update notes

pages/CSE442T/Exam_reviews/CSE442T_E1.md

@@ -151,10 +151,15 @@ A group $G$ is a set of elements with a binary operator $\oplus:G\times G\to G$
 
 $$
 \Phi(p)=p-1
-$$ if $p$ is prime
+$$ 
+
+if $p$ is prime
+
 $$
 \Phi(N)=(p-1)(q-1)
-$$ if $N=pq$ and $p,q$ are primes
+$$ 
+
+if $N=pq$ and $p,q$ are primes
 
 #### Theorem 47.10
 

pages/CSE442T/fun.py

@@ -0,0 +1,65 @@
+from math import gcd
+
+def euclidean_algorithm(a,b):
+    if a<b: return euclidean_algorithm(b,a)
+    if b==0: return a
+    return euclidean_algorithm(b,a%b)
+
+def get_generator(p):
+    """
+    p should be a prime
+    """
+    f=3
+    g=[]
+    for i in range(1,p):
+        sg=[]
+        step=p
+        k=i
+        while k!=1 and step>0:
+            if k==0:
+                break
+                # raise ValueError(f"Damn, {i} generates 0 for group {p}")
+            sg.append(k)
+            k=(k**f)%p
+            step-=1
+        sg.append(1)
+        # if len(sg)!=(p-1): continue
+        g.append((i,[j for j in sg]))
+    return g
+
+def __list_print(arr):
+    for i in arr:print(i)
+
+def factorization(n):
+    # Pollard's rho integer factorization algorithm
+    # https://stackoverflow.com/questions/32871539/integer-factorization-in-python
+    factors = []
+
+    def get_factor(n):
+        x_fixed = 2
+        cycle_size = 2
+        x = 2
+        factor = 1
+
+        while factor == 1:
+            for count in range(cycle_size):
+                if factor > 1: break
+                x = (x * x + 1) % n
+                factor = gcd(x - x_fixed, n)
+
+            cycle_size *= 2
+            x_fixed = x
+
+        return factor
+
+    while n > 1:
+        next = get_factor(n)
+        factors.append(next)
+        n //= next
+
+    return factors
+
+if __name__=='__main__':
+    print(euclidean_algorithm(285,(10**9+7)*5))
+    __list_print(get_generator(23))
+    print(factorization(162000))