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Cryptography Number Theory Divisibility and Primes Modular Number Theory Arithmetic Fermat’s and Euler’s Theorems Discrete Cryptography Logarithms Computationally Hard Problems School of Engineering and Technology CQUniversity Australia Prepared by Steven Gordon on 19 Feb 2020, number.tex, r1791 1/67

Cryptography Contents Number Theory Divisibility and Primes Divisibility and Primes Modular Arithmetic Fermat’s and Euler’s Theorems Modular Arithmetic Discrete Logarithms Computationally Hard Problems Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 2/67

Cryptography Divides (definition) Number Theory b divides a if a = mb for some m , where a , b and m are Divisibility and integers. We can also say b is a divisor of a , or b | a . Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 3/67

Cryptography Divides (example) Number Theory 3 divides 12, since 12 = 4 × 3. Also, 3 is a divisor of 12, or Divisibility and 3 | 12. Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 4/67

Cryptography Greatest Common Divisor (definition) Number Theory gcd( a , b ) returns the greatest common divisor of integers a Divisibility and and b . There are efficient algorithms for finding the gcd, i.e. Primes Euclidean algorithm. Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 5/67

Cryptography Greatest Common Divisor (example) Number Theory gcd(12 , 20) = 4, since the divisors of 12 are (1, 2, 3, 4 , 6, Divisibility and 12) and the divisors of 20 are (1, 2, 4 , 5, 10, 20). Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 6/67

Cryptography Relatively Prime (definition) Number Theory Two integers, a and b , are relatively prime if gcd( a , b ) = 1. Divisibility and Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 7/67

Cryptography Relatively Prime (example) Number Theory gcd(7 , 12) = 1, since the divisors of 7 are (1, 7) and the Divisibility and divisors of 12 are (1, 2, 3, 4, 6, 12). Therefore 7 and 12 are Primes relatively prime to each other. Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 8/67

Cryptography Relatively Prime (exercise) Number Theory How many positive integers less than 10 are relatively prime Divisibility and with 10? Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 9/67

Cryptography Prime Number (definition) Number Theory An integer p > 1 is a prime number if and only if its only Divisibility and divisors are +1, − 1, + p and − p . Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 10/67

Cryptography Prime Number (example) Number Theory The divisors of 13 are (1, 13), that is, 1 and itself. Therefore Divisibility and 13 is a prime number. The divisors of 15 are (1, 3, 5, 15). Primes Since the divisors include numbers other than 1 and itself, Modular Arithmetic 15 is not prime. Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 11/67

Cryptography Primes Under 2000 Number Theory Divisibility and Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems Credit: Table 1.4 in Stallings, Cryptography and Network Security , 5th Ed., Pearson 2011 12/67

Cryptography Prime Factors (definition) Number Theory Any integer a > 1 can be factored as: Divisibility and Primes a = p a 1 2 × p a 2 2 × · · · × p a t Modular t Arithmetic Fermat’s and where p 1 < p 2 < . . . < p t are prime numbers and where Euler’s Theorems each a i is a positive integer Discrete Logarithms Computationally Hard Problems 13/67

Cryptography Prime Factors (example) Number Theory The following are examples of integers expressed as prime Divisibility and factors: Primes Modular Arithmetic 13 = 13 1 Fermat’s and 15 = 3 1 × 5 1 Euler’s Theorems Discrete 24 = 2 3 × 3 1 Logarithms 50 = 2 1 × 5 2 Computationally Hard Problems 560 = 2 4 × 5 1 × 7 1 2800 = 2 4 × 5 2 × 7 1 14/67

Cryptography Integers as Prime Factors (exercise) Number Theory Find the prime factors of 12870, 12936 and 30607. Divisibility and Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 15/67

Cryptography Prime Factorization Problem (definition) Number Theory There are no known efficient, non-quantum algorithms that Divisibility and can find the prime factors of a sufficiently large number. Primes Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 16/67

