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1	Sequences and Summation and Product of Series Lin, Yi-Ping 23 June 2010 (has links) This paper investigates four important methods of solving summation and product problems in mathematics competitions. Chapter 1 presents the basic concepts of sequence and series, including arithmetic sequence (series), geometric sequence (series) and infinite geometric sequence (series). Chapter 2 handles the binomial coefficients and binomial theorem and show they how can be applied to compute series sum. Chapter 3 deals with power series, including interchanging summation and differentiation; interchanging summation and integration; and generating function which expresses a sequence as coefficients arising from a power series in variables. Chapter 4 provides four methods of telescoping sum, including antidifference, partial fractions, trigonometric functions, and factorial functions. Chapter 5 discusses the telescoping product which the main ideas and techniques are analogous to telescoping sum. Two types of telescoping product including difference of two squares and trigonometric functions are investigated. trigonometric functions telescoping sum telescoping product Taylor's theorem power series partial fraction series sequence Newton theorem geometric sequence (series) arithmetic sequence (series) generating function falling factorial binomial theorem antidifference
2	Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem Yoshida, Kayo January 2009 (has links) The Boneh-Boyen signature scheme is a short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman (SDH) assumption. The primary objective of this thesis is to examine the relationship between the Boneh-Boyen signature scheme and SDH. The secondary objective is to survey surrounding topics such as the generic group model, related signature schemes, intractability assumptions, and the relationship to identity-based encryption (IBE) schemes. Along these lines, we analyze the plausibility of the SDH assumption using the generic bilinear group model. We present the security proofs for the Boneh-Boyen signature scheme, with the addition of a small improvement in one of the probability bounds. Our main contribution is to give the reduction in the reverse direction; that is, to show that if the SDH problem can be solved then the Boneh-Boyen signature scheme can be forged. This contribution represents the first known proof of equivalence between the SDH problem and Boneh-Boyen signatures. We also discuss the algorithm of Cheon for solving the SDH problem. We analyze the implications of Cheon's algorithm for the security of the Boneh-Boyen signature scheme, accompanied by a brief discussion on how to counter the attack. cryptography digital signatures signature scheme Diffie-Hellman Strong Diffie-Hellman SDH Boneh-Boyen signatures Pollard rho Pollard lambda baby-step giant-step assumptions short signatures random oracle generic group BLS Waters Okamoto identity-based encryption IBE Cheon's algorithm partial fraction security existential forgery Combinatorics and Optimization
3	Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem Yoshida, Kayo January 2009 (has links) The Boneh-Boyen signature scheme is a short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman (SDH) assumption. The primary objective of this thesis is to examine the relationship between the Boneh-Boyen signature scheme and SDH. The secondary objective is to survey surrounding topics such as the generic group model, related signature schemes, intractability assumptions, and the relationship to identity-based encryption (IBE) schemes. Along these lines, we analyze the plausibility of the SDH assumption using the generic bilinear group model. We present the security proofs for the Boneh-Boyen signature scheme, with the addition of a small improvement in one of the probability bounds. Our main contribution is to give the reduction in the reverse direction; that is, to show that if the SDH problem can be solved then the Boneh-Boyen signature scheme can be forged. This contribution represents the first known proof of equivalence between the SDH problem and Boneh-Boyen signatures. We also discuss the algorithm of Cheon for solving the SDH problem. We analyze the implications of Cheon's algorithm for the security of the Boneh-Boyen signature scheme, accompanied by a brief discussion on how to counter the attack. cryptography digital signatures signature scheme Diffie-Hellman Strong Diffie-Hellman SDH Boneh-Boyen signatures Pollard rho Pollard lambda baby-step giant-step assumptions short signatures random oracle generic group BLS Waters Okamoto identity-based encryption IBE Cheon's algorithm partial fraction security existential forgery Combinatorics and Optimization
4	Applications of recurrence relation Chuang, Ching-hui 26 June 2007 (has links) Sequences often occur in many branches of applied mathematics. Recurrence relation is a powerful tool to characterize and study sequences. Some commonly used methods for solving recurrence relations will be investigated. Many examples with applications in algorithm, combination, algebra, analysis, probability, etc, will be discussed. Finally, some well-known contest problems related to recurrence relations will be addressed. characteristic equation characteristic root constant coefficients conjugate root distinct root Fibonacci numbers general solution homogeneous method of annihilator linear method of binary number system method of change of variable method of divide-and-conquer relation method of generating function multiple root method of logarithmic transformation nonhomogeneous nonlinear operator partial-fraction decomposition particular solution recurrence relation recurrence sequences

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