Rotation symmetric Boolean functions---count and cryptographic properties

The article of record as published may be located at http://dx.doi.org/10.1.1.137.6388Rotation symmetric (RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnsideﾒs lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2gn, where gn = 1 nPt|n (t) 2n t , and (.) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in Fn2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree > 2. Further, we studied the RotS functions on 5, 6, 7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier

Algorithm 959: VBF: A Library of C plus plus Classes for Vector Boolean Functions in Cryptography

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks

On applications of simulated annealing to cryptology

Boolean functions are critical building blocks of symmetric-key ciphers. In most cases, the security of a cipher against a particular kind of attacks can be explained by the existence of certain properties of its underpinning Boolean functions. Therefore, the design of appropriate functions has received significant attention from researchers for several decades. Heuristic methods have become very powerful tools for designing such functions. In this thesis, we apply simulated annealing methods to construct Boolean functions with particular properties. Our results meet or exceed the best results of available theoretical constructions and/or heuristic searches in the literature, including a 10-variable balanced Boolean function with resiliency degree 2, algebraic degree 7, and nonlinearity 488 for the first time. This construction affirmatively answers the open problem about the existence of such functions. This thesis also includes results of cryptanalysis for symmetric ciphers, such as Geffe cipher and TREYFER cipher

Algebraic approaches for coded caching and distributed computing

This dissertation examines the power of algebraic methods in two areas of modern interest: caching for large scale content distribution and straggler mitigation within distributed computation. Caching is a popular technique for facilitating large scale content delivery over the Internet. Traditionally, caching operates by storing popular content closer to the end users. Recent work within the domain of information theory demonstrates that allowing coding in the cache and coded transmission from the server (referred to as coded caching) to the end users can allow for significant reductions in the number of bits transmitted from the server to the end users. The first part of this dissertation examines problems within coded caching. The original formulation of the coded caching problem assumes that the server and the end users are connected via a single shared link. In Chapter 2, we consider a more general topology where there is a layer of relay nodes between the server and the users. We propose novel schemes for a class of such networks that satisfy a so-called resolvability property and demonstrate that the performance of our scheme is strictly better than previously proposed schemes. Moreover, the original coded caching scheme requires that each file hosted in the server be partitioned into a large number (i.e., the subpacketization level) of non-overlapping subfiles. From a practical perspective, this is problematic as it means that prior schemes are only applicable when the size of the files is extremely large. In Chapter 3, we propose a novel coded caching scheme that enjoys a significantly lower subpacketization level than prior schemes, while only suffering a marginal increase in the transmission rate. We demonstrate that several schemes with subpacketization levels that are exponentially smaller than the basic scheme can be obtained. The second half of this dissertation deals with large scale distributed matrix computations. Distributed matrix multiplication is an important problem, especially in domains such as deep learning of neural networks. It is well recognized that the computation times on distributed clusters are often dominated by the slowest workers (called stragglers). Recently, techniques from coding theory have found applications in straggler mitigation in the specific context of matrix-matrix and matrix-vector multiplication. The computation can be completed as long as a certain number of workers (called the recovery threshold) complete their assigned tasks. In Chapter 4, we consider matrix multiplication under the assumption that the absolute values of the matrix entries are sufficiently small. Under this condition, we present a method with a significantly smaller recovery threshold than prior work. Besides, the prior work suffers from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices; this condition number grows exponentially in the number of workers. In Chapter 5, we present a novel approach that leverages the properties of circulant permutation matrices and rotation matrices for coded matrix computation. In addition to having an optimal recovery threshold, we demonstrate an upper bound on the worst case condition number of our recovery matrices grows polynomially in the number of workers

On Achieving Unconditionally Secure Communications Via the Physical Layer Approaches

