X-Git-Url: https://git.distorted.org.uk/u/mdw/catacomb/blobdiff_plain/45c0fd363937c6e9b05da04a9167e9912c05ca0c..ea932d59b3071ce00f9e510aad014ad64a3dc48c:/mp-jacobi.c

diff --git a/mp-jacobi.c b/mp-jacobi.c
index d0a67b6..3674f22 100644
--- a/mp-jacobi.c
+++ b/mp-jacobi.c
@@ -35,27 +35,88 @@
 
 /* --- @mp_jacobi@ --- *
  *
- * Arguments:	@mp *a@ = an integer less than @n@
- *		@mp *n@ = an odd integer
+ * Arguments:	@mp *a@ = an integer
+ *		@mp *n@ = another integer
  *
  * Returns:	@-1@, @0@ or @1@ -- the Jacobi symbol %$J(a, n)$%.
  *
- * Use:		Computes the Jacobi symbol.  If @n@ is prime, this is the
- *		Legendre symbol and is equal to 1 if and only if @a@ is a
- *		quadratic residue mod @n@.  The result is zero if and only if
- *		@a@ and @n@ have a common factor greater than one.
+ * Use:		Computes the Kronecker symbol %$\jacobi{a}{n}$%.  If @n@ is
+ *		prime, this is the Legendre symbol and is equal to 1 if and
+ *		only if @a@ is a quadratic residue mod @n@.  The result is
+ *		zero if and only if @a@ and @n@ have a common factor greater
+ *		than one.
+ *
+ *		If @n@ is composite, then this computes the Kronecker symbol
+ *
+ *		  %$\jacobi{a}{n}=\jacobi{a}{u}\prod_i\jacobi{a}{p_i}^{e_i}$%
+ *
+ *		where %$n = u p_0^{e_0} \ldots p_{n-1}^{e_{n-1}}$% is the
+ *		prime factorization of %$n$%.  The missing bits are:
+ *
+ *		  * %$\jacobi{a}{1} = 1$%;
+ *		  * %$\jacobi{a}{-1} = 1$% if @a@ is negative, or 1 if
+ *		    positive;
+ *		  * %$\jacobi{a}{0} = 0$%;
+ *	          * %$\jacobi{a}{2}$ is 0 if @a@ is even, 1 if @a@ is
+ *		    congruent to 1 or 7 (mod 8), or %$-1$% otherwise.
+ *
+ *		If %$n$% is positive and odd, then this is the Jacobi
+ *		symbol.  (The Kronecker symbol is a consistant domain
+ *		extension; the Jacobi symbol was implemented first, and the
+ *		name stuck.)
  */
 
 int mp_jacobi(mp *a, mp *n)
 {
   int s = 1;
+  size_t p2;
+
+  /* --- Handle zero specially --- *
+   *
+   * I can't find any specific statement for what to do when %$n = 0$%; PARI
+   * opts to set %$\jacobi{\pm1}{0} = \pm 1$% and %$\jacobi{a}{0} = 0$% for
+   * other %$a$%.
+   */
+
+  if (MP_ZEROP(n)) {
+    if (MP_EQ(a, MP_ONE)) return (+1);
+    else if (MP_EQ(a, MP_MONE)) return (-1);
+    else return (0);
+  }
+
+  /* --- Deal with powers of two --- *
+   *
+   * This implicitly takes a copy of %$n$%.  Copy %$a$% at the same time to
+   * make cleanup easier.
+   */
+
+  MP_COPY(a);
+  n = mp_odd(MP_NEW, n, &p2);
+  if (p2) {
+    if (MP_EVENP(a)) {
+      s = 0;
+      goto done;
+    } else if ((p2 & 1) && ((a->v[0] & 7) == 3 || (a->v[0] & 7) == 5))
+      s = -s;
+  }
+
+  /* --- Deal with negative %$n$% --- */
+
+  if (MP_NEGP(n)) {
+    n = mp_neg(n, n);
+    if (MP_NEGP(a))
+      s = -s;
+  }
+
+  /* --- Check for unit %$n$% --- */
 
-  assert(MP_ODDP(n));
+  if (MP_EQ(n, MP_ONE))
+    goto done;
 
-  /* --- Take copies of the arguments --- */
+  /* --- Reduce %$a$% modulo %$n$% --- */
 
-  a = MP_COPY(a);
-  n = MP_COPY(n);
+  if (MP_NEGP(a) || MP_CMP(a, >=, n))
+    mp_div(0, &a, a, n);
 
   /* --- Main recursive mess, flattened out into something nice --- */