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Zadanie-4/poly.py
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83
Zadanie-4/poly.py
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import sys
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import ast
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class Polynomial:
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n = 0
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def __init__(self, coeff_list):
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self.degree = len(coeff_list) - 1
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self.coefficients = [x % Polynomial.n for x in coeff_list]
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def __pow__(self, n):
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result = self
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for i in range(n):
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result = Polynomial.multiply(result, result)
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return result
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@staticmethod
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def add(p1, p2):
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result = []
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f = p1.coefficients
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g = p2.coefficients
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if len(f) >= len(g):
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result = f
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for i in range(len(g)):
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result[i] = f[i] + g[i]
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else:
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result = g
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for i in range(len(f)):
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result[i] = f[i] + g[i]
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result = [x % int(Polynomial.n) for x in result]
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return Polynomial(result)
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@staticmethod
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def multiply(p1, p2):
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result = [0] * (p1.degree + p2.degree + 1)
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f = p1.coefficients
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g = p2.coefficients
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for i in range(len(f)):
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for j in range(len(g)):
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result[i+j] += f[i] * g[j]
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result = [x % int(Polynomial.n) for x in result]
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return Polynomial(result)
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@staticmethod
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def divide(p1, p2):
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def inverse(x):
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for i in range(1, int(Polynomial.n)):
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r = (i * x) % int(Polynomial.n)
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if r == 1:
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break
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else:
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raise ZeroDivisionError
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return i
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if p1.degree < p2.degree:
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return p1
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f = p1.coefficients
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g = p2.coefficients
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g_lead_coef = g[-1]
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g_deg = p2.degree
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while len(f) >= len(g):
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f_lead_coef = f[-1]
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tmp_coef = f_lead_coef * inverse(g_lead_coef)
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tmp_exp = len(f) - 1 - g_deg
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tmp = []
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for _ in range(tmp_exp):
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tmp.append(0)
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tmp.append(tmp_coef)
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tmp_poly = Polynomial(tmp)
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sub = Polynomial.multiply(p2, tmp_poly)
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f = [x - y for x, y in zip(f, sub.coefficients)]
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f = [x % int(Polynomial.n) for x in f]
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while f and f[-1] == 0:
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f.pop()
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return Polynomial(f)
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@staticmethod
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def gcd(p1, p2):
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if len(p2.coefficients) == 0:
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return p1
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return Polynomial.gcd(p2, Polynomial.divide(p1, p2))
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88
Zadanie-4/quotient_ring.py
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88
Zadanie-4/quotient_ring.py
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from poly import Polynomial as P
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import sys
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import ast
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class QuotientRing:
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def __init__(self, coeffs):
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self.fx = P(coeffs)
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self.remainder_set = self.create_remainder_set()
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self.invertible_elements = self.get_invertible_elements()
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self.zero_divisors = self.get_zero_divisors()
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self.nilpotent_elements = self.get_nilpotent_elements()
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self.idempotent_elements = self.get_idempotent_elements()
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def create_remainder_set(self):
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remainders = []
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rem = [0]
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i = 0
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while len(rem) < len(self.fx.coefficients):
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remainders.append(P(rem))
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i = (i + 1) % P.n
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rem[0] = i
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if i == 0:
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if len(rem) == 1:
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rem.append(1)
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else:
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rem[1] += 1
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for j in range(1, len(rem)):
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if rem[j] == 0 or rem[j] % P.n != 0:
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break
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tmp = rem[j] % P.n
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rem[j] = 0
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if tmp == 0:
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if (j + 1) < len(rem):
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rem[j+1] += 1
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else:
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rem.append(1)
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return remainders
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def get_invertible_elements(self):
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invertible_elements = []
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for i in self.remainder_set:
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if i.coefficients != [0] and len(P.gcd(self.fx, i).coefficients) == 1:
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invertible_elements.append(i)
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return invertible_elements
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def get_zero_divisors(self):
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zero_diviors = []
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for i in self.remainder_set:
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if i not in self.invertible_elements:
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zero_diviors.append(i)
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return zero_diviors
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def get_nilpotent_elements(self):
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nilpotent_elements = []
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for i in self.zero_divisors:
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for j in range(1, len(self.invertible_elements) + 1):
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if i.coefficients == [0] or len(P.divide(i**j, self.fx).coefficients) == 0:
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nilpotent_elements.append(i)
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break
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return nilpotent_elements
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def get_idempotent_elements(self):
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idempotent_elements = []
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for i in self.remainder_set:
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if P.divide(i**2, self.fx).coefficients == P.divide(i, self.fx).coefficients:
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idempotent_elements.append(i)
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return idempotent_elements
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def main():
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P.n = int(sys.argv[1])
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coeffs = ast.literal_eval(sys.argv[2])
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Q = QuotientRing(coeffs)
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ans = [
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[x.coefficients for x in Q.invertible_elements],
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[x.coefficients for x in Q.zero_divisors],
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[x.coefficients for x in Q.nilpotent_elements],
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[x.coefficients for x in Q.idempotent_elements]
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]
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for i in range(len(ans)):
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print(ans[i])
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if __name__ == '__main__':
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main()
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