Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Modular Arithmetic
- 3 The Addition Cypher, an Insecure Block Cypher
- 4 Functions
- 5 Probability Theory
- 6 Perfect Secrecy and Perfectly Secure Cryptosystems
- 7 Number Theory
- 8 Euclid's Algorithm
- 9 Some Uses of Perfect Secrecy
- 10 Computational Problems, Easy and Hard
- 11 Modular Exponentiation, Modular Logarithm, and One-Way Functions
- 12 Diffie and Hellman's Exponential-Key-Agreement Protocol
- 13 Computationally Secure Single-Key Cryptosystems
- 14 Public-Key Cryptosystems and Digital Signatures
- Further Reading
- Index
7 - Number Theory
Published online by Cambridge University Press: 05 July 2014
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Modular Arithmetic
- 3 The Addition Cypher, an Insecure Block Cypher
- 4 Functions
- 5 Probability Theory
- 6 Perfect Secrecy and Perfectly Secure Cryptosystems
- 7 Number Theory
- 8 Euclid's Algorithm
- 9 Some Uses of Perfect Secrecy
- 10 Computational Problems, Easy and Hard
- 11 Modular Exponentiation, Modular Logarithm, and One-Way Functions
- 12 Diffie and Hellman's Exponential-Key-Agreement Protocol
- 13 Computationally Secure Single-Key Cryptosystems
- 14 Public-Key Cryptosystems and Digital Signatures
- Further Reading
- Index
Summary
Divisibility
We say an integer b evenly divides another integer c if c/b is a whole number. Actually, nobody in mathematics ever says that b “evenly divides” c – people just say b “divides” c. Another way to say the same thing is to say that b is a divisor of c. The divisors of c are the numbers that (evenly) divide c. Finally, one can also say that b is divisible by c.
Examples:
• 3 divides 12.
• 3 is a divisor of 9.
• 40 is not a divisor of 20.
• 40 is divisible by 20.
• 4 divides 4.
• 5 is a divisor of −10.
• 12 divides 60.
• The positive divisors of 50 are 1, 2, 5, 10, 25, and 50.
Relative primality
Two numbers r and s are relatively prime if there is no integer bigger than 1 that is both a divisor of r and a divisor of s. We also say in this case that r is relatively prime to s. For example, 18 and 8 are not relatively prime because 2 is a divisor of both of them. On the other hand, 9 and 8 are relatively prime because the only divisors common to both of them are 1 and −1. We never count 1 and −1 as common divisors when determining relative primality.
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- Information
- A Cryptography PrimerSecrets and Promises, pp. 82 - 88Publisher: Cambridge University PressPrint publication year: 2014