On the Analysis of Public-Key Cryptologic Algorithms

The RSA cryptosystem introduced in 1977 by Ron Rivest, Adi Shamir and Len Adleman is the most commonly deployed public-key cryptosystem. Elliptic curve cryptography (ECC) introduced in the mid 80's by Neal Koblitz and Victor Miller is becoming an increasingly popular alternative to RSA offering competitive performance due the use of smaller key sizes. Most recently hyperelliptic curve cryptography (HECC) has been demonstrated to have comparable and in some cases better performance than ECC. The security of RSA relies on the integer factorization problem whereas the security of (H)ECC is based on the (hyper)elliptic curve discrete logarithm problem ((H)ECDLP). In this thesis the practical performance of the best methods to solve these problems is analyzed and a method to generate secure ephemeral ECC parameters is presented. The best publicly known algorithm to solve the integer factorization problem is the number field sieve (NFS). Its most time consuming step is the relation collection step. We investigate the use of graphics processing units (GPUs) as accelerators for this step. In this context, methods to efficiently implement modular arithmetic and several factoring algorithms on GPUs are presented and their performance is analyzed in practice. In conclusion, it is shown that integrating state-of-the-art NFS software packages with our GPU software can lead to a speed-up of 50%. In the case of elliptic and hyperelliptic curves for cryptographic use, the best published method to solve the (H)ECDLP is the Pollard rho algorithm. This method can be made faster using classes of equivalence induced by curve automorphisms like the negation map. We present a practical analysis of their use to speed up Pollard rho for elliptic curves and genus 2 hyperelliptic curves defined over prime fields. As a case study, 4 curves at the 128-bit theoretical security level are analyzed in our software framework for Pollard rho to estimate their practical security level. In addition, we present a novel many-core architecture to solve the ECDLP using the Pollard rho algorithm with the negation map on FPGAs. This architecture is used to estimate the cost of solving the Certicom ECCp-131 challenge with a cluster of FPGAs. Our design achieves a speed-up factor of about 4 compared to the state-of-the-art. Finally, we present an efficient method to generate unique, secure and unpredictable ephemeral ECC parameters to be shared by a pair of authenticated users for a single communication. It provides an alternative to the customary use of fixed ECC parameters obtained from publicly available standards designed by untrusted third parties. The effectiveness of our method is demonstrated with a portable implementation for regular PCs and Android smartphones. On a Samsung Galaxy S4 smartphone our implementation generates unique 128-bit secure ECC parameters in 50 milliseconds on average

Dynamic block encryption with self-authenticating key exchange

One of the greatest challenges facing cryptographers is the mechanism used for key exchange. When secret data is transmitted, the chances are that there may be an attacker who will try to intercept and decrypt the message. Having done so, he/she might just gain advantage over the information obtained, or attempt to tamper with the message, and thus, misguiding the recipient. Both cases are equally fatal and may cause great harm as a consequence. In cryptography, there are two commonly used methods of exchanging secret keys between parties. In the first method, symmetric cryptography, the key is sent in advance, over some secure channel, which only the intended recipient can read. The second method of key sharing is by using a public key exchange method, where each party has a private and public key, a public key is shared and a private key is kept locally. In both cases, keys are exchanged between two parties. In this thesis, we propose a method whereby the risk of exchanging keys is minimised. The key is embedded in the encrypted text using a process that we call `chirp coding', and recovered by the recipient using a process that is based on correlation. The `chirp coding parameters' are exchanged between users by employing a USB flash memory retained by each user. If the keys are compromised they are still not usable because an attacker can only have access to part of the key. Alternatively, the software can be configured to operate in a one time parameter mode, in this mode, the parameters are agreed upon in advance. There is no parameter exchange during file transmission, except, of course, the key embedded in ciphertext. The thesis also introduces a method of encryption which utilises dynamic blocks, where the block size is different for each block. Prime numbers are used to drive two random number generators: a Linear Congruential Generator (LCG) which takes in the seed and initialises the system and a Blum-Blum Shum (BBS) generator which is used to generate random streams to encrypt messages, images or video clips for example. In each case, the key created is text dependent and therefore will change as each message is sent. The scheme presented in this research is composed of five basic modules. The first module is the key generation module, where the key to be generated is message dependent. The second module, encryption module, performs data encryption. The third module, key exchange module, embeds the key into the encrypted text. Once this is done, the message is transmitted and the recipient uses the key extraction module to retrieve the key and finally the decryption module is executed to decrypt the message and authenticate it. In addition, the message may be compressed before encryption and decompressed by the recipient after decryption using standard compression tools

