On Using Proportional Representation Methods as Alternatives to Pro-Rata Based Order Matching Algorithms in Stock Exchanges

The main observation of this short note is that methods for determining proportional representation in electoral systems may be suitable as alternatives to the pro-rata order matching algorithm used in stock exchanges. Our simulation studies provide strong evidence that the Jefferson/D'Hondt and the Webster/Saint-Lagu\"{e} proportional representation methods provide order allocations which are closer to proportionality than the order allocations obtained from the pro-rata algorithm

A Direct Construction of Intergroup Complementary Code Set for CDMA

A collection of mutually orthogonal complementary codes (CCs) is said to be complete complementary codes (CCCs) where the number of CCs are equal to the number of constituent sequences in each CC. Intergroup complementary (IGC) code set is a collection of multiple disjoint code groups with the following correlation properties: (1) inside the zero-correlation zone (ZCZ), the aperiodic autocorrelation function (AACF) of any IGC code is zero for all nonzero time shifts; (2) the aperiodic cross-correlation function (ACCF), of two distinct IGC codes, is zero for all time shifts inside the ZCZ when they are taken from the same code groups; and (3) the ACCF, for two IGC codes from two different code groups, is zero everywhere. IGC code set has a larger set size than CCC, and both can be applicable in multicarrier code-division multiple access (CDMA). In this chapter, we present a direct construction of IGC code set by using second-order generalized Boolean functions (GBFs), and our IGC code set can support interference-free code-division multiplexing. We also relate our construction with a graph where the ZCZ width depends on the number of isolated vertices present in a graph after the deletion of some vertices. Here, the construction that we propose can generate IGC code set with more flexible parameters

Influence of a Set of Variables on a Boolean Function

The influence of a variable is an important concept in the analysis of Boolean functions. The more general notion of influence of a set of variables on a Boolean function has four separate definitions in the literature. In the present work, we introduce a new definition of influence of a set of variables which is based on the auto-correlation function and develop its basic theory. Among the new results that we obtain are generalisations of the Poincar\'{e} inequality and the edge expansion property of the influence of a single variable. Further, we obtain new characterisations of resilient and bent functions using the notion of influence. We show that the previous definition of influence due to Fischer et al. (2002) and Blais (2009) is half the value of the auto-correlation based influence that we introduce. Regarding the other prior notions of influence, we make a detailed study of these and show that each of these definitions do not satisfy one or more desirable properties that a notion of influence may be expected to satisfy

Reducing Communication Overhead of the Subset Difference Scheme

In Broadcast Encryption (BE) systems like Pay-TV, AACS, online content sharing and broadcasting, reducing the header length (communication overhead per session) is of practical interest. The Subset Difference (SD) scheme due to Naor-Naor-Lotspiech (NNL) is the most popularly used BE scheme. We introduce the (a, b, γ) augmented binary tree subset difference ( (a, b, γ) -ABTSD) scheme which is a generalization of the NNL-SD scheme. By varying the parameters (a, b, γ) , it is possible to obtain O(n log n) different schemes. The average header length achieved by the new schemes is smaller than all known schemes having the same decryption time as that of the NNL-SD scheme and achieving non-trivial trade-offs between the user storage and the header size. The amount of key material that a user is required to store increases. For the earlier mentioned applications, reducing header size and achieving fast decryption is perhaps more of a concern than the user storage

Pseudo-Random Functions and Parallelizable Modes of Operations of a Block Cipher

This paper considers the construction and analysis of pseudo-random functions (PRFs) with specific reference to modes of operations of a block cipher. In the context of message authentication codes (MACs), earlier independent work by Bernstein and Vaudenay show how to reduce the analysis of relevant PRFs to some probability calculations. In the first part of the paper, we revisit this result and use it to prove a general result on constructions which use a PRF with a ``small\u27\u27 domain to build a PRF with a ``large\u27\u27 domain. This result is used to analyse two new parallelizable PRFs which are suitable for use as MAC schemes. The first scheme, called {\iPMAC}, is based on a block cipher and improves upon the well-known PMAC algorithm. The improvements consist in faster masking operations and the removal of a design stage discrete logarithm computation. The second scheme, called {\VPMAC}, uses a keyed compression function rather than a block cipher. The only previously known compression function based parallelizable PRF is called the protected counter sum (PCS) and is due to Bernstein. {\VPMAC} improves upon PCS by requiring lesser number of calls to the compression function. The second part of the paper takes a new look at the construction and analysis of modes of operations for authenticated encryption (AE) and for authenticated encryption with associated data (AEAD). Usually, the most complicated part in the security analysis of such modes is the analysis of authentication security. Previous work by Liskov, Rivest and Wagner and later Rogaway had suggested that this analysis is simplified by using a primitive called a tweakable block cipher (TBC). In contrast, we take a direct approach. We prove a general result which shows that the authentication security of an AE scheme can be proved from the privacy of the scheme and by showing a certain associated function to be a PRF. Two new AE schemes \sym{PAE} and \sym{PAE}-1 are described and analysed using this approach. In particular, it is shown that the authentication security of \sym{PAE} follows easily from the security of {\iPMAC}. As a result, no separate extensive analysis of the authentication security of \sym{PAE} is required. An AEAD scheme can be obtained by combining an AE scheme and an authentication scheme and it has been suggested earlier that a TBC based approach simplifies the analysis. Again, in contrast to the TBC based approach, we take a direct approach based on a simple masking strategy. Our idea uses double encryption of a fixed string and achieves the same effect of mask separation as in the TBC based approach. Using this idea, two new AEAD schemes \sym{PAEAD} and \sym{PAEAD}-1 are described. An important application of AEAD schemes is in the encryption of IP packets. The new schemes offer certain advantages over previously well known schemes such as the offset codebook (OCB) mode. These improvements include providing a wider variety of easily reconfigurable family of schemes, a small speed-up, a smaller size decryption algorithm for hardware implementation and uniform processing of only full-block messages