Fermat Theorem and Euler Theorem

版权声明：转载请附上文章地址 https://blog.csdn.net/weixin_38134491/article/details/85008052

Fermat Theorem

a^(p-1) = 1 mod p , where p is prime and gcd(a, p)=1

Also known as Fermat's litte Theorem 

Useful in public key and primality testing

Alternative form of Fermat's theorm:

a^p = a  mod p ，not require gcd(a,p)=1

example:   p=5, a=3  --->  3^5 = 3 mod 5

p=5,a=10-----> 10^5= 10 mod 5 

Euler Totient Function Ø(n)

扫描二维码关注公众号，回复： 4520076 查看本文章

Number of elements in reduced set of residues is called the Euler Tolient Function Ø(n)

By convention, Ø(1)=1 约定

when doing arithmetic modulo n:

• Complete set of residues is : 0....n-1
• Reduced set of residues is those numbers which are relatively prime to n 

for example:

n=10,

complete set of residues is {0123456789}

reduced set of residues is {1 3 7 9}

for a prime p ,   Ø(p) =p -1

for a composite n= p*q (p,q are prime)

 Ø(n)=  Ø(p*q )=  Ø(p) Ø(q)= (p-1)(q-1)

怎么求一个大数的欧拉函数值？？？？？方法如下，拆分

example:  Ø(37)= 36

Ø(21)=Ø(3*7)=Ø(3)Ø(7)=2*6=12

Euler's Theorem 

A generalisation 归纳  of Fermat Theorem

a^Ø(n)= 1 mod n,   for any a,n  where gcd(a,n)=1

Alternative form:  a^(Ø(n)+1) = a(mod n) , not require gcd(a,n)=1