Synchronization of multi-carrier CDMA signals and security on internet.

by Yooh Ji Heng.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 119-128).Appendix in Chinese.Chapter I --- Synchronization of Multi-carrier CDMA Signals --- p.1Chapter 1 --- Introduction --- p.2Chapter 1.1 --- Spread Spectrum CDMA --- p.4Chapter 1.1.1 --- Direct Sequence/SS-CDMA --- p.5Chapter 1.1.2 --- Frequency Hopping/SS-CDMA --- p.5Chapter 1.1.3 --- Pseudo-noise Sequence --- p.6Chapter 1.2 --- Synchronization for CDMA signal --- p.7Chapter 1.2.1 --- Acquisition of PN Sequence --- p.7Chapter 1.2.2 --- Phase Locked Loop --- p.8Chapter 2 --- Multi-carrier CDMA --- p.10Chapter 2.1 --- System Model --- p.11Chapter 2.2 --- Crest Factor --- p.12Chapter 2.3 --- Shapiro-Rudin Sequence --- p.14Chapter 3 --- Synchronization and Detection by Line-Fitting --- p.16Chapter 3.1 --- Unmodulated Signals --- p.16Chapter 3.2 --- Estimating the Time Shift by Line-Fitting --- p.19Chapter 3.3 --- Modulated Signals --- p.22Chapter 4 --- Matched Filter --- p.23Chapter 5 --- Performance and Conclusion --- p.27Chapter 5.1 --- Line Fitting Algorithm --- p.27Chapter 5.2 --- Matched Filter --- p.28Chapter 5.3 --- Conclusion --- p.30Chapter II --- Security on Internet --- p.31Chapter 6 --- Introduction --- p.32Chapter 6.1 --- Introduction to Cryptography --- p.32Chapter 6.1.1 --- Classical Cryptography --- p.33Chapter 6.1.2 --- Cryptanalysis --- p.35Chapter 6.2 --- Introduction to Internet Security --- p.35Chapter 6.2.1 --- The Origin of Internet --- p.35Chapter 6.2.2 --- Internet Security --- p.36Chapter 6.2.3 --- Internet Commerce --- p.37Chapter 7 --- Elementary Number Theory --- p.39Chapter 7.1 --- Finite Field Theory --- p.39Chapter 7.1.1 --- Euclidean Algorithm --- p.40Chapter 7.1.2 --- Chinese Remainder Theorem --- p.40Chapter 7.1.3 --- Modular Exponentiation --- p.41Chapter 7.2 --- One-way Hashing Function --- p.42Chapter 7.2.1 --- MD2 --- p.43Chapter 7.2.2 --- MD5 --- p.43Chapter 7.3 --- Prime Number --- p.44Chapter 7.3.1 --- Listing of Prime Number --- p.45Chapter 7.3.2 --- Primality Testing --- p.45Chapter 7.4 --- Random/Pseudo-Random Number --- p.47Chapter 7.4.1 --- Examples of Random Number Generator --- p.49Chapter 8 --- Private Key and Public Key Cryptography --- p.51Chapter 8.1 --- Block Ciphers --- p.51Chapter 8.1.1 --- Data Encryption Standard (DES) --- p.52Chapter 8.1.2 --- International Data Encryption Algorithm (IDEA) --- p.54Chapter 8.1.3 --- RC5 --- p.55Chapter 8.2 --- Stream Ciphers --- p.56Chapter 8.2.1 --- RC2 and RC4 --- p.57Chapter 8.3 --- Public Key Cryptosystem --- p.58Chapter 8.3.1 --- Diffie-Hellman --- p.60Chapter 8.3.2 --- Knapsack Algorithm --- p.60Chapter 8.3.3 --- RSA --- p.62Chapter 8.3.4 --- Elliptic Curve Cryptosystem --- p.63Chapter 8.3.5 --- Public Key vs. Private Key Cryptosystem --- p.64Chapter 8.4 --- Digital Signature --- p.65Chapter 8.4.1 --- ElGamal Signature Scheme --- p.66Chapter 8.4.2 --- Digital Signature Standard (DSS) --- p.67Chapter 8.5 --- Cryptanalysis to Current Cryptosystems --- p.68Chapter 8.5.1 --- Differential Cryptanalysis --- p.68Chapter 8.5.2 --- An Attack to RC4 in Netscapel.l --- p.69Chapter 8.5.3 --- "An Timing Attack to Diffie-Hellman, RSA" --- p.71Chapter 9 --- Network Security and Electronic Commerce --- p.73Chapter 9.1 --- Network Security --- p.73Chapter 9.1.1 --- Password --- p.73Chapter 9.1.2 --- Network Firewalls --- p.76Chapter 9.2 --- Implementation for Network Security --- p.79Chapter 9.2.1 --- Kerberos --- p.79Chapter 9.2.2 --- Privacy-Enhanced Mail (PEM) --- p.80Chapter 9.2.3 --- Pretty Good Privacy (PGP) --- p.82Chapter 9.3 --- Internet Commerce --- p.83Chapter 9.3.1 --- Electronic Cash --- p.85Chapter 9.4 --- Internet Browsers --- p.87Chapter 9.4.1 --- Secure NCSA Mosaic --- p.87Chapter 9.4.2 --- Netscape Navigator --- p.89Chapter 9.4.3 --- SunSoft HotJava --- p.91Chapter 10 --- Examples of Electronic Commerce System --- p.94Chapter 10.1 --- CyberCash --- p.95Chapter 10.2 --- DigiCash --- p.97Chapter 10.3 --- The Financial Services Technology Consortium --- p.98Chapter 10.3.1 --- Electronic Check Project --- p.99Chapter 10.3.2 --- Electronic Commerce Project --- p.101Chapter 10.4 --- FirstVirtual --- p.103Chapter 10.5 --- Mondex --- p.104Chapter 10.6 --- NetBill --- p.106Chapter 10.7 --- NetCash --- p.108Chapter 10.8 --- NetCheque --- p.111Chapter 11 --- Conclusion --- p.113Chapter A --- An Essay on Chinese Remainder Theorem and RSA --- p.115Bibliography --- p.11

