Properties: Let p be an odd prime.

        

The properties above can be used to build a recursive algorithm to compute the Jacobi symbol
efficiently. In fact, the algorithm is strongly reminiscent of Euclid’s algorithm for the gcd. Here is how the algorithm applies to compute :

• If m > n then use the invariance property: return  .
• If m=0 or m=1, then use(7) : return 0 or 1
• Factor  m as 2^kl, where l is odd. If  k >0 use formulas (7) and (3) : return  .
• Use reciprocity : if m=n=3 mod 4 then return  -; otherwise return .
  

As this method is similar to Euclidean GCD algorithm, its complexity too is O ().

The Jacobi symbol extends the Legendre symbol from primes p to composite odd integers n. One might define the symbol to be +1 if a is a square mod n and -1 if not. However, this would cause the property (3) to fail.
     In order to preserve property (3), we define the Jacobi symbol as follows. Let n be an odd positive integer and let a be a nonzero integer with gcd (a, n) =1. Let

                          n = p₁^ap₂^bp₃^c……p_r^q

be prime factorization of n. Then

The symbols on the right side are Legendre symbols introduced earlier. Note that if n=p, the right side is simply one Legendre symbol, so the Jacobi symbol reduces to the Legendre symbol.