X-Git-Url: https://git.distorted.org.uk/~mdw/catacomb/blobdiff_plain/0f00dc4c8eb47e67bc0f148c2dd109f73a451e0a..8f2287ef5c05d496fcb9b012629af007fe56f897:/math/mpbarrett.h

diff --git a/math/mpbarrett.h b/math/mpbarrett.h
index 40e0fe42..ec022516 100644
--- a/math/mpbarrett.h
+++ b/math/mpbarrett.h
@@ -40,6 +40,28 @@
  * modexp routine provided which uses Barrett reduction rather than
  * Montgomery reduction.  This is handy when you're working on indices in an
  * even-order cyclic group or something.
+ *
+ * In more detail: suppose that %$b^{k-1} \le m < b^k$%.  Let %$\mu = {}$%
+ * %$\lfloor b^{2k}/m \rfloor$%; %$\mu$% is a scaled approximation to the
+ * reciprocal %$1/m$%.  Now, suppose we're given some %$a$% with
+ * %$0 \le a < b^{2k}$%.  The first step is to calculate an approximation
+ * %$q = \lfloor \mu \lfloor a/b^{k-1} \rfloor/b^{k+1} \rfloor$% to the
+ * quotient %$a/m$%.  Then we have:
+ *
+ *	%$\lfloor a/m - a/b^{2k} - b^{k-1}/m + 1/b^{k+1} \rfloor \le {}$%
+ *		%$q \le \lfloor a/m \rfloor
+ *
+ * But by assumption %$a < b^{2k}$% and %$2^{k-1} \le m$% so
+ *
+ *	%$\lfloor a/m \rfloor - 2 \le q \le \lfloor a/m \rfloor$%
+ *
+ * Now we approximate the remainder by calculating %$r = a - q m$%.
+ * Certainly %$r \equiv a \pmod{m}$%; and
+ *
+ *	%$0 \le r \le (a - m \lfloor a/m \rfloor) + 2 m < 3 m$%.
+ *
+ * So the remainder can be fixed up with at most two conditional
+ * subtractions.
  */
 
 #ifndef CATACOMB_MPBARRETT_H
@@ -91,7 +113,7 @@ extern void mpbarrett_destroy(mpbarrett */*mb*/);
 
 /* --- @mpbarrett_reduce@ --- *
  *
- * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
+ * Arguments:	@const mpbarrett *mb@ = pointer to Barrett reduction context
  *		@mp *d@ = destination for result
  *		@mp *m@ = number to reduce
  *
@@ -101,11 +123,11 @@ extern void mpbarrett_destroy(mpbarrett */*mb*/);
  * Use:		Performs an efficient modular reduction.
  */
 
-extern mp *mpbarrett_reduce(mpbarrett */*mb*/, mp */*d*/, mp */*m*/);
+extern mp *mpbarrett_reduce(const mpbarrett */*mb*/, mp */*d*/, mp */*m*/);
 
 /* --- @mpbarrett_exp@ --- *
  *
- * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
+ * Arguments:	@const mpbarrett *mb@ = pointer to Barrett reduction context
  *		@mp *d@ = fake destination
  *		@mp *a@ = base
  *		@mp *e@ = exponent
@@ -113,11 +135,12 @@ extern mp *mpbarrett_reduce(mpbarrett */*mb*/, mp */*d*/, mp */*m*/);
  * Returns:	Result, %$a^e \bmod m$%.
  */
 
-extern mp *mpbarrett_exp(mpbarrett */*mb*/, mp */*d*/, mp */*a*/, mp */*e*/);
+extern mp *mpbarrett_exp(const mpbarrett */*mb*/, mp */*d*/,
+			 mp */*a*/, mp */*e*/);
 
 /* --- @mpbarrett_mexp@ --- *
  *
- * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
+ * Arguments:	@const mpbarrett *mb@ = pointer to Barrett reduction context
  *		@mp *d@ = fake destination
  *		@const mp_expfactor *f@ = pointer to array of factors
  *		@size_t n@ = number of factors supplied
@@ -129,7 +152,7 @@ extern mp *mpbarrett_exp(mpbarrett */*mb*/, mp */*d*/, mp */*a*/, mp */*e*/);
  *		%$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$%
  */
 
-extern mp *mpbarrett_mexp(mpbarrett */*mb*/, mp */*d*/,
+extern mp *mpbarrett_mexp(const mpbarrett */*mb*/, mp */*d*/,
 			  const mp_expfactor */*f*/, size_t /*n*/);
 
 /*----- That's all, folks -------------------------------------------------*/