Design and Analysis of Multi-Block-Length Hash Functions

Cryptographic hash functions are used in many cryptographic applications, and the design of provably secure hash functions (relative to various security notions) is an active area of research. Most of the currently existing hash functions use the Merkle-Damgård paradigm, where by appropriate iteration the hash function inherits its collision and preimage resistance from the underlying compression function. Compression functions can either be constructed from scratch or be built using well-known cryptographic primitives such as a blockcipher. One classic type of primitive-based compression functions is single-block-length : It contains designs that have an output size matching the output length n of the underlying primitive. The single-block-length setting is well-understood. Yet even for the optimally secure constructions, the (time) complexity of collision- and preimage-finding attacks is at most 2n/2, respectively 2n ; when n = 128 (e.g., Advanced Encryption Standard) the resulting bounds have been deemed unacceptable for current practice. As a remedy, multi-block-length primitive-based compression functions, which output more than n bits, have been proposed. This output expansion is typically achieved by calling the primitive multiple times and then combining the resulting primitive outputs in some clever way. In this thesis, we study the collision and preimage resistance of certain types of multi-call multi-block-length primitive-based compression (and the corresponding Merkle-Damgård iterated hash) functions : Our contribution is three-fold. First, we provide a novel framework for blockcipher-based compression functions that compress 3n bits to 2n bits and that use two calls to a 2n-bit key blockcipher with block-length n. We restrict ourselves to two parallel calls and analyze the sufficient conditions to obtain close-to-optimal collision resistance, either in the compression function or in the Merkle-Damgård iteration. Second, we present a new compression function h: {0,1}3n → {0,1}2n ; it uses two parallel calls to an ideal primitive (public random function) from 2n to n bits. This is similar to MDC-2 or the recently proposed MJH by Lee and Stam (CT-RSA'11). However, unlike these constructions, already in the compression function we achieve that an adversary limited (asymptotically in n) to O (22n(1-δ)/3) queries (for any δ > 0) has a disappearing advantage to find collisions. This is the first construction of this type offering collision resistance beyond 2n/2 queries. Our final contribution is the (re)analysis of the preimage and collision resistance of the Knudsen-Preneel compression functions in the setting of public random functions. Knudsen-Preneel compression functions utilize an [r,k,d] linear error-correcting code over 𝔽2e (for e > 1) to build a compression function from underlying blockciphers operating in the Davies-Meyer mode. Knudsen and Preneel show, in the complexity-theoretic setting, that finding collisions takes time at least 2(d-1)n2. Preimage resistance, however, is conjectured to be the square of the collision resistance. Our results show that both the collision resistance proof and the preimage resistance conjecture of Knudsen and Preneel are incorrect : With the exception of two of the proposed parameters, the Knudsen-Preneel compression functions do not achieve the security level they were designed for