key.

The Diffie-Hellman key exchange protocol (Diffie and Hellman, 1976) is a primary example of a public key exchange protocol. In its most basic form, the protocol chooses a finite cyclic group of order N, with generator g, where denotes the group operation. In what follows we chose the multiplicative operation to denote the operation in the group, and thus the group G is generated by the powers of g (i.e., ), symbolically . Note that G, g and N are public information.

The participants in the information transfer, call them A and B, each randomly choose an integer and independently. Then A computes , B computes and exchange these elements of G over the insecure channel. Since each of A and B knows their respective values chosen (a and b) they can both compute, which, or a publicly known derivation of that, becomes the public key.

Any method of converting to is publicly known, and the security of the key is directly dependent on the security of thus, most articles consider as the established key of the exchange.

Key Terms in this Chapter

Statistically Indistinguishable Random Variables: Are two or more random variable whose distribution is identical almost everywhere (with the possible exception of a set of probability measure zero).

Prime Group: A group that contains no subgroups except for the trivial subgroup. A prime subgroup is a subgroup of a group that contains no further subgroups except for the trivial subgroup. An example is included in.

Cryptographic Key: A piece of information that controls the operation of a cryptographic algorithm.

Generator of a Cyclic Group: An element g such that all the elements of the group are generated by successive applications of the group operation to g itself. Not all the elements in a group are generators.

p-Value of a Test: the probability of obtaining as extreme or more extreme values as the result of the experiment assuming that the null hypothesis is true. Numbers close to 0 are evidence against the null hypothesis (it is unlikely to see such numbers if the null hypothesis would be true).

Encryption Key: A piece of information used to specify the particular transformation of plaintext into ciphertext, or vice versa during the encryption/decryption process.

Subgroup of a Group: A set of elements from the initial group which together form a smaller goup structure included in the original group (i.e, the operation stays in the subgroup, the identity and the inverse elements are in the subgroup) . An example is the trivial subgroup .

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Table of Contents
Foreword
Borko Furht
Preface
Shiguo Lian, Yan Zhang
Acknowledgment
Shiguo Lian, Yan Zhang
Chapter 1
Pramod A. Jamkhedkar, Gregory L. Heileman
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Rights Expression Languages
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Chapter 2
Deepali Brahmbhatt, Mark Stamp
This chapter presents a digital rights management (DRM) system designed for streaming media. A brief, general introduction to DRM is also provided... Sample PDF
Digital Rights Management for Streaming Media
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Chapter 3
Jean-Henry Morin
This chapter introduces and discusses much needed alternatives to the traditional either/or debate on total security of secure multimedia... Sample PDF
Rethinking DRM Using Exception Management
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Chapter 4
Mercè Serra Joan, Bert Greevenbosch, Anja Becker, Harald Fuchs
This chapter gives an overview of the Open Mobile AllianceTM Digital Rights Management (OMA DRM) standard, which allows for the secure distribution... Sample PDF
Overview of OMA Digital Rights Management
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Chapter 5
Hugo Jonker, Sjouke Mauw
The use of Digital Rights Management (DRM) systems involves several stakeholders, such as the content provider, the license provider, and the user... Sample PDF
Discovering the Core Security Requirements of DRM Systems by Means of Objective Trees
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Chapter 6
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Peer-to-peer (P2P) networks have proliferated and become ubiquitous. A school of thought has emerged that harnessing the established user-base and... Sample PDF
Digital Rights Management for Untrusted Peer-to-Peer Networks
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Chapter 7
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Pervasive Video Surveillance Systems Over TCP/IP Networks
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Chapter 8
Ramya Venkataramu, Mark Stamp
Digital Rights Management (DRM) technology is used to control access to copyrighted digital content. Apple employs a DRM system known as Fairplay in... Sample PDF
P2PTunes: A Peer-to-Peer Digital Rights Management System
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Chapter 9
Nicolas Anciaux, Luc Bouganim, Philippe Pucheral
This chapter advocates the convergence between Access Control (AC) models, focusing on the granularity of sharing, and Digital Right Management... Sample PDF
A Hardware Approach for Trusted Access and Usage Control
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Chapter 10
Ionut Florescu
Regarding fundamental protocols in cryptography, the Diffie-Hellman (Diffie and Hellman, 1976) public key exchange protocol is one of the oldest and... Sample PDF
A Summary of Recent and Old Results on the Security of the Diffie-Hellman Key Exchange Protocol in Finite Groups
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Chapter 11
Guojun Wang, Yirong Wu, Geyong Min, Ronghua Shi
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Secret Sharing with k-Dimensional Access Structure
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Chapter 12
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Wireless Video Transmission
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Chapter 13
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A Survey of Information Hiding
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Chapter 14
Fan Zhang
The digital multimedia, including text, image, graphics, audio, video, and so forth, has become a main way for information communication along with... Sample PDF
Digital Watermarking Capacity and Detection Error Rate
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Chapter 15
Digital Watermarking  (pages 277-297)
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Digital Watermarking
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Chapter 16
Pradeep K. Atrey, Abdulmotaleb El Saddik, Mohan Kankanhalli
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Digital Video Authentication
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Chapter 17
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Flexible Multimedia Stream Authentication
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Chapter 18
K-G Stenborg
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Scalable Distribution of Watermarked Media
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Chapter 19
Hafiz Malik
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Critical Analysis of Digital Steganography
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Chapter 20
Esther Palomar, Juan M.E. Tapiador, Julio C. Hernandez-Castro, Arturo Ribagorda
Perhaps the most popular feature offered by Peer-to-Peer (P2P) networks is the possibility of having several replicas of the same content... Sample PDF
Secure Content Distribution in Pure P2P
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Chapter 21
Andreas U. Schmidt, Nicolai Kuntze
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Trust in the Value-Creation Chain of Multimedia Goods
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Chapter 22
Goo-Rak Kwon, Sung-Jea Ko
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Chapter 23
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Steganography is the art of hiding secret data inside other innocent media file. Steganalysis is the process of detecting hidden data which are... Sample PDF
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Chapter 24
Guangjie Liu, Shiguo Lian, Yuewei Dai, Zhiquan Wang
Image steganography is a common form of information hiding which embeds as many message bits into images and keep the introduced distortion... Sample PDF
Adaptive Image Steganography Based on Structural Similarity Metric
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Chapter 25
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Video watermarking technique embeds some information into videos by modifying video content slightly. The embedded information, named watermark, may... Sample PDF
A Survey on Video Watermarking
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Chapter 26
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Digital watermarking is a useful and powerful tool for multimedia security such as copyright protection, tamper proofing and assessment, broadcast... Sample PDF
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Chapter 27
Hsuan T. Chang, Chih-Chung Hsu
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Fractal-Based Secured Multiple-Image Compression and Distribution
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