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## Number Theory in the Spirit of Ramanujan

by: Elaina Osinski

43

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# Number Theory in the Spirit of Ramanujan MATH 497A

Marketplace > Pennsylvania State University > Mathematics (M) > MATH 497A > Number Theory in the Spirit of Ramanujan
Elaina Osinski
Penn State
GPA 3.88

Staff

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## Popular in Mathematics (M)

This 0 page Study Guide was uploaded by Elaina Osinski on Sunday November 1, 2015. The Study Guide belongs to MATH 497A at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/233012/math-497a-pennsylvania-state-university in Mathematics (M) at Pennsylvania State University.

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Date Created: 11/01/15
Math 497A Elliptic Curves and Applications to Cryptography Fall 2008 List of theoretical questions for the nal exam 1 Explain how to nd all rational points on a nondegenerate conic de ned over Q 2 State the LutZ Nagell Theorem and give an outline of its proof 3 State two different methods of computing the torsion group of an elliptic curve over Q and explain howwhy they work 4 Let E be an elliptic curve de ned over Q given by an equation with integer coef cients7 and let p be a prime Prove that the set Epk O U E EQ ordp S 72k70rdpy 3 73k forms a subgroup of 5 State and prove the Descent Theorem 6 State the Weak Mordell Weil Theorem and give an outline of its proof 7 Let E be an elliptic curve de ned over Q Explain how the index of 2EQ in EQ7 2EQ7 is related to the rank of the elliptic curve 8 State a theorem that allows you to compute the rank of elliptic curves of the form yz 3 azz bx Give an outline of its proof 9 State properties of the reduction map pp EQ a Ele7 and prove them 10 State Hasse7s theorem about the number of rational points on an elliptic curve de ned over a nite eld and give an outline of the proof ll De ne what the characteristic polynomial of the Frobenius en domorphism is and stateprove the theorem that tells you what its coef cients are 12 What is the Weil pairing on an elliptic curve What are the properties of the Weil pairing7 and what theorems can you prove using the Weil pairing 13 H 4 V H CH H CT H H H 00 19 D O V 21 Give two di erent de nitions ofthe Weil pairing Explain which of the two de nitions is more suitable for explicitly computing the pairing What is the Weil pairing on an elliptic curve Give at least two applications of the Weil pairing to cryptography and ex plain whyhow they work What are the underlying hardness assumptions in each case Explain Lenstra7s method for factoring integers that uses ellip tic curves In this context7 what can you say about elliptic curves rnodulo 71 when n is not necessarily prirne Explain how elliptic curves can be used for Prirnality Testing Explain why this works Explain a related prirnality test7 the Pocklington Lehrner test State a theorem that characterizes the n torsion on an elliptic curve You should have di erent cases depending on the char acteristic of the eld over which your curve is de ned Explain how to prove this theorem De ne what an endornorphisrn of an elliptic curve is7 and de ne the endornorphisrn ring of an elliptic curve De ne what the degree of an endornorphisrn is7 and what it means for an endornorphisrn to be separable Now let 04 be a separable en domorphism of an elliptic curve E de ned over a nite eld E Stateprove a theorem that lets you relate the degree of Oz to a determinant calculation Let E be an elliptic curve de ned over Q Prove that Z EQ for n 2 3 Let E be an elliptic curve de ned over a nite eld K Let n be an integer that is coprirne to the characteristic of K Let T1T2 be a basis for the n torsion on E State what proper ties the Weil pairing en has and use them that enT1T2 is a primitive n th root of unity Let E be an elliptic curve de ned over Q and let P be a point on E State several equivalent criteria that hold if and only if P is a point of order 3 on E Prove your statement

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