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Integers, Primes, and some
Cryptography
Rosen 3.4, 3.5, 3.
Integers
Integer Division
If a and b are integers with a ≠ 0 , we say that a divides b if there is an integer c such that b = ac. When a divides b we say that a is a factor of b and b is a multiple of a. The notation a | b denotes that a divides b. Examples: Does 5 | 10? Does 5 | 15? Does 2 | 5?
Division Theorem
Theorem : Let a , b , and c be integers. Then
I. if a | b and a | c , then a | (b + c) ;
II. if a | b , then a | bc for all integers c ;
III. if a | b and b | c , then a | c.
Division Theorem
Corollary : If a , b , and c be integers such
that a | b and a | c , then a | mb + nc
whenever m and n are integers.
Integer division
The Division Algorithm Let a be an integer and d a positive integer. Then there are unique integers q and r , with 0 ≤ r < d , such that a = dq + r. d is called the divisor , q is called the quotient , q = a div d r is called the remainder. r = a mod d (a % d in Java) Examples 101 = 11 * 9 + 2. 101 = 11 * 8 + 13? -11 = 3 (-4) + 1. -11 = 3 (-3) – 2
Integer division Examples : Given a = 13 and d = 4 , what are the values of q and r? Since 13/4 = 3 + 1/4 , we know that q = 13 div 4 = 3 and r = 13 mod 4 = 1. Given a = 96 and d = 15 , what are the values of q and r? Since 97/15 = 6 + 7/15 , we know that q = 97 div 15 = 6 and r = 97 mod 15 = 7 Modular Arithemetic If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a-b Notation: a ≡ b (mod m ) Example: Is 17 congruent to 5 modulo 6? As 6 divides 17-5, they are congruent Example: Is 24 congruent to 14 modulo 6? As 6 does not divide 24-14 = 10, they are not congruent distinguish between b (mod m) and b mod m Modular Arithmetic (cont) Theorem : Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m ) if and only if a mod m = b mod m a % m = b % m (in Java notation) Example: Is 17 congruent to 5 modulo 6? Rephrased: does 17 ≡ 5 (mod 6)? 17 mod 6 = 5 mod 6 Example: Is 24 congruent to 14 modulo 6? Rephrased: 24 ≡ 14 (mod 6) 24 mod 6 ≠ 14 mod 6 Modular Arithmetic (cont)
Let m be a positive integer. The integers a
and b are congruent modulo m if and only if
there is an integer k such that a = b + km
Example:
17 and 5 are congruent modulo 6
Applications of modular arithmetic: Hash functions
Hash functions provides fast access to data
of the form (key, value). Common hash
function:
h(k) = k mod m
image from http://en.wikipedia.org/wiki/Image:HASHTB08.svg Pseudorandom Numbers A linear congruential methods uses the following rule: xn+1 = (a xn + c) mod m m – modulus a – multiplier c – increment x 0 – seed Example: m=9, a=7, c=4, x 0 =3 generates 3,7,8,6,1,2,0,4,5,3,7,8,6,1,2,0,4,5,3…
Example
Is 899 prime or composite? Solution Divide 899 by successively larger primes < √899 , starting with 2 We find that 29 and 31 divide 899 FUN FACT: On a unix system, enter “factor 899”
factor 899 899: 29 31
Greatest Common Divisors
and Least Common Multiples
Greatest Common Divisor (GCD)
The greatest common divisor of two integers a and b is the largest integer d such that d | a and d | b Denoted by gcd(a,b) Examples gcd (24, 36) = 12 gcd (17, 22) = 1 gcd (100, 17) = 1 Easy way to find gcd: Find all positive common divisors of both integers then take the largest
Relatively Prime
Two numbers are relatively prime if they don’t
have any common factors
(other than 1)
Rephrased: a and b are relatively prime if gcd (a,b) = 1
gcd (25, 39) = 1,
so 25 and 39 are relatively prime
Least Common Multiple
The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. Denoted by lcm ( a , b ) Example: lcm(10, 25) = 50 What is lcm (95256, 432)? 95256 = 2^33572 , 432=2^433 lcm (2^33572 , 2^433 ) = 2max(3,4) 3 max(5,3) 7 max(2,0)^ = 2^43572 = 190512 And in general: Theorem: Let a and b be positive integers. Then ab = gcd(a,b) * lcm(a,b) Example: gcd (10,25) = 5, lcm (10,25) = 50 1025 = 5
An Application of Number
Theory: The RSA
Cryptosystem
Ceasar Cipher revisited
You encrypt using c = (p + k) mod 26 k is the encryption key and decrypt by shifting by -k: p = (c – k ) mod 26 i.e. -k is the decryption key If you know the encryption key you know the decryption key Problem: Need a safe way to send the encryption key!
Public key cryptography
Eliminate the need for securely sending
the encryption key: use a public key!
I generate public and private keys I publish my public key You encode a message into numbers and encrypt them Only I, who know the private key, can decrypt it
Public key cryptography
We use such technology on a daily basis
whenever we visit a secure website
(https:… address)
RSA encryption (Rivest, Shamir, Adleman)
uses ideas from number theory to make all
that possible.
RSA encryption
The encryption process:
C = Me^ mod n
M – integer representing the plaintext C – the encrypted message n – a product of two large prime numbers p,q e – a number which is relatively prime to (p-1)(q-1) Public key: (e, n)
RSA decryption
How to decrypt a message:
M = Cd^ mod n
n = pq, where p,q are prime d – can be computed knowing e,p,q
Private key: (d, n)
If you could factor n you could find out the private key! But it turns out that it’s practically impossible to factor very large numbers
