fermat and euler theorems
TRANSCRIPT
-
7/24/2019 Fermat and Euler Theorems
1/14
Fermat and Eulers Theorems
Presentation by Chris Simons
-
7/24/2019 Fermat and Euler Theorems
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)
-
7/24/2019 Fermat and Euler Theorems
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#
-
7/24/2019 Fermat and Euler Theorems
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" !" $" 1- and !1 hasfa(tors of 1" #" %" !1)
The 3reatest Common 4ivisor 53C46 of t+o
relatively prime numbers (an be determined by(omparin0 their prime fa(tori*ations and sele(tin0
the least po+ers)
-
7/24/2019 Fermat and Euler Theorems
5/14
2elatively Prime Numbers Cont)
For example" 1!$ . $#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)
-
7/24/2019 Fermat and Euler Theorems
6/14
A 8ittle 9it ,f istory
Pierre de Fermat 51;-1>? that thee@uation xn yn. *nhas no non
-
7/24/2019 Fermat and Euler Theorems
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)
-
7/24/2019 Fermat and Euler Theorems
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)
-
7/24/2019 Fermat and Euler Theorems
9/14
Euler Totient Fun(tion: 5n6
5n6 . ho+ many numbers there are
bet+een 1 and n
-
7/24/2019 Fermat and Euler Theorems
10/14
Euler Totient Fun(tion Cont)
As you (an see from 5$6 and 5%6" 5n6 +ill
be n
-
7/24/2019 Fermat and Euler Theorems
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)
-
7/24/2019 Fermat and Euler Theorems
12/14
Eulers Totient Theorem Cont)
8ets test the theorem:
7f a. $ andp. ;
Then 5;6 . 5!
-
7/24/2019 Fermat and Euler Theorems
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
-
7/24/2019 Fermat and Euler Theorems
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