Zolotarev's lemma
In number theory, Zolotarev's lemma states that the Legendre symbol
for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:
where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a.
For example, take a = 2 and p = 7. The nonzero squares mod 7 are 1, 2, and 4, so (2|7) = 1 and (6|7) = −1. Multiplication by 2 on the nonzero numbers mod 7 has the cycle decomposition (1,2,4)(3,6,5), so the sign of this permutation is 1, which is (2|7). Multiplication by 6 on the nonzero numbers mod 7 has cycle decomposition (1,6)(2,5)(3,4), whose sign is −1, which is (6|7).
In general, for any finite group G of order n, it is easy to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.
We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1. The jth power of a primitive root modulo p will by index calculus have index the greatest common divisor
The condition for a nonzero number mod p to be an quadratic non-residue is to be an odd power of a primitive root. The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (p is odd).
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