Attribute-based functional encryption on lattices

We introduce a broad lattice manipulation technique for expressive cryptography, and use it to realize functional encryption for access structures from post-quantum hardness assumptions. Specifically, we build an efficient key-policy attribute-based encryption scheme, and prove its security in the selective sense from learning-with-errors intractability in the standard model.

Algebraic analysis of Trivium-like ciphers

Trivium is a bit-based stream cipher in the final portfolio of the eSTREAM project. In this paper, we apply the algebraic attack approach of Berbain et al. to Trivium-like ciphers and perform new analyses on them. We demonstrate a new algebraic attack on Bivium-A. This attack requires less time and memory than previous techniques to recover Bivium-A's initial state. Though our attacks on Bivium-B, Trivium and Trivium-N are worse than exhaustive keysearch, the systems of equations which are constructed are smaller and less complex compared to previous algebraic analyses. We also answer an open question posed by Berbain et al. on the feasibility of applying their technique on Trivium-like ciphers. Factors which can affect the complexity of our attack on Trivium-like ciphers are discussed in detail. Analysis of Bivium-B and Trivium-N are omitted from this manuscript. The full paper is available on the IACR ePrint Archive.

Functional encryption for threshold functions (or fuzzy IBE) from lattices

Cryptosystems based on the hardness of lattice problems have recently acquired much importance due to their average-case to worst-case equivalence, their conjectured resistance to quantum cryptanalysis, their ease of implementation and increasing practicality, and, lately, their promising potential as a platform for constructing advanced functionalities. In this work, we construct “Fuzzy” Identity Based Encryption from the hardness of the Learning With Errors (LWE) problem. We note that for our parameters, the underlying lattice problems (such as gapSVP or SIVP) are assumed to be hard to approximate within supexponential factors for adversaries running in subexponential time. We give CPA and CCA secure variants of our construction, for small and large universes of attributes. All our constructions are secure against selective-identity attacks in the standard model. Our construction is made possible by observing certain special properties that secret sharing schemes need to satisfy in order to be useful for Fuzzy IBE. We also discuss some obstacles towards realizing lattice-based attribute-based encryption (ABE).

Expressive encryption systems from lattices

In this survey, we review a number of the many “expressive” encryption systems that have recently appeared from lattices, and explore the innovative techniques that underpin them.

An algebraic approach for determination of DG parameters to support voltage profiles in radial distribution networks

Rapidly increasing electricity demands and capacity shortage of transmission and distribution facilities are the main driving forces for the growth of Distributed Generation (DG) integration in power grids. One of the reasons for choosing a DG is its ability to support voltage in a distribution system. Selection of effective DG characteristics and DG parameters is a significant concern of distribution system planners to obtain maximum potential benefits from the DG unit. This paper addresses the issue of improving the network voltage profile in distribution systems by installing a DG of the most suitable size, at a suitable location. An analytical approach is developed based on algebraic equations for uniformly distributed loads to determine the optimal operation, size and location of the DG in order to achieve required levels of network voltage. The developed method is simple to use for conceptual design and analysis of distribution system expansion with a DG and suitable for a quick estimation of DG parameters (such as optimal operating angle, size and location of a DG system) in a radial network. A practical network is used to verify the proposed technique and test results are presented.

An algebraic analysis of trivium ciphers based on the boolean satisfiability problem

Trivium is a stream cipher candidate of the eStream project. It has successfully moved into phase three of the selection process under the hardware category. No attacks faster than the exhaustive search have so far been reported on Trivium. Bivium-A and Bivium-B are simplified versions of Trivium that are built on the same design principles but with two registers. The simplified design is useful in investigating Trivium type ciphers with a reduced complexity and provides insight into effective attacks which could be extended to Trivium. This paper focuses on an algebraic analysis which uses the boolean satisfiability problem in propositional logic. For reduced variants of the cipher, this analysis recovers the internal state with a minimal amount of keystream observations.

Colour by numbers in Excel 2007 : solving algebraic equations without algebra

This is an update of an earlier paper, and is written for Excel 2007. A series of Excel 2007 models is described. The more advanced versions allow solution of f(x)=0 by examining change of sign of function values. The function is graphed and change of sign easily detected by a change of colour. Relevant features of Excel 2007 used are Names, Scatter Chart and Conditional Formatting. Several sample Excel 2007 models are available for download, and the paper is intended to be used as a lesson plan for students having some familiarity with derivatives. For comparison and reference purposes, the paper also presents a brief outline of several common equation-solving strategies as an Appendix.

The Number Crunch game : a simple vehicle for building algebraic reasoning skills

A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.

