Let $a\in\mathbb Z$ be an integer, $b > 0$ be an odd and positive integer. If $|a|\le b,$ the following algorithm calculates correctly the Jacobi symbol $\left(\frac ab\right)$ in the time $\mathcal O(\log^2(b)),$ which corresponds to $$\mathcal O(\log^3 |b|)$$ bit operations.

# Algorithm: Jacobi Symbol (Python)

class NotOddException(Exception): def init(self, x): print(str(x) + " is not odd")

class NotPositiveException(Exception): def init(self, x): print(str(x) + " is not > 0")

class JacobiSymbol: """Calculates the JacobiSymbol (a|b)

  Keyword arguments:
a -- integer
b -- positive odd integer
"""

def __init__(self, a, b):
if b < 0:
raise NotPositiveException(b)
self.a = a
self.b = b
if (b % 2) == 0:
raise NotOddException(b)
self.a = a
self.b = b
if gcd(self.a, self.b) != 1:
self.notCoPrime = True
else:
self.notCoPrime = False

@staticmethod
def __supplementary2__(x):
if x % 8 == 1 or x % 8 == 7:
return 1
else:
return -1

@staticmethod
def __supplementary1__(x, y):
if x % 4 == 1 or y % 4 == 1:
return 1
else:
return -1

def calculate(self):
if self.notCoPrime:
return 0
elif self.a == 1:
return 1
elif self.a == 2:
return self.__supplementary2__(self.b)
elif self.a % 2 == 0:
return JacobiSymbol(self.a // 2, self.b).calculate() * JacobiSymbol(2, self.b).calculate()
else:
return JacobiSymbol(self.b % self.a, self.a).calculate() * self.__supplementary1__(self.a, self.b)


# Usage

b=21 for i in range(0, b): print("(" + str(i), "|", str(b) + ")=" + str(JacobiSymbol(i, b).calculate()))

''' will output (0 | 21)=0 (1 | 21)=1 (2 | 21)=-1 (3 | 21)=0 (4 | 21)=1 (5 | 21)=1 (6 | 21)=0 (7 | 21)=0 (8 | 21)=-1 (9 | 21)=0 (10 | 21)=-1 (11 | 21)=-1 (12 | 21)=0 (13 | 21)=-1 (14 | 21)=0 (15 | 21)=0 (16 | 21)=1 (17 | 21)=1 (18 | 21)=0 (19 | 21)=-1 (20 | 21)=1 '''

Proofs: 1

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### References

#### Bibliography

1. Hermann, D.: "Algorithmen Arbeitsbuch", Addison-Wesley Publishing Company, 1992
2. BlĂ¶mer, J.: "Lecture Notes Algorithmen in der Zahlentheorie", Goethe University Frankfurt, 1997