Due to the broadcast nature, wireless links are open to malicious intrusions from outsiders, which makes the security issues a critical concern in the wireless communicationsover them. Physical-layer security techniques, which are based on the Shannon’s unconditional secrecy model, are effective in addressing the security issue while meeting the required performance level. According to the Wyner’s wiretap channel model, to achieve unconditionally security communication, the first step is to build up a wiretap channel with better channel quality between the legitimate communication peers than that of the eavesdropper; and the second step is to employ a robust security code to ensure that the legitimate users experience negligible errors while the eavesdropper is subject to 0.5 error probability. Motivated by this idea, in this thesis, we build wiretap channels for the single antenna systems without resorting to the spatial degree in commonly observed the multiple-input multiple-output (MIMO) systems. Firstly, to build effective wiretap channels, we design a novel scheme, called multi-round two-way communications (MRTWC). By taking feedback mechanisms into the design of Low Density Parity Check (LDPC) codes, our scheme adds randomness to the feedback signals from the destination to keep the eavesdropper ignorant while adding redundancy with the LDPC codes so that the legitimate receiver can correctly receive and decode the signals. Then, the channel BERs are specifically quantified according to the crossover probability in the case of Binary Symmetric Channel (BSC), or the Signal to Noise Ratio (SNR) in the case of AWGN and Rayleigh channels. Thus, the novel scheme can be utilized to address the security and reliability. Meanwhile, we develop a cross-layer approach to building the wiretap channel, which is suitable for high dynamic scenarios. By taking advantage of multiple parameters freedom in the discrete fractional Fourier transform (DFRFT) for single antenna systems, the proposed scheme introduces a distortion parameter instead of a general signal parameter for wireless networks based on DFRFT. The transmitter randomly flip-flops the uses of the distortion parameter and the general signal parameter to confuse the eavesdropper. An upper-layer cipher sequence will be employed to control the flip-flops. This cryptographic sequence in the higher layer is combined with the physical layer security scheme with random parameter fipping in DFRFT to guarantee security advantages over the main communication channel. As the efforts on the second step, this thesis introduces a novel approach to generate security codes, which can be used for encoding with low complexity by taking advantage of a matrix general inverse algorithm. The novel constructions of the security codes are based on binary and non-binary resilient functions. With the proposed security codes, we prove that our novel security codes can ensure 0.5 error probability seen by the wiretapper while close to zero by the intended receiver if the error probability of the wiretapper’s channel is over a derived threshold. Therefore, the unconditionally secure communication of legitimate partners can be guaranteed. It has been proved mathematically that the non-binary security codes could achieve closer to the security capacity bound than any other reported short-length security codes under BSC. Finally, we develop the framework of associating the wiretap channel building approach with the security codes. The advantages between legitimate partners are extended via developing the security codes on top of our cross-layer DFRFT and feedback MRTWC security communication model. In this way, the proposed system could ensure almost zero information obtained by the eavesdroppers while still keeping rather lower error transmissions for legitimate users. Extensive experiments are carried out to verify the proposed security schemes and demonstrate the feasibility and implement ability. An USRP testbed is also constructed, under which the physical layer security mechanisms are implemented and tested. Our study shows that our proposed security schemes can be implemented in practical communications settings

D.STVL.9 - Ongoing Research Areas in Symmetric Cryptography

This report gives a brief summary of some of the research trends in symmetric cryptography at the time of writing (2008). The following aspects of symmetric cryptography are investigated in this report: • the status of work with regards to different types of symmetric algorithms, including block ciphers, stream ciphers, hash functions and MAC algorithms (Section 1); • the algebraic attacks on symmetric primitives (Section 2); • the design criteria for symmetric ciphers (Section 3); • the provable properties of symmetric primitives (Section 4); • the major industrial needs in the area of symmetric cryptography (Section 5)

Tamper-Resistant Arithmetic for Public-Key Cryptography

Cryptographic hardware has found many uses in many ubiquitous and pervasive security devices with a small form factor, e.g. SIM cards, smart cards, electronic security tokens, and soon even RFIDs. With applications in banking, telecommunication, healthcare, e-commerce and entertainment, these devices use cryptography to provide security services like authentication, identification and confidentiality to the user. However, the widespread adoption of these devices into the mass market, and the lack of a physical security perimeter have increased the risk of theft, reverse engineering, and cloning. Despite the use of strong cryptographic algorithms, these devices often succumb to powerful side-channel attacks. These attacks provide a motivated third party with access to the inner workings of the device and therefore the opportunity to circumvent the protection of the cryptographic envelope. Apart from passive side-channel analysis, which has been the subject of intense research for over a decade, active tampering attacks like fault analysis have recently gained increased attention from the academic and industrial research community. In this dissertation we address the question of how to protect cryptographic devices against this kind of attacks. More specifically, we focus our attention on public key algorithms like elliptic curve cryptography and their underlying arithmetic structure. In our research we address challenges such as the cost of implementation, the level of protection, and the error model in an adversarial situation. The approaches that we investigated all apply concepts from coding theory, in particular the theory of cyclic codes. This seems intuitive, since both public key cryptography and cyclic codes share finite field arithmetic as a common foundation. The major contributions of our research are (a) a generalization of cyclic codes that allow embedding of finite fields into redundant rings under a ring homomorphism, (b) a new family of non-linear arithmetic residue codes with very high error detection probability, (c) a set of new low-cost arithmetic primitives for optimal extension field arithmetic based on robust codes, and (d) design techniques for tamper resilient finite state machines

Ongoing Research Areas in Symmetric Cryptography

This report is a deliverable for the ECRYPT European network of excellence in cryptology. It gives a brief summary of some of the research trends in symmetric cryptography at the time of writing. The following aspects of symmetric cryptography are investigated in this report: • the status of work with regards to different types of symmetric algorithms, including block ciphers, stream ciphers, hash functions and MAC algorithms (Section 1); • the recently proposed algebraic attacks on symmetric primitives (Section 2); • the design criteria for symmetric ciphers (Section 3); • the provable properties of symmetric primitives (Section 4); • the major industrial needs in the area of symmetric cryptography (Section 5)