Algebraic attacks on SOBER-t32 and SOBER-t16 without stuttering

This paper presents algebraic attacks on SOBER-t32 and SOBER-t16 without stuttering. For unstuttered SOBER-t32, two different attacks are implemented. In the first attack, we obtain multivariate equations of degree 10. Then, an algebraic attack is developed using a collection of output bits whose relation to the initial state of the LFSR can be described by low-degree equations. The resulting system of equations contains 2^69 equations and monomials, which can be solved using the Gaussian elimination with the complexity of 2^196.5. For the second attack, we build a multivariate equation of degree 14. We focus on the property of the equation that the monomials which are combined with output bit are linear. By applying the Berlekamp-Massey algorithm, we can obtain a system of linear equations and the initial states of the LFSR can be recovered. The complexity of attack is around O(2^100) with 2^92 keystream observations. The second algebraic attack is applicable to SOBER-t16 without stuttering. The attack takes around O(2^85) CPU clocks with 2^78 keystream observations.

On algebraic immunity and annihilators

Algebraic immunity AI(f) defined for a boolean function f measures the resistance of the function against algebraic attacks. Currently known algorithms for computing the optimal annihilator of f and AI(f) are inefficient. This work consists of two parts. In the first part, we extend the concept of algebraic immunity. In particular, we argue that a function f may be replaced by another boolean function f^c called the algebraic complement of f. This motivates us to examine AI(f ^c ). We define the extended algebraic immunity of f as AI *(f)= min {AI(f), AI(f^c )}. We prove that 0≤AI(f)–AI *(f)≤1. Since AI(f)–AI *(f)= 1 holds for a large number of cases, the difference between AI(f) and AI *(f) cannot be ignored in algebraic attacks. In the second part, we link boolean functions to hypergraphs so that we can apply known results in hypergraph theory to boolean functions. This not only allows us to find annihilators in a fast and simple way but also provides a good estimation of the upper bound on AI *(f).

Implementation of an integrated, technology-based, discovery mode assessment item involving an incubation period to enhance learning outcomes for engineering maths students

In this paper we discuss our current efforts to develop and implement an exploratory, discovery mode assessment item into the total learning and assessment profile for a target group of about 100 second level engineering mathematics students. The assessment item under development is composed of 2 parts, namely, a set of "pre-lab" homework problems (which focus on relevant prior mathematical knowledge, concepts and skills), and complementary computing laboratory exercises which are undertaken within a fixed (1 hour) time frame. In particular, the computing exercises exploit the algebraic manipulation and visualisation capabilities of the symbolic algebra package MAPLE, with the aim of promoting understanding of certain mathematical concepts and skills via visual and intuitive reasoning, rather than a formal or rigorous approach. The assessment task we are developing is aimed at providing students with a significant learning experience, in addition to providing feedback on their individual knowledge and skills. To this end, a noteworthy feature of the scheme is that marks awarded for the laboratory work are primarily based on the extent to which reflective, critical thinking is demonstrated, rather than the amount of CBE-style tasks completed by the student within the allowed time. With regard to student learning outcomes, a novel and potentially critical feature of our scheme is that the assessment task is designed to be intimately linked to the overall course content, in that it aims to introduce important concepts and skills (via individual student exploration) which will be revisited somewhat later in the pedagogically more restrictive formal lecture component of the course (typically a large group plenary format). Furthermore, the time delay involved, or "incubation period", is also a deliberate design feature: it is intended to allow students the opportunity to undergo potentially important internal re-adjustments in their understanding, before being exposed to lectures on related course content which are invariably delivered in a more condensed, formal and mathematically rigorous manner. In our presentation, we will discuss in more detail our motivation and rationale for trailing such a scheme for the targeted student group. Some of the advantages and disadvantages of our approach (as we perceived them at the initial stages) will also be enumerated. In a companion paper, the theoretical framework for our approach will be more fully elaborated, and measures of student learning outcomes (as obtained from eg. student provided feedback) will be discussed.

Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

The effect of language models on phonetic decoding for spoken term detection

Spoken term detection (STD) popularly involves performing word or sub-word level speech recognition and indexing the result. This work challenges the assumption that improved speech recognition accuracy implies better indexing for STD. Using an index derived from phone lattices, this paper examines the effect of language model selection on the relationship between phone recognition accuracy and STD accuracy. Results suggest that language models usually improve phone recognition accuracy but their inclusion does not always translate to improved STD accuracy. The findings suggest that using phone recognition accuracy to measure the quality of an STD index can be problematic, and highlight the need for an alternative that is more closely aligned with the goals of the specific detection task.

Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.