symm/chacha-x86ish-sse2.S: Fix mathematical errors in commentary.

[catacomb] / math / gfreduce.c

diff --git a/math/gfreduce.c b/math/gfreduce.c
index 99f3bea..f29c104 100644 (file)
--- a/math/gfreduce.c
+++ b/math/gfreduce.c
@@ -554,12 +554,16 @@ mp *gfreduce_quadsolve(const gfreduce *r, mp *d, mp *x)
      *
      * Choose %$\rho \inr \gf{2^m}$% with %$\Tr(\rho) = 1$%.  Let
      * %$z = \sum_{0\le i<m} \rho^{2^i} \sum_{0\le j<i} x^{2^j} = {}$%
+     * %$\sum_{1\le i<m} \rho^{2^i} (x + \sum_{1\le j<i} x^{2^j} = {}$%
      * %$\rho^2 x + \rho^4 (x + x^2) + \rho^8 (x + x^2 + x^4) + \cdots + {}$%
      * %$\rho^{2^{m-1}} (x + x^2 + x^{2^{m-2}})$%.  Then %$z^2 = {}$%
      * %$\sum_{0\le i<m} \rho^{2^{i+1}} \sum_{0\le j<i} x^{2^{j+1}} = {}$%
-     * %$\sum_{1\le i\le m} \rho^{2^i} \sum_{1\le j\le i} x^{2^j}$% and,
-     * somewhat miraculously, %$z^2 + z = \sum_{0\le i<m} \rho^{2^i} x + {}$%
-     * %$\rho \sum_{1\le i<m} x^{2^i} = x \Tr(\rho) + \rho \Tr(x)$%.  Again,
+     * %$\sum_{1\le i\le m} \rho^{2^i} \sum_{1\le j<i} x^{2^j} = {}$%
+     * %$\sum_{1\le i<m} \rho^{2^i} \sum_{1\le j<i} x^{2^j} + {}$%
+     * %$\rho^{2^m} \sum_{1\le j<m} x^{2^j}$%; and, somewhat miraculously,
+     * %$z^2 + z = \sum_{1\le i<m} \rho^{2^i} x + {}$%
+     * %$\rho \sum_{1\le i<m} x^{2^i} = x (\Tr(\rho) + \rho) + {}$%
+     * %$\rho (\Tr(x) + x) = x \Tr(\rho) + \rho \Tr(x)$%.  Again,
      * this gives us the root we want whenever %$\Tr(x) = 0$%.
      *
      * The loop below calculates %$w = \Tr(\rho)$% and %$z$% simultaneously,