Number-Theory

gcd is short for greatest common divisor

If a , b are co-primes, we denote as $(a,b)=1, \text{which means } gcd(a,b)=1$

We can use Euclid algorithm to calculate gcd of two numbers.

def gcd(a,b):
while b!=0:
a,b=b,a%b
return a


0.1.1. Bezout’s identity

Let a and b be integers with greatest common divisor d . Then, there exist integers x and y such that ax + by = d . More generally, the integers of the form ax + by are exactly the multiples of d .

we can use extended euclid algorithm to calculate x,y,gcd(a,b)

def xgcd(a,b):
'''return gcd(a,b),  x,y  where  ax+by=gcd(a,b)'''
if b==0:return a,1,0
g,x,y = xgcd(b,a%b)
return g,y,x-a//b*y


0.2. primality_test

0.2.1. Prime Sieve

class primeSieve:
'''sieve of Eratosthenes, It will be more efficient when judging many times'''
primes = [2,3,5,7,11,13]
def isPrime(self,x):
if x<=primes[-1]:
return twoDivideFind(x,self.primes)
while x>self.primes[-1]:
left = self.primes[-1]
right = (left+1)**2
lst = []
for i in range(left,right):
for j in self.primes:
if i%j==0:break
else:lst.append(i)
self.primes+=lst
return twoDivideFind(x,lst)

def twoDivideFind(x,li):
a,b = 0, len(li)
while a<=b:
mid = (a+b)//2
if li[mid]<x:a=mid+1
elif li[mid]>x: b= mid-1
else:return mid
return -1


0.2.2. Miller-Rabin

Excerpted from wikipedia:Miller_Rabin_primality_test

Just like the Fermat and Solovay–Strassen tests, the Miller–Rabin test relies on an equality or set of equalities that hold true for prime values, then checks whether or not they hold for a number that we want to test for primality.

First, a lemma “Lemma (mathematics)”) about square roots of unity in the finite field Z / p** Z *, where p is prime and p > 2\. Certainly 1 and −1 always yield 1 when squared modulo p ; call these trivial “Trivial (mathematics)”) square roots of 1\. There are no nontrivial square roots of 1 modulo p (a special case of the result that, in a field, a polynomial has no more zeroes than its degree). To show this, suppose that x is a square root of 1 modulo p*. Then:

In other words, prime p divides the product ( x − 1)( x + 1). By Euclid’s lemma it divides one of the factors x − 1 or x + 1, implying that x is congruent to either 1 or −1 modulo p .

Now, let n be prime, and odd, with n > 2\. It follows that n − 1 is even and we can write it as 2 s · d , where s and d are positive integers and d is odd. For each a in ( Z / n** Z *) , either

or

To show that one of these must be true, recall Fermat’s little theorem , that for a prime number n:

By the lemma above, if we keep taking square roots of a n −1 , we will get either 1 or −1\. If we get −1 then the second equality holds and it is done. If we never get −1, then when we have taken out every power of 2, we are left with the first equality.

The Miller–Rabin primality test is based on the contrapositive of the above claim. That is, if we can find an a such that

and

then n is not prime. We call a a witness “Witness (mathematics)”) for the compositeness of n (sometimes misleadingly called a strong witness , although it is a certain proof of this fact). Otherwise a is called a strong liar , and n is a strong probable prime to base a . The term “strong liar” refers to the case where n is composite but nevertheless the equations hold as they would for a prime.

Every odd composite n has many witnesses a , however, no simple way of generating such an a is known. The solution is to make the test probabilistic : we choose a non-zero a in Z / n** Z * randomly, and check whether or not it is a witness for the compositeness of n . If n is composite, most of the choices for a will be witnesses, and the test will detect n as composite with high probability. There is, nevertheless, a small chance that we are unlucky and hit an a which is a strong liar for n . We may reduce the probability of such error by repeating the test for several independently chosen a*.

