Number_Theory

Ø Primes numbers

A positive integer p is prime iff it is evenly divisible by exactly two positive integers: itself and 1.

· The single digits primes are 4 numbers:

2 3 5 7

· The two digits primes are 21 numbers:

11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

· The three digits primes are 143 numbers:

101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

· A list of the 10,000 smallest prime numbers is available at:

http://www.math.utah.edu/~pa/math/p10000.html

· There are infinitely many primes.

· Primes do thin out as numbers get bigger:

o The density of primes < 2 digit numbers is 1 / 4.

o The density of primes < 3 digit numbers is 1 / 6.

o The density of primes < 10 digit numbers is 1 / 23.

o The density of primes < 100 digit numbers is 1 / 230.

In general, the density is inversely proportional to their length in digits.

· In RSA a common size prime is 100 digit and thus a random 100 digit has a chance of 1 in 230 of being prime.

i.e., on average it will be necessary to select 230 hundred digit numbers before a prime is found.

Ø Euclid’s Algorithm

Finding the multiplicative inverse m^{-1 of} m (mod n) ,

where m and n are relatively primes (the GCD of m and n is 1).

Initial Step:

r_{-2 =} m, u_-2 = 1, v_-2 = 0

r_{-1 =} n, u_-1 = 0, v_-1 = 1

i=0

Step i:

If r_{i-1 =} 1 then u_i-1 is the multiplicative inverse m^-1 of m (mod n).

Otherwise:

Divide r_i-2 by r_i-1 to get quotient qi and remainder r_i

Set u_i = u_i-2 - q_i u_i-1 & v_i = v_i-2 - q_i v_i-1

Set i = i + 1

Repeat

Example 1:

m=7, n = 10

i	qi	ri	ui	vi
-2		7	1	0
-1		10	0	1
0	0	7	1	0
1	1	3	-1	1
2	2	1	3	-2

Since r₂ is 1 then m^-1 is 3

Example 2:

m=7, n = 15

i	qi	ri	ui	vi
-2		7	1	0
-1		15	0	1
0	0	7	1	0
1	2	1	-2	1

Since r₁ is 1 then m^-1 is -2 mod 15 = 13

Example 3:

m=797, n = 1047

i	qi	ri	ui	vi
-2		797	1	0
-1		1047	0	1
0	0	797	1	0
1	1	250	-1	1
2	3	47	4	-3
3	5	15	-21	16
4	3	2	67	-51
5	7	1	-490	373

Since r₅ is 1 then m^-1 is -490 mod 1047 = 557

Ø The Chinese Remainder Theorem (CRT)

If p and q are relatively prime, then any number x has two equivalent representations:

Standard: x mod pq

Decomposed: a=x mod p & b=x mod q

To covert from decomposed to standard we use Garner’s Formula:

x = (((a-b)( q^-1 mod p)) mod p).q + b

The CRT can be used to speed up the computation is RSA and other cryptography applications.

Ø Euler’s Theorem:

§ Z_n*

Z_n = {i >=0: i <n}

Example: Z₁₀ = {0,1,2,3,4,5,6,7,8,9}

Z_n* = {i > 0: i <n is relatively prime to n}

Example: Z₁₀* = {1,3,7,9}

ü If you multiply any two numbers in Z₁₀* you get another number in Z₁₀*

ü Each raw & column in Z₁₀* multiplication table contains all elements of Z₁₀* with no repeats.

ü Z_n* is closed under multiplication mod n.

	1	3	7	9
1	1	3	7	9
3	3	9	1	7
7	7	1	9	3
9	9	7	3	1

Z₁₀* Multiplication table

§ Euler’s Totient Function

Ø(n) (called Euler’s Totient Function) is the number of elements in Z_n*.

Example: Ø(10) = 4.

· If n is prime, then Ø(n) = n-1

· If n = pq where p and q are relatively prime, then

Ø(n) = Ø(p) Ø(q)

· If n= pq where p and q are primes, then Ø(n) = (p-1)(q-1).

Example: 10 = 2.5 and thus Ø(10) = (2-1).(5-1)=1.4=4

§ Euler’s Theorem:

For all a in Z_n*: a^Ø(n) = 1 mod n

§ Generalized Euler’s Theorem:

If n = pq where p and q are distinct primes, then

For all a in Z_n (not just Z_n*): a^kØ(n)+1 = a mod n