hw4.py

@@ -0,0 +1,84 @@
+from poly import Polynomial
+from sys import argv
+from ast import literal_eval
+from fractions import gcd
+
+class QuotientRing():
+    def __init__(self, f, m):
+        self.f = Polynomial(f, m)
+        self.m = m
+        self.remainders = self.remainders()
+        self.reversibles = self.reversibles() 
+        self.zero_divisors = self.zero_divisors()
+        self.idempotent = self.idempotent()
+        self.nilpotent = self.nilpotent()
+   
+    def remainders(self): #n - exponent
+        rems = [] #lista reszt
+        m = self.m
+        t = [0] 
+        i = 0
+        while len(t) < len(self.f.poly):
+            rems.append(Polynomial(t, m))
+            i = (i + 1) % m
+            t[0] = i 
+            if i == 0:
+                if len(t) == 1:
+                    t.append(1)
+                else: 
+                    t[1] += 1
+                    for j in range(1, len(t)):
+                        if t[j] == 0 or t[j] % m != 0:
+                            break
+                        temp = t[j] % m
+                        t[j] = 0
+                        if temp == 0:
+                            if (j + 1) < len(t):
+                                t[j+1] += 1
+                            else:
+                                t.append(1)
+        return rems
+    
+    def reversibles(self):
+        return [ rem for rem in self.remainders if len(rem.poly_gcd(self.f).poly) == 1 ]
+
+    #dopelnienie elementow odwracalnych
+    def zero_divisors(self):
+        return [ rem for rem in self.remainders if rem not in self.reversibles ]
+    
+    def idempotent(self):
+        idems = []
+        for rem in self.remainders:
+            if (rem * rem % self.f) == (rem % self.f):
+                idems.append(rem)
+        try:
+            if idems[0].poly == []:     #implementacja wielomianow ucina zera
+                idems[0].poly = [0]
+        except IndexError:
+            return idems
+        return idems
+
+    def nilpotent(self):
+        nils = []
+        phi = len([ i for i in range(1, self.m) if gcd(i, self.m) == 1 ])
+        for zero_div in self.zero_divisors:
+            for i in range(phi+1):
+               if len((zero_div ** i % self.f).poly) == 0:
+                   nils.append(zero_div)
+                   break
+        return nils
+
+def main():
+    m = int(argv[1])
+    f = literal_eval(argv[2])
+    qr = QuotientRing(f, m)
+    out = [
+            [ rev.poly for rev in qr.reversibles ],
+            [ zero_div.poly for zero_div in qr.zero_divisors ],
+            [ nil.poly for nil in qr.nilpotent ],
+            [ idem.poly for idem in qr.idempotent ] 
+           ]
+    print(*out, sep='\n')
+
+if __name__ == '__main__':
+    main()

poly.py

@@ -0,0 +1,115 @@
+from fractions import gcd
+class Polynomial():
+    def __init__(self, lst, mod):
+        self.poly = list(map(lambda x: x % mod, lst))    
+        self.mod = mod
+        self.normalize()
+    def normalize(self):
+        while self.poly and self.poly[-1] == 0:
+            self.poly.pop() 
+
+    #zwraca jednomian stopnia n 
+    @staticmethod
+    def Monomial(n, c, mod):
+        zeros = [0]*n
+
+        zeros.append(c)
+        return Polynomial(zeros, mod)
+
+    def __add__(self, p2):
+        p1 = self
+        len_p1, len_p2= len(p1.poly), len(p2.poly)
+        res = [0] * max(len_p1, len_p2)
+        if len_p1 > len_p2:
+            for _ in range(len_p1-len_p2):
+                p2.poly.append(0)
+        else:
+            for _ in range(len_p2-len_p1):
+                p1.poly.append(0)
+
+        for i in range(len(res)):
+            res[i] = (p1.poly[i] + p2.poly[i]) % self.mod
+        return Polynomial(res, self.mod)
+
+    def __sub__(self, p2):
+        p1 = self
+        res = []
+        len_p2 = len(p2.poly)
+        for i in range(len(p1.poly)):
+            if i < len_p2:
+                res.append(p1.poly[i] - p2.poly[i] % self.mod)
+            else:
+                res.append(p1.poly[i])
+        return Polynomial(res, self.mod)
+
+    def __mul__(self, p2):
+        res = [0]*(len(self.poly)+len(p2.poly)-1)
+        for i, x1 in enumerate(self.poly):
+            for j, x2 in enumerate(p2.poly):
+                res[i+j] += x1 * x2 % self.mod
+        return Polynomial(res, self.mod)
+
+    def __eq__(self, p2):
+        p1 = self
+        return p1.poly == p2.poly and p1.mod == p2.mod
+
+    def __pow__(self, n):
+        p1 = self
+        for i in range(n):
+            p1 = p1 * p1
+        return p1
+    
+    def __mod__(self, p2):
+        p1 = self
+        m = self.mod
+        
+        if len(p1.poly) < len(p2.poly):
+            return p1
+        
+        if len(p2.poly) == 0:
+            raise ZeroDivisionError
+            
+        divisor_coeff = p2.poly[-1]
+        divisor_exp = len(p2.poly) - 1 
+        while len(p1.poly) >= len(p2.poly):   
+            max_coeff_p1 = p1.poly[-1] #wspolczynnik przy najwyzszej potedze
+            try:
+              tmp_coeff = modDiv(max_coeff_p1, divisor_coeff, m)
+            except ZeroDivisionError as e:
+                raise e
+            
+            tmp_exp = len(p1.poly)-1 - divisor_exp
+            tmp = [0] * tmp_exp
+            tmp.append(tmp_coeff)
+            sub = Polynomial(tmp, m) * p2
+            p1 = p1 - sub
+        p1.normalize()
+        return Polynomial(p1.poly, m)
+
+    def poly_gcd(self, p2):
+        p1 = self
+        try:
+            divisible = p2
+        except ZeroDivisionError as e:
+            raise e
+        if p2.poly == []:
+            return p1
+        
+        return p2.poly_gcd(p1 % p2)
+
+def modDiv(a, b, m): # a*b^-1 (mod m)
+    if gcd(b, m) != 1:
+        raise ZeroDivisionError
+    else:
+        return (a * modinv(b, m)) % m
+#rozszerzony algorytm euklidesa
+def egcd(a, b):
+    if a == 0:
+        return (b, 0, 1)
+    else:
+        g, y, x = egcd(b % a, a)
+        return (g, x - (b // a) * y, y)
+
+def modinv(a, m):
+    g, x, y = egcd(a, m)
+    return x % m