For testing large numbers, it is common to choose random bases a , as, a priori, we don’t know the distribution of witnesses and liars among the numbers 1, 2, …, n − 1\. In particular, Arnault [4] gave a 397-digit composite number for which all bases a less than 307 are strong liars. As expected this number was reported to be prime by the Maple “Maple (software)”) isprime() function, which implemented the Miller–Rabin test by checking the specific bases 2,3,5,7, and 11\. However, selection of a few specific small bases can guarantee identification of composites for n less than some maximum determined by said bases. This maximum is generally quite large compared to the bases. As random bases lack such determinism for small n , specific bases are better in some circumstances.

python implementation

from random import sample
def quickMulMod(a,b,m):
'''a*b%m,  quick'''
ret = 0
while b:
if b&1:
ret = (a+ret)%m
b//=2
a = (a+a)%m
return ret

def quickPowMod(a,b,m):
'''a^b %m, quick,  O(logn)'''
ret =1
while b:
if b&1:
ret =quickMulMod(ret,a,m)
b//=2
a = quickMulMod(a,a,m)
return ret

def isPrime(n,t=5):
'''miller rabin primality test,  a probability result
t is the number of iteration(witness)
'''
if n<2:
print('[Error]: {} can\'t be classed with prime or composite'.format(n))
return
if n==2: return True
d = n-1
r = 0
while d%2==0:
r+=1
d//=2
t = min(n-3,t)
for a in sample(range(2,n-1),t):
x= quickPowMod(a,d,n)
if x==1 or x==n-1: continue  #success,
for j in range(r-1):
x= quickMulMod(x,x,n)
if x==n-1:break
else:
return False
return True


0.3. Factorization

0.3.1. Pollard’s rho algorithm

Excerpted from wikipedia:Pollard’s rho algorithm

Suppose we need to factorize a number

$n=pq$, where $p$ is a non-trivial factor. A polynomial modulo $n$, called

where c is a chosen number ,eg 1.

is used to generate a pseudo-random sequence : A starting value, say 2, is chosen, and the sequence continues as

,

The sequence is related to another sequence$\{x_k\ mod \ p\}$ . Since $p$ is not known beforehand, this sequence cannot be explicitly computed in the algorithm. Yet, in it lies the core idea of the algorithm.

Because the number of possible values for these sequences are finite, both the$\{x_n\}$ sequence, which is mod $n$ , and $\{x_n\ mod\ p\}$ sequence will eventually repeat, even though we do not know the latter. Assume that the sequences behave like random numbers. Due to the birthday paradox , the number of$x_k$before a repetition occurs is expected to be $O(\sqrt{N})$ , where $N$ is the number of possible values. So the sequence $\{x_n\ mod\ p\}$ will likely repeat much earlier than the sequence $x_k$. Once a sequence has a repeated value, the sequence will cycle, because each value depends only on the one before it. This structure of eventual cycling gives rise to the name “Rho algorithm”, owing to similarity to the shape of the Greek character ρ when the values

\begin{aligned}

\phi(n) &=\phi( \prod _{i=1}^{l} p_i^{k_i}) \\

&=\prod _{i=1}^{l} \phi( p_i^{k_i}) \\

&=\prod _{i=1}^{l} ( p_i^{k_i}-p_i^{ {k_i}-1})\\

&=\prod _{i=1}^{l}p_i^{k_i} \prod _{i=1}^{l} ( 1-\frac{1}{p_i})\\

&=n \prod _{i=1}^{l} ( 1-\frac{1}{p_i})\\

\end{aligned}

\begin{aligned}

\sigma(n) &= \prod _{i=1}^{l} \sum_{j=0}^{k_i} p_i^j \\

&=\prod _{i=1}^{l} \frac{p_i^{k_i+1}-1}{p_i-1}\\

\end{aligned}



A perfect number is defined as $\sigma(n) = 2n$

The following is the implementation of this two functions.

from factor import factor
from collections import Counter
from functools import reduce
from operator import mul
def phi(n):
st  = set(factor(n))
return round(reduce(mul,(1-1/p for p in st),n))

def sigma(n):
ct = Counter(factor(n))
return reduce(mul,(round((p**(ct[p]+1)-1)/(p-1)) for p in ct),1)


0.5. Modulo equation

The following codes can solve a linear, group modulo equation. More details and explanations will be supplied if I am not too busy.

