On the basis of the number of secondary school on password

1. modular arithmetic

Modular arithmetic modulo operation may also be referred to as, e.g. 23≡11 (mod12), so if a = kn + b, may be expressed as a ≡ b (mod n), computing rules:

（a+b) mod n = ((a mod n) + (b mod n))mod n

(a*b) mod n = ((a mod n) * (b mod n)) mod n

Complete set of the remaining

1 ~ n-1 constitute a complete set of remaining natural number n, for any integer m% n are present in the collection of 1 ~ n.

Adder chain configuration

In the encryption algorithm, applied to a large number of modulo operation mode for a k bit number n, all operations such as addition, subtraction of the intermediate structure will not exceed 2k bits, for example, A ^X MOD n-calculation will be greatly simplified when the complexity.

E.g. A ^{. 8} MOD n-can be calculated in the calculation ((A ² MOD n-) ² MOD n-) ² MOD n-

When the index x is not a multiple of 2 is required adder chain configuration, e.g. 25, 25 = 16 + 8 + 1 = 2 ^{. 4} + 2 ^{3 +} 2 ⁰

Thus A ²⁵ MOD n-= (A * A ^{. 8} * A ¹⁶ ) n-MOD = ((((A ² * A) ² ) ² ) ² * A) n-MOD

C language representation

unsigned long ss(unsigned long x , unsigned long y , unsigned long n) {
unsigned long s,t,u;
int i ;
s = 1; t =x ; u=y;
while(u){
if(u&1) s = (s*t)&n;
u>>=1;
t = (t*t) % n ;
}
return (s);
}

Prime numbers

The greatest common divisor (GCD)

gcd(4,2)=2 gcd(4,3)=1 gcd(6,4)=2

c Language:

int gcd(int x,int y){
int m;
while(x>0){
m = x;
x = x%y;
y = m;
} 
return m

Inverse mode

Inverse:

Equation $ax\equiv 1(mod\, \: p)$ Solutions called $a$ modulo $p$ inverse, when $gcd(a,p)=1$ (i.e. $a$ , $p$ prime), then the equation has a unique solution, or no solution.

Then the inverse can be used to do it, for example $(a/b)\, mod\: p$ , and no $((a\: mod\: p)/(b\: mod\:p))\, mod\: p$ , but in addition will direct the explosion accuracy, then we can use the inverse, assuming a $inv(b)$ representative of $b$ inverse, then $(a/b)\,mod\:p=(a*inv(b))\,mod\:p$ .

Inverse mode:

4 * x ≡ 1 (mod 7), i.e. 4x = 7k + 1

More common problems are: 1 = (a * x) mod n can be written A ^-1 ≡ X (n-MOD)

Therefore, there is a solution set of problems here:

When the GCD (A, n-) =. 1, A ^-1 ≡ X (n-MOD) a unique solution

When the GCD (A, n-). 1 A ≠ ^-1 ≡ X (n-MOD) no solution

How to find the inverse of a% n, and expand the use of the Euclidean algorithm

void e_gcd(int a, int b, int &gcd, int &x, int &y)
{
    if (b == 0)
    {
        x = 1;
        y = 0;
        gcd = a;
    }
    else
    {
        e_gcd(b, a % b, gcd, y, x);
        y -= x * (a / b)
    }
}

Fermat's Little Theorem

This theorem is often encountered in junior high school math league, before seemingly had also seen the high school league, but the CMO as a basis for writing a bit.

Content: if m is a prime number, and is not a multiple of m, then there is a theorem ^{m. 1-} ≡. 1 (MOD m)

Euler function

Euler function: φ (n) indicates the number mutually prime number with n from 1 ~ n-1. φ (1) = 1, the general term formulas: φ (n) = n * (1-1 / p1) * (1-1 / p2) * (1-1 / p3) * (1-1 / p4) ... .. (1-1 / pn)

Wherein p1 ~ pn prime factors of n.

In some encryption algorithms, secret key generation process is the process of computing the Euler function.

Discrete logarithm over finite fields

Die index is frequently used in another way function Cryptography

For example:. 3 ^X ≡ MOD. 17 15, X =. 6

Not all of the discrete logarithm has a corresponding solution for example. 3 ^x ≡ 13 is no integer x MOD. 7 may comply with the formula, so that no solution to the discrete logarithm. On the discrete logarithm of domain present in two cases:

1. The multiplicative group of prime domain

2. The elliptic curve group of finite field