Prime Numbers,
Cryptography
and Network
Security
José Bermúdez, Ph.D.

Cryptography
Alan Turing:
 Born: 23 June 1912 · Maida Vale, London, England
 Died: 7 June 1954 (aged 41) · Wilmslow, Cheshire, England
 Cause of death Cyanide poisoning
Father of Modern Computer.
Father of Artificial Intelligence.
Father of many things.

Outcomes:
Upon the completion of this session, the learners will be able to:
 Understand about prime numbers and composite numbers.
 Know some facts about prime numbers.
 Know the role to prime numbers in cryptography.

Prime Numbers
 Have exactly two divisors.
 If "X" is prime, then the divisor are 1 and "X".
 All numbers have prime factors
Example:
Numbers 2 3 4 5 6 7 8 9
Prime
Factorization
1¹x2¹ 3¹x1¹ 2² 5¹x1¹ 2¹x3¹ 7¹x1¹ 2³x1¹ 3²x1¹
Prime Numbers 2 3 2 5 2, 3 7 2 3

Prime Numbers
 Have exactly two divisors.
 If "X" is prime, then the divisor are 1 and "X".
 All numbers have prime factors
Example:
Numbers 2 3 4 5 6 7 8 9
Prime
Factorization
1¹x2¹ 3¹x1¹ 2² 5¹x1¹ 2¹x3¹ 7¹x1¹ 2³x1¹ 3²x1¹
Prime Numbers 2 3 2 5 2, 3 7 2 3
Prime
Numbers
Composite
Numbers

Prime Numbers
 A prime number is a number greater than 1 with only two factors, itself and one.
 It cannot be divided further by any other number without leaving a remainder.

Why prime numbers
are important for
cryptography
 Many encryption algorithms are based on
prime numbers.
 Very fast to multiply two large prime numbers.
 Extreme computer-intensive to do the reverse
(found the prime numbers)
 Factoring very large numbers is very hard.
86331288713595159341392743491288713595
1359583051879012887

Prime Numbers
Conjectures
 Goldbach's conjecture
 Twin prime conjecture
 Andrica's conjecture,
 Brocard's conjecture,
 Legendre's conjecture,
 Oppermann's conjecture
 Cramér conjecture
 ...

Summary:
 Learned about prime numbers and composite numbers.
 Know some facts about prime numbers.
 Know the role to prime numbers in cryptography.

Prime Numbers,
Cryptography
and Network
Security
José Bermúdez, Ph.D.