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The Quadratic Sieve Factoring Algorithm
, 2001
"... Mathematicians have been attempting to find better and faster ways to factor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factorization. This trial division was not improved upon until Fermat applied the ..."
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Mathematicians have been attempting to find better and faster ways to factor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factorization. This trial division was not improved upon until Fermat applied the
Three Connections to Continued Fractions
, 2003
"... It is often the case that seemingly unrelated parts of mathematics turn out to have unexpected connections. In this paper, we explore three puzzles and see how they are related to continued fractions, an area of mathematics with a distinguished history within the world of number theory. ..."
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It is often the case that seemingly unrelated parts of mathematics turn out to have unexpected connections. In this paper, we explore three puzzles and see how they are related to continued fractions, an area of mathematics with a distinguished history within the world of number theory.
An Accurate and Efficient TwoStage Channel Estimation Method Utilizing Training Sequences with Closed Form Expressions
"... Abstract—A novel twostage frequency domain channel estimation method especially suitable for the estimation of long channels such as ultra wide band channels is proposed. The proposed method can efficiently use the sequences with closed form analytical expressions such as the Legendre sequences. ( ..."
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Abstract—A novel twostage frequency domain channel estimation method especially suitable for the estimation of long channels such as ultra wide band channels is proposed. The proposed method can efficiently use the sequences with closed form analytical expressions such as the Legendre sequences. (The suggested method does not require a computationally intense search for good training sequences which is infeasible for long training sequences.) The method is shown to present a minor improvement in the total estimation error variance when compared with the conventional single stage frequency domain channel estimation. In addition, the proposed method has a very efficient time domain implementation requiring at most 2
Comparison study for Primality testing using Mathematica
"... Abstract: In this paper, we present four primality testings. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests are also discussed and this leads to the concept of pseudoprime. We also discussed some resul ..."
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Abstract: In this paper, we present four primality testings. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests are also discussed and this leads to the concept of pseudoprime. We also discussed some results on pseudoprimes with certain range and do quantitative comparison and in addition to this, we discussed the existence of carmichael numbers and Mersenne primes. 1
Mobile SMS Banking Security Using Elliptic Curve
"... Over the past many years several leading banks in India have launched SMS banking services. However, the security of mobile SMS payment and banking has topped the list of concerns for most of the customers. In this paper, I investigate the security loopholes in SMS banking and propose a system to ma ..."
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Over the past many years several leading banks in India have launched SMS banking services. However, the security of mobile SMS payment and banking has topped the list of concerns for most of the customers. In this paper, I investigate the security loopholes in SMS banking and propose a system to make mobile SMS banking secure using Elliptic Curve Cryptosystem(ECC). Also, another aim is to design an API to implement ECC algorithm. Key words:
1. The World’s Oldest Algorithm. Arithmetic and numbers have always fascinated
"... me. Even in the first grade, my friend Coleman and I used to race each other to see who could finish our pages of addition and subtraction and get the most answers right. (I usually won, but he’s now president of a very large jewelry store. Oh, well....) In third grade, my friend Ron and I decided t ..."
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me. Even in the first grade, my friend Coleman and I used to race each other to see who could finish our pages of addition and subtraction and get the most answers right. (I usually won, but he’s now president of a very large jewelry store. Oh, well....) In third grade, my friend Ron and I decided to write down all the Roman numerals up to 1000, just for the fun of it. And after long division in the fourth grade, I stood atop Arithmetic with no new worlds to conquer. Or so I thought. Right at the end of the fifth grade, we were told to read the Iliad and the Odyssey over the summer, and that in sixth grade math, Mrs. Garrison was going to teach us how to take square roots by hand. “Great! ” I said. But what was a square root? I soon found out. We children who grew up in New Orleans and who rode on truck floats on Mardi Gras Day all knew about perfect squares. A gross was a square, namely 12 times 12. You bought carnival throws and beads by the gross, and every kid knew that a gross was a dozen dozen, or 12 times 12, or 144. We soon learned that a square root was a number, such as 12, that you multiplied by
Juxtaposition of RSA and Elliptic Curve Cryptosystem
, 2009
"... This paper presents an indepth juxtaposition of RSA and Elliptic Curve Cryptosystem (ECC) and provides an overview of the different tradeoff involved in choosing between cryptosystems based on them. We offer ECC as a suitable alternative to RSA. We also present experimental results quantifying the ..."
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This paper presents an indepth juxtaposition of RSA and Elliptic Curve Cryptosystem (ECC) and provides an overview of the different tradeoff involved in choosing between cryptosystems based on them. We offer ECC as a suitable alternative to RSA. We also present experimental results quantifying the benefits of using ECC for public cryptosystems. Key words:
Elliptic Curve Cryptography
, 1999
"... Elliptic curves and the elliptic curve discrete logarithm problem have been used in cryptographic systems for the last 12 years. These applications were independently realized by Neal Koblitz [4] and Victor Miller [9]. Part 1 describes elliptic curves as mathematical structures and provide the n ..."
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Elliptic curves and the elliptic curve discrete logarithm problem have been used in cryptographic systems for the last 12 years. These applications were independently realized by Neal Koblitz [4] and Victor Miller [9]. Part 1 describes elliptic curves as mathematical structures and provide the necessary background to understand the cryptosystems that utilize them [5] [6]. Part 2 presents the elliptic curve analogs of various public key cryptosystems [5]. Part 3 covers the elliptic curve discrete logarithm problem (ECDLP) and presents algorithms for solving it [13] [7] [3] [10]. The difficulty of this problem is compared to other commonly used hard problems. Part 4 examines the order of an elliptic curve [11]. 1 Math An elliptic curve over any field F is the set of (x; y) with x; y 2 F satisfying the general equation y 2 + axy + by = x 3 + cx 2 + dx + e (1) and an identity element O ("point at infinity"); the cubic on the right must have 3 distinct roots. When the fie...
Implementing PublicKey Cryptography in
, 2001
"... In this paper we describe how the RSA publickey cipher can be implemented in the functional programming language Haskell and we present a complete working implementation. This implementation serves two useful pedagogical purposes: (1) it is a very clear, highlevel implementation of RSA that can e ..."
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In this paper we describe how the RSA publickey cipher can be implemented in the functional programming language Haskell and we present a complete working implementation. This implementation serves two useful pedagogical purposes: (1) it is a very clear, highlevel implementation of RSA that can easily be understood and manipulated by students of cryptography, and (2) it is an excellent example of how a functional programming language such as Haskell can be used to implement real applications. While the main goal of our implementation is clarity rather than efficiency, no treatment of publickey cryptography can ignore efficiency and we show how wellknown techniques for improving the efficiency of RSA can be implemented. We also compare the performance of our implementation with an equivalent implementation in Java, and show that it is also practical to implement publickey cryptography in Haskell.