A Running Time Improvement for Two Thresholds Two Divisors Algorithm

Chunking algorithms play an important role in data de-duplication systems. The Basic Sliding Window (BSW) algorithm is the first prototype of the content-based chunking algorithm which can handle most types of data. The Two Thresholds Two Divisors (TTTD) algorithm was proposed to improve the BSW algorithm in terms of controlling the variations of the chunk-size. In this project, we investigate and compare the BSW algorithm and TTTD algorithm from different factors by a series of systematic experiments. Up to now, no paper conducts these experimental evaluations for these two algorithms. This is the first value of this paper. According to our analyses and the results of experiments, we provide a running time improvement for the TTTD algorithm. Our new solution reduces about 7 % of the total running time and also reduces about 50 % of the large-sized chunks while comparing with the original TTTD algorithm and make average chunk-size closer to the expected chunk-size. These significant results are the second important value of this project

Computing π(N)\pi(N): An elementary approach in O~(N)\tilde{O}(\sqrt{N}) time

We present an efficient and elementary algorithm for computing the number of primes up to $N$ in $\tilde{O}(\sqrt N)$ time, improving upon the existing combinatorial methods that require $\tilde{O}(N ^ {2/3})$ time. Our method has a similar time complexity to the analytical approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. While the most time-efficient version of our algorithm requires $\tilde{O}(\sqrt N)$ space, we present a continuous space-time trade-off, showing, e.g., how to reduce the space complexity to $\tilde{O}(\sqrt[3]{N})$ while slightly increasing the time complexity to $\tilde{O}(N^{8/15})$. We apply our techniques to improve the state-of-the-art complexity of elementary algorithms for computing other number-theoretic functions, such as the the Mertens function (in $\tilde{O}(\sqrt N)$ time compared to the known $\tilde{O}(N^{0.6})$), summing Euler's totient function, counting square-free numbers and summing primes. Implementation code is provided

COMPRESSION OF WEARABLE BODY SENSOR NETWORK DATA USING IMPROVED TWO-THRESHOLD-TWO-DIVISOR DATA CHUNKING ALGORITHM

Compression plays a significant role in Body Sensor Networks (BSN) data since the sensors in BSNs have limited battery power and memory. Also, data needs to be transmitted fast and in a lossless manner to provide near real-time feedback. The paper evaluates lossless data compression algorithms like Run Length Encoding (RLE), Lempel Zev Welch (LZW) and Huffman on data from wearable devices and compares them in terms of Compression Ratio, Compression Factor, Savings Percentage and Compression Time. It also evaluates a data deduplication technique used for Low Bandwidth File Systems (LBFS) named Two Thresholds Two Divisors (TTTD) algorithm to determine if it could be used for BSN data. By changing the parameters and running the algorithm multiple times on the data, it arrives at a set of values that give \u3e50 compression ratio on BSN data. This is the first value of the paper. Based on these performance evaluation results of TTTD and various classical compression algorithms, it proposes a technique to combine multiple algorithms in sequence. Upon comparison of the performance, it has been found that the new algorithm, TTTD-H, which does TTTD and Huffman in sequence, improves the Savings Percentage by 23 percent over TTTD, and 31 percent over Huffman when executed independently. Compression Factor improved by 142 percent over TTTD, 52 percent over LZW, 178 percent over Huffman for a file of 3.5 MB. These significant results are the second important value of the project

Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm

The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most efficient way to trade space against time for the SUBSET SUM problem. In the random-instance setting, however, improved tradeoffs exist. In particular, the recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a significantly improved space--time tradeoff curve for instances with strong randomness properties. Our main result is that these strong randomness assumptions can be removed, obtaining the same space--time tradeoffs in the worst case. We also show that for small space usage the dissection algorithm can be almost fully parallelized. Our strategy for dealing with arbitrary instances is to instead inject the randomness into the dissection process itself by working over a carefully selected but random composite modulus, and to introduce explicit space--time controls into the algorithm by means of a "bailout mechanism"

Security systems based on Gaussian integers : Analysis of basic operations and time complexity of secret transformations

Many security algorithms currently in use rely heavily on integer arithmetic modulo prime numbers. Gaussian integers can be used with most security algorithms that are formulated for real integers. The aim of this work is to study the benefits of common security protocols with Gaussian integers. Although the main contribution of this work is to analyze and improve the application of Gaussian integers for various public key (PK) algorithms, Gaussian integers were studied in the context of image watermarking as well. The significant benefits of the application of Gaussian integers become apparent when they are used with Discrete Logarithm Problem (DLP) based PK algorithms. In order to quantify the complexity of the Gaussian integer DLP, it is reduced to two other well known problems: DLP for Lucas sequences and the real integer DLP. Additionally, a novel exponentiation algorithm for Gaussian integers, called Lucas sequence Exponentiation of Gaussian integers (LSEG) is introduced and its performance assessed, both analytically and experimentally. The LSEG achieves about 35% theoretical improvement in CPU time over real integer exponentiation. Under an implementation with the GMP 5.0.1 library, it outperformed the GMP\u27s mpz_powm function (the particularly efficient modular exponentiation function that comes with the GMP library) by 40% for bit sizes 1000-4000, because of low overhead associated with LSEG. Further improvements to real execution time can be easily achieved on multiprocessor or multicore platforms. In fact, over 50% improvement is achieved with a parallelized implementation of LSEG. All the mentioned improvements do not require any special hardware or software and are easy to implement. Furthermore, an efficient way for finding generators for DLP based PK algorithms with Gaussian integers is presented. In addition to DLP based PK algorithms, applications of Gaussian integers for factoring-based PK cryptosystems are considered. Unfortunately, the advantages of Gaussian integers for these algorithms are not as clear because the extended order of Gaussian integers does not directly come into play. Nevertheless, this dissertation describes the Extended Square Root algorithm for Gaussian integers used to extend the Rabin Cryptography algorithm into the field of Gaussian integers. The extended Rabin Cryptography algorithm with Gaussian integers allows using fewer preset bits that are required by the algorithm to guard against various attacks. Additionally, the extension of RSA into the domain of Gaussian integers is analyzed. The extended RSA algorithm could add security only if breaking the original RSA is not as hard as factoring. Even in this case, it is not clear whether the extended algorithm would increase security. Finally, the randomness property of the Gaussian integer exponentiation is utilized to derive a novel algorithm to rearrange the image pixels to be used for image watermarking. The new algorithm is more efficient than the one currently used and it provides a degree of cryptoimmunity. The proposed method can be used to enhance most picture watermarking algorithms

On Near-Linear-Time Algorithms for Dense Subset Sum

In the Subset Sum problem we are given a set of $n$ positive integers $X$ and a target $t$ and are asked whether some subset of $X$ sums to $t$. Natural parameters for this problem that have been studied in the literature are $n$ and $t$ as well as the maximum input number $\rm{mx}_X$ and the sum of all input numbers $\Sigma_X$. In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in $n$. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time $n^{O(1)}$. Our main question is: When can dense Subset Sum be solved in near-linear time $\tilde{O}(n)$? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters $n,t,\rm{mx}_X,\Sigma_X$ for which dense Subset Sum is in time $\tilde{O}(n)$. For notational convenience we assume without loss of generality that $t \ge \rm{mx}_X$ (as larger numbers can be ignored) and $t \le \Sigma_X/2$ (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time $\tilde{O}(n)$ if $t \gg \rm{mx}_X \Sigma_X/n^2$. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with $t \ll \rm{mx}_X \Sigma_X/n^2$, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds