Analysis of linear relationships in block ciphers

This thesis is devoted to the study of linear relationships in symmetric block ciphers. A block cipher is designed so that the ciphertext is produced as a nonlinear function of the plaintext and secret master key. However, linear relationships within the cipher can still exist if the texts and components of the cipher are manipulated in a number of ways, as shown in this thesis. There are four main contributions of this thesis. The first contribution is the extension of the applicability of integral attacks from word-based to bitbased block ciphers. Integral attacks exploit the linear relationship between texts at intermediate stages of encryption. This relationship can be used to recover subkey bits in a key recovery attack. In principle, integral attacks can be applied to bit-based block ciphers. However, specific tools to define the attack on these ciphers are not available. This problem is addressed in this thesis by introducing a refined set of notations to describe the attack. The bit patternbased integral attack is successfully demonstrated on reduced-round variants of the block ciphers Noekeon, Present and Serpent. The second contribution is the discovery of a very small system of equations that describe the LEX-AES stream cipher. LEX-AES is based heavily on the 128-bit-key (16-byte) Advanced Encryption Standard (AES) block cipher. In one instance, the system contains 21 equations and 17 unknown bytes. This is very close to the upper limit for an exhaustive key search, which is 16 bytes. One only needs to acquire 36 bytes of keystream to generate the equations. Therefore, the security of this cipher depends on the difficulty of solving this small system of equations. The third contribution is the proposal of an alternative method to measure diffusion in the linear transformation of Substitution-Permutation-Network (SPN) block ciphers. Currently, the branch number is widely used for this purpose. It is useful for estimating the possible success of differential and linear attacks on a particular SPN cipher. However, the measure does not give information on the number of input bits that are left unchanged by the transformation when producing the output bits. The new measure introduced in this thesis is intended to complement the current branch number technique. The measure is based on fixed points and simple linear relationships between the input and output words of the linear transformation. The measure represents the average fraction of input words to a linear diffusion transformation that are not effectively changed by the transformation. This measure is applied to the block ciphers AES, ARIA, Serpent and Present. It is shown that except for Serpent, the linear transformations used in the block ciphers examined do not behave as expected for a random linear transformation. The fourth contribution is the identification of linear paths in the nonlinear round function of the SMS4 block cipher. The SMS4 block cipher is used as a standard in the Chinese Wireless LAN Wired Authentication and Privacy Infrastructure (WAPI) and hence, the round function should exhibit a high level of nonlinearity. However, the findings in this thesis on the existence of linear relationships show that this is not the case. It is shown that in some exceptional cases, the first four rounds of SMS4 are effectively linear. In these cases, the effective number of rounds for SMS4 is reduced by four, from 32 to 28. The findings raise questions about the security provided by SMS4, and might provide clues on the existence of a flaw in the design of the cipher.

Linearity within the SMS4 Block Cipher

We present several new observations on the SMS4 block cipher, and discuss their cryptographic significance. The crucial observation is the existence of fixed points and also of simple linear relationships between the bits of the input and output words for each component of the round functions for some input words. This implies that the non-linear function T of SMS4 does not appear random and that the linear transformation provides poor diffusion. Furthermore, the branch number of the linear transformation in the key scheduling algorithm is shown to be less than optimal. The main security implication of these observations is that the round function is not always non-linear. Due to this linearity, it is possible to reduce the number of effective rounds of SMS4 by four. We also investigate the susceptibility of SMS4 to further cryptanalysis. Finally, we demonstrate a successful differential attack on a slightly modified variant of SMS4. These findings raise serious questions on the security provided by SMS4.

Cryptanalysis of the 10-Round Hash and Full Compression Function of SHAvite-3-512

In this paper, we analyze the SHAvite-3-512 hash function, as proposed and tweaked for round 2 of the SHA-3 competition. We present cryptanalytic results on 10 out of 14 rounds of the hash function SHAvite-3-512, and on the full 14 round compression function of SHAvite-3-512. We show a second preimage attack on the hash function reduced to 10 rounds with a complexity of 2497 compression function evaluations and 216 memory. For the full 14-round compression function, we give a chosen counter, chosen salt preimage attack with 2384 compression function evaluations and 2128 memory (or complexity 2448 without memory), and a collision attack with 2192 compression function evaluations and 2128 memory.

Rotational cryptanalysis of round-reduced Keccak

In this paper we attack round-reduced Keccak hash function with a technique called rotational cryptanalysis. We focus on Keccak variants proposed as SHA-3 candidates in the NIST’s contest for a new standard of cryptographic hash function. Our main result is a preimage attack on 4-round Keccak and a 5-round distinguisher on Keccak-f[1600] permutation — the main building block of Keccak hash function.

An approximate method for the solution of an influence function foil rolling model

Fleck and Johnson (Int. J. Mech. Sci. 29 (1987) 507) and Fleck et al. (Proc. Inst. Mech. Eng. 206 (1992) 119) have developed foil rolling models which allow for large deformations in the roll profile, including the possibility that the rolls flatten completely. However, these models require computationally expensive iterative solution techniques. A new approach to the approximate solution of the Fleck et al. (1992) Influence Function Model has been developed using both analytic and approximation techniques. The numerical difficulties arising from solving an integral equation in the flattened region have been reduced by applying an Inverse Hilbert Transform to get an analytic expression for the pressure. The method described in this paper is applicable to cases where there is or there is not a flat region.

Smoothing a Discrete Hazard Function for the Number of Patients Colonized with Methicillin-Resistance Staphylococcus Aureus in an Intensive Care Unit

A new method for estimating the time to colonization of Methicillin-resistant Staphylococcus Aureus (MRSA) patients is developed in this paper. The time to colonization of MRSA is modelled using a Bayesian smoothing approach for the hazard function. There are two prior models discussed in this paper: the first difference prior and the second difference prior. The second difference prior model gives smoother estimates of the hazard functions and, when applied to data from an intensive care unit (ICU), clearly shows increasing hazard up to day 13, then a decreasing hazard. The results clearly demonstrate that the hazard is not constant and provide a useful quantification of the effect of length of stay on the risk of MRSA colonization which provides useful insight.