Cryptography Prime Factorization Problem (example) Number Theory RSA Challenge involved researchers attempting to factor Divisibility and large numbers. Largest number measured in number of bits Primes or decimal digits. Some records held over time are: Modular Arithmetic 1991: 330 bits or 100 digits Fermat’s and 2005: 640 bits or 193 digits Euler’s Theorems Discrete 2009: 768 bits or 232 digits Logarithms Equivalent of 2000 years on single core 2.2 GHz computer Computationally Hard Problems to factor 768 bit Current algorithms such as RSA rely on numbers of 1024, 2048 and even 4096 bits in length 17/67

Cryptography Euler’s Totient Function (definition) Number Theory Euler’s totient function, φ ( n ), is the number of positive Divisibility and integers less than n and relatively prime to n . Also written Primes as ϕ ( n ) or Tot( n ). Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 18/67

Cryptography Properties of Euler’s Totient (definition) Number Theory Several useful properties of Euler’s totient are: Divisibility and Primes φ (1) = 1 Modular Arithmetic Fermat’s and For prime p , φ ( p ) = p − 1 Euler’s Theorems Discrete For primes p and q , φ ( px × q ) = φ ( p ) × φ ( q ) = ( p − 1) × ( q − 1) Logarithms Computationally Hard Problems 19/67

Cryptography Euler’s Totient Function (example) Number Theory The integers relatively prime to 10, and less than 10, are: 1, Divisibility and 3, 7, 9. There are 4 such numbers. Therefore φ (10) = 4. Primes The integers relatively prime to 11, and less than 11, are: Modular Arithmetic 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. There are 10 such numbers. Fermat’s and Therefore φ (11) = 10. The property could also be used since Euler’s Theorems Discrete 11 is prime. Logarithms Since 7 is prime, φ (7) = 6. Computationally Hard Problems Since 77 = 7 × 11, then φ (77) = φ (7 × 11) = 6 × 10 = 60. 20/67

Cryptography Contents Number Theory Divisibility and Primes Divisibility and Primes Modular Arithmetic Fermat’s and Euler’s Theorems Modular Arithmetic Discrete Logarithms Computationally Hard Problems Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 21/67

Cryptography Modular arithmetic simple (definition) Number Theory Modular arithmetic is similar to normal arithmetic (addition, Divisibility and subtraction, multiplication, division) but the answers “wrap Primes around”. Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 22/67

Cryptography mod operator (definition) Number Theory If a is an integer and n is a positive integer, then a mod n is Divisibility and defined as the remainder when a is divided by n . n is called Primes the modulus . Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 23/67

Cryptography mod operator (example) Number Theory The following are several examples of mod: Divisibility and Primes 3 mod 7 = 3, since 0 × 7 + 3 = 3 Modular Arithmetic Fermat’s and 9 mod 7 = 2, since 1 × 7 + 2 = 9 Euler’s Theorems Discrete 10 mod 7 = 3, since 1 × 7 + 3 = 10 Logarithms Computationally ( − 3) mod 7 = 4, since ( − 1) × 7 + 4 = − 3 Hard Problems 24/67

Cryptography Congruent modulo (definition) Number Theory Two integers a and b are congruent modulo n if Divisibility and ( a mod n ) = ( b mod n ). The congruence relation is written Primes as: Modular Arithmetic a ≡ b (mod n ) Fermat’s and When the modulus is known from the context, it may be Euler’s Theorems Discrete written simply as a ≡ b . Logarithms Computationally Hard Problems 25/67

Cryptography Congruent modulo (example) Number Theory The following are examples of congruence: Divisibility and Primes 3 ≡ 10 (mod 7) Modular Arithmetic Fermat’s and 14 ≡ 4 (mod 10) Euler’s Theorems Discrete 3 ≡ 11 (mod 8) Logarithms Computationally Hard Problems 26/67

Cryptography Modular arithmetic (definition) Number Theory Modular arithmetic with modulus n performs arithmetic Divisibility and operations within the confines of set Primes Z n = { 0 , 1 , 2 , . . . , ( n − 1) } . Modular Arithmetic Fermat’s and Euler’s Theorems Discrete Logarithms Computationally Hard Problems 27/67