Chapter 9: Integer Properties

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9.1 The Division Algorithm

Definition 9.1.1: Divides.

Proof.

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Quotients and remainders

Definition 9.1.2: Integer division definitions.

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9.2 Modular arithmetic

Application 9.2.1: Hash functions.

Application 9.2.2: Pseudo-random number generators.

Congruence mod m

Proof.

Precedence of the mod operation

Arithmetic expressions mod m

Proof.

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9.3 Prime factorizations

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Greatest common divisors and least common multiples

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9.4 Factoring and primality testing

Proof.

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The Prime Number Theorem

Proof.

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EXERCISE

9.5 Greatest common divisor and Euclid’s algorithm ©zyBooks 04/08/25 22:07 2 James Ding

Proof.

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The extended Euclidean algorithm

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EXERCISE

9.6 Number representation

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Hexadecimal numbers

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Converting decimal numbers to base b

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The number of digits required to represent a number

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EXERCISE

9.7 Fast exponentiation

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9.8 Introduction to cryptography

Messages as numbers

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Encryption and decryption functions

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9.9 The RSA cryptosystem

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Figure 9.9.1: Preparation of public and private keys in RSA.

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Proof.

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EXERCISE

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