stein w. elementary number theory and elliptic curves (web draft, sept. 2004)(182s)_mt_

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1、ThisispageiPrinter:OpaquethisElementaryNumberTheoryWilliamSteinSeptember2004iiTomystudentsandmywife,ClaritaLefthand.ThisispageiiiPrinter:OpaquethisContentsPreface31PrimeNumbers51.1PrimeFactorization......................51.2TheSequenceofPrimeNumbers...............131.3Exercises...........

2、.................192TheRingofIntegersModulon212.1CongruencesModulon.....................212.2TheChineseRemainderTheorem...............272.3QuicklyComputingInversesandHugePowers........292.4FindingPrimes.........................332.5TheStructureof(Z/pZ)∗...................342.6Exercises...

3、.........................383Public-KeyCryptography433.1TheDiffie-HellmanKeyExchange..............463.2TheRSACryptosystem....................513.3AttackingRSA.........................543.4Exercises............................584QuadraticReciprocity594.1StatementoftheQuadraticReciprocityLaw..

4、......604.2Euler’sCriterion........................62Contents14.3FirstProofofQuadraticReciprocity.............634.4AProofofQuadraticReciprocityUsingGaussSums....684.5FindingSquareRoots.....................724.6Exercises............................745ContinuedFractions775.1FiniteContinuedF

5、ractions..................785.2InfiniteContinuedFractions..................835.3TheContinuedFractionofe..................885.4QuadraticIrrationals......................915.5RecognizingRationalNumbers................965.6SumsofTwoSquares.....................975.7Exercises...................

6、.........1006EllipticCurves1036.1TheGroupStructureonanEllipticCurve..........1046.2IntegerFactorizationUsingEllipticCurves.........1076.3EllipticCurveCryptography.................1126.4EllipticCurvesOvertheRationalNumbers.........1166.5Exercises............................1217Computationa

7、lNumberTheory1237.1PrimeNumbers.........................1257.2TheRingofIntegersModulon................1317.3Public-KeyCryptography...................1397.4QuadraticReciprocity.....................1457.5ContinuedFractions......................1487.6EllipticCurves....