/* --- @gfreduce_dump@ --- *
*
- * Arguments: @gfreduce *r@ = structure to dump
+ * Arguments: @const gfreduce *r@ = structure to dump
* @FILE *fp@ = file to dump on
*
* Returns: ---
* Use: Dumps a reduction context.
*/
-extern void gfreduce_dump(gfreduce */*r*/, FILE */*fp*/);
+extern void gfreduce_dump(const gfreduce */*r*/, FILE */*fp*/);
/* --- @gfreduce_do@ --- *
*
- * Arguments: @gfreduce *r@ = reduction context
+ * Arguments: @const gfreduce *r@ = reduction context
* @mp *d@ = destination
* @mp *x@ = source
*
* Returns: Destination, @x@ reduced modulo the reduction poly.
*/
-extern mp *gfreduce_do(gfreduce */*r*/, mp */*d*/, mp */*x*/);
+extern mp *gfreduce_do(const gfreduce */*r*/, mp */*d*/, mp */*x*/);
/* --- @gfreduce_sqrt@ --- *
*
- * Arguments: @gfreduce *r@ = pointer to reduction context
+ * Arguments: @const gfreduce *r@ = pointer to reduction context
* @mp *d@ = destination
* @mp *x@ = some polynomial
*
* Returns: The square root of @x@ modulo @r->p@, or null.
*/
-extern mp *gfreduce_sqrt(gfreduce */*r*/, mp */*d*/, mp */*x*/);
+extern mp *gfreduce_sqrt(const gfreduce */*r*/, mp */*d*/, mp */*x*/);
/* --- @gfreduce_trace@ --- *
*
- * Arguments: @gfreduce *r@ = pointer to reduction context
+ * Arguments: @const gfreduce *r@ = pointer to reduction context
* @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*/);
+extern int gfreduce_trace(const gfreduce */*r*/, mp */*x*/);
/* --- @gfreduce_halftrace@ --- *
*
- * Arguments: @gfreduce *r@ = pointer to reduction context
+ * Arguments: @const gfreduce *r@ = pointer to reduction context
* @mp *d@ = destination
* @mp *x@ = some polynomial
*
* if %$x \in \gf{2^m}$% with %$m$% odd).
*/
-extern mp *gfreduce_halftrace(gfreduce */*r*/, mp */*d*/, mp */*x*/);
+extern mp *gfreduce_halftrace(const gfreduce */*r*/, mp */*d*/, mp */*x*/);
/* --- @gfreduce_quadsolve@ --- *
*
- * Arguments: @gfreduce *r@ = pointer to reduction context
+ * Arguments: @const gfreduce *r@ = pointer to reduction context
* @mp *d@ = destination
* @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*/);
+extern mp *gfreduce_quadsolve(const gfreduce */*r*/, mp */*d*/, mp */*x*/);
/* --- @gfreduce_exp@ --- *
*
- * Arguments: @gfreduce *gr@ = pointer to reduction context
+ * Arguments: @const gfreduce *gr@ = pointer to reduction context
* @mp *d@ = fake destination
* @mp *a@ = base
* @mp *e@ = exponent
* Returns: Result, %$a^e \bmod m$%.
*/
-extern mp *gfreduce_exp(gfreduce */*gr*/, mp */*d*/, mp */*a*/, mp */*e*/);
+extern mp *gfreduce_exp(const gfreduce */*gr*/, mp */*d*/,
+ mp */*a*/, mp */*e*/);
/*----- That's all, folks -------------------------------------------------*/