X-Git-Url: https://git.distorted.org.uk/~mdw/catacomb/blobdiff_plain/c144ab4f7d780f14f8f5c83c72b17a7c54c01150..5a19b5dfdde3f31feef4443442cc1c5c0bad6484:/math/gfreduce.h

diff --git a/math/gfreduce.h b/math/gfreduce.h
index 9f69585d..ef075c45 100644
--- a/math/gfreduce.h
+++ b/math/gfreduce.h
@@ -129,7 +129,10 @@ extern mp *gfreduce_sqrt(gfreduce */*r*/, mp */*d*/, mp */*x*/);
  *		@mp *x@ = some polynomial
  *
  * Returns:	The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
- *		if %$x \in \gf{2^m}$%).
+ *		if %$x \in \gf{2^m}$%).  Since the trace is invariant under
+ *		the Frobenius automorphism (i.e., %$\Tr(x)^2 = \Tr(x)$%), it
+ *		must be an element of the base field, i.e., %$\gf{2}$%, and
+ *		we only need a single bit to represent it.
  */
 
 extern int gfreduce_trace(gfreduce */*r*/, mp */*x*/);
@@ -154,6 +157,17 @@ extern mp *gfreduce_halftrace(gfreduce */*r*/, mp */*d*/, mp */*x*/);
  *		@mp *x@ = some polynomial
  *
  * Returns:	A polynomial @y@ such that %$y^2 + y = x$%, or null.
+ *
+ * Use:		Solves quadratic equations in a field with characteristic 2.
+ *		Suppose we have an equation %$y^2 + A y + B = 0$% where
+ *		%$A \ne 0$%.  (If %$A = 0$% then %$y = \sqrt{B}$% and you
+ *		want @gfreduce_sqrt@ instead.)  Use this function to solve
+ *		%$z^2 + z = B/A^2$%; then set %$y = A z$%, since
+ *		%$y^2 + y = A^2 z^2 + A^2 z = A^2 (z^2 + z) = B$% as
+ *		required.
+ *
+ *		The two roots are %$z$% and %$z + 1$%; this function always
+ *		returns the one with zero scalar coefficient.
  */
 
 extern mp *gfreduce_quadsolve(gfreduce */*r*/, mp */*d*/, mp */*x*/);