GNU MP 6.1.0: Jacobi Symbol
Description: How to install and use the GNU multiple precision arithmetic library, version 6.1.0.

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#### 15.3.5 Jacobi Symbol

[This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.]

`mpz_jacobi` and `mpz_kronecker` are currently implemented with a simple binary algorithm similar to that described for the GCDs (see Binary GCD). They’re not very fast when both inputs are large. Lehmer’s multi-step improvement or a binary based multi-step algorithm is likely to be better.

When one operand fits a single limb, and that includes `mpz_kronecker_ui` and friends, an initial reduction is done with either `mpn_mod_1` or `mpn_modexact_1_odd`, followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient.

In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps.