fermat and euler theorems

7/24/2019 Fermat and Euler Theorems

1/14

Fermat and Eulers Theorems

Presentation by Chris Simons


2/14

Prime Numbers

A prime number is divisible only by 1 and

itself

For example: !" #" $" %" 11" 1#" 1%" &' 1 (ould also be (onsidered prime" but its

not very useful)


3/14

Prime Fa(tori*ation

To fa(tor a number nis to +rite it as a

produ(t of other numbers)

n = a * b * c,r" 1-- . $ / $ / ! / !

Prime fa(tori*ation of a number nis +ritin0

it as a produ(t of prime numbers) 1# . 11 / 1#


4/14

2elatively Prime Numbers

T+o numbers are relatively prime if they have no

(ommon divisors other than 1)

1- and !1 are relatively prime" in respe(t to ea(h

other" as 1- has fa(tors of 1" !" $" 1- and !1 hasfa(tors of 1" #" %" !1)

The 3reatest Common 4ivisor 53C46 of t+o

relatively prime numbers (an be determined by(omparin0 their prime fa(tori*ations and sele(tin0

the least po+ers)


5/14

2elatively Prime Numbers Cont)

For example" 1!$ . $#and !-- . !#/ $!

3C451!$" !--6 . !-/ $!. !$

7f the t+o numbers are relatively prime the 3C4

+ill be 1) Consider the follo+in0: 1-51" !" $" 1-6 and !151"

#" %" !16

3C451-" !16 . 1

7t then follo+s" that a prime number is alsorelatively prime to any other number other thanitself and 1)


6/14

A 8ittle 9it ,f istory

Pierre de Fermat 51;-1>? that thee@uation xn yn. *nhas no non


7/14

istory Cont)

,ne of Fermats boo=s (ontained a

hand+ritten note in the mar0in de(larin0

that he had a proof for this e@uation" but it

+ould not fit in the mar0in) e never

published his proof" nor +as it found after

his death) 7n 1>> Andre+ iles +or=ed

out a proof of this e@uation usin0 advan(edmodern te(hni@ues)


8/14

Fermats 8ittle Theorem

7fpis prime and ais an inte0er not divisible by

p" then ) ) )

ap-1 15modp6)

And for every inte0er a

ap a5modp6)

This theorem is useful in publi( =ey 52SA6 and

primality testin0)


9/14

Euler Totient Fun(tion: 5n6

5n6 . ho+ many numbers there are

bet+een 1 and n


10/14

Euler Totient Fun(tion Cont)

As you (an see from 5$6 and 5%6" 5n6 +ill

be n


11/14

Eulers Totient Theorem

This theorem 0enerali*es Fermats theorem and is

an important =ey to the 2SA al0orithm)

7f 3C45a"p6 . 1" and a Dp" then

a5p6 15modp6) 7n other +ords" 7f aandpare relatively prime"

+ith abein0 the smaller inte0er" then +hen +e

multiply a+ith itself 5p6 times and divide the

result byp" the remainder +ill be 1)


12/14

Eulers Totient Theorem Cont)

8ets test the theorem:

7f a. $ andp. ;

Then 5;6 . 5!


13/14

,=aaaay ) ) ) So hat

Eulers theorem uses modulus arithmeti(+hi(h helps to lay the foundation for 2SAen(ryption) To (onstru(t a personal (ipher

=ey +e need an appropriate value +e +ill(all variableR) So" +e sele(t t+o very lar0eprime numbers Uand Vand multiply them)

.B 5R6 . 5U


14/14

So hat Cont)

e also define the variablesP and Q) P is an arbitrarynumber that is relatively prime to 5R). Qis the(al(ulated inverse ofPin 5mod 5R66)

e usePandRto (reate a publi( =ey" and QandRto

(reate a private =ey) This yieldsP/Q 15mod 5R6 6)

The result is that too mu(h information is lost in theen(ryption due to the modulus arithmeti( to de(ipher a

privately en(rypted 2SA messa0e +ithout the use of thepubli( =ey) Gnless the +ould

fermat and euler theorems

Documents