Note that I use -- to represent $\equiv$ in the python codes.

import re

from gcd import xgcd
from euler import phi
from isPrime import isPrime
from factor import factor

def  ind(m,g):
''' mod m ,primary root g  ->  {n:indg n}'''
return {j:i for i in range(m-1) \
for j in range(m) if (g**i-j)%m==0}

def gs(m,num=100):
'''return list of  m's  primary roots below num'''
p = phi(m)
mp = factor(p)
checkLst = [p//i for i in mp]
return [i for i in range(2,num) if all((i**n-1)%m !=0  for n in checkLst)]

def minG(m):
p = phi(m)
mp = factor(p)
checkLst = [p//i for i in mp]
i=2
while  1:
if all((i**n-1)%m !=0  for n in checkLst):return i
i+=1

class solve:
def __init__(self,equ=None):
self.linearPat= re.compile(r'\s*(\d+)\s*--\s*(\d+)[\s\(]*mod\s*(\d+)')
self.sol  = []
#self.m = m
#self.ind_mp = ind(m,minG(m))
def noSol(self):
print('equation {equ} has no solution'.format(equ=self.equ))
def error(self):
print("Error! The divisor m must be postive integer")
def solveLinear(self,a,b,m):
'''ax--b(mod m): solve linear equation with one unknown
return  ([x1,x2,...],m)
'''
a,b,m = self.check(a,b,m)
g,x,y=xgcd(a,m)
if a*b%g!=0:
self.noSol()
return None
sol=x*b//g
m0 = m//g
sols = [(sol+i*m0)%m for i in range(g)]
print('{}x--{}(mod {}), solution: {} mod {}'.format(a,b,m,sols,m))
return (sols,m)
def check(self,a,b,m):
if m<=0:
self.error()
return None
if a<0:a,b=-a,-b  ## important
if b<0:b+= -b//m * m
return a,b,m

def solveHigh(self,a,n,b,m):
''' ax^n -- b (mod m)  ind_mp is a dict of  m's {n: indg n}'''
ind_mp = ind(m,minG(m))
tmp = ind_mp[b] - ind_mp[a]
if tmp < 0:tmp+=m
sol = self.solveLinear(n,tmp,phi(m))
re_mp = {j:i for i ,j in ind_mp.items()}
sols = [re_mp[i] for i in sol[0]]
print('{}x^{}--{}(mod {}),  solution: {} mod {}'.format(a,n,b,m,sols,m))
return sols,m

def solveGroup(self,tups):
'''tups is a list of tongyu equation groups, like
[(a1,b1,m1),(a2,b2,m2)...]
and, m1,m2... are all primes
'''
mp = {}
print('solving group of equations: ')
for a,b,m in tups:
print('{}x--{}(mod {})'.format(a,b,m))
if m in mp:
if mp[m][0]*b!=mp[m][1]*a:
self.noSol()
return
else:mp[m] = (a,b)
product = 1
for i in mp.keys():
product *=i
sol = [0]
for i in mp:
xs,m = self.solveLinear(product//i*mp[i][0],1,i)
new = []
for x in xs:
cur = x*product//i*mp[i][1]
for old in sol:
new.append(old+cur)
sol = new
sol= [i%product for i in sol]
print('final solution: {} mod {}'.format(sol,product))
return sol,product
def __call__(self):
s=input('輸入同餘方程，用--代表同於號，形如3--5(mod 7)代表3x模7同餘於5')
li= self.linearPat.findall(s)
li = [(int(a),int(b),int(m)) for a,b,m in li]
print(self.solveLinear(li[0]))

if __name__ == '__main__':
solver  = solve()
res = solver.solveLinear(3,6,9)
print()
res = solver.solveHigh(1,8,3,11)
print()
res = solver.solveGroup([(5,11,2),(3,8,5),(4,1,7)])