Merge bitcoin-core/secp256k1#1340: clean up in-comment Sage code (ref…

…er to secp256k1_params.sage, update to Python3)

600c5ad clean up in-comment Sage code (refer to secp256k1_params.sage, update to Python3) (Sebastian Falbesoner)

Pull request description:

  Some of the C source files contain contain in-comment Sage code calculating secp256k1 parameters that are already defined in the file secp256k1_params.sage.  Replace that by a corresponding load instruction and access the necessary variables. In ecdsa_impl.h, update the comment to use a one-line shell command calling sage to get the values.

  The remaining code (test `test_add_neg_y_diff_x` in tests.c) is updated to work with a current version based on Python3 (Sage 9.0+, see https://wiki.sagemath.org/Python3-Switch).

  The latter can be seen as a small follow-up to PR bitcoin#849 (commit 13c88ef).

ACKs for top commit:
  sipa:
    ACK 600c5ad
  real-or-random:
    ACK 600c5ad

Tree-SHA512: a9e52f6afbce65edd9ab14203612c3d423639f450fe8f0d269a3dda04bebefa95b607f7aa0faec864cb78b46d49f281632bb1277118749b7d8613e9f5dcc8f3d

src/ecdsa_impl.h

@@ -16,17 +16,8 @@
 #include "ecdsa.h"
 
 /** Group order for secp256k1 defined as 'n' in "Standards for Efficient Cryptography" (SEC2) 2.7.1
- *  sage: for t in xrange(1023, -1, -1):
- *     ..   p = 2**256 - 2**32 - t
- *     ..   if p.is_prime():
- *     ..     print '%x'%p
- *     ..     break
- *   'fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f'
- *  sage: a = 0
- *  sage: b = 7
- *  sage: F = FiniteField (p)
- *  sage: '%x' % (EllipticCurve ([F (a), F (b)]).order())
- *   'fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141'
+ *  $ sage -c 'load("secp256k1_params.sage"); print(hex(N))'
+ *  0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
  */
 static const secp256k1_fe secp256k1_ecdsa_const_order_as_fe = SECP256K1_FE_CONST(
     0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL,
@@ -35,12 +26,8 @@ static const secp256k1_fe secp256k1_ecdsa_const_order_as_fe = SECP256K1_FE_CONST
 
 /** Difference between field and order, values 'p' and 'n' values defined in
  *  "Standards for Efficient Cryptography" (SEC2) 2.7.1.
- *  sage: p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
- *  sage: a = 0
- *  sage: b = 7
- *  sage: F = FiniteField (p)
- *  sage: '%x' % (p - EllipticCurve ([F (a), F (b)]).order())
- *   '14551231950b75fc4402da1722fc9baee'
+ *  $ sage -c 'load("secp256k1_params.sage"); print(hex(P-N))'
+ *  0x14551231950b75fc4402da1722fc9baee
  */
 static const secp256k1_fe secp256k1_ecdsa_const_p_minus_order = SECP256K1_FE_CONST(
     0, 0, 0, 1, 0x45512319UL, 0x50B75FC4UL, 0x402DA172UL, 0x2FC9BAEEUL

src/tests.c

@@ -4035,22 +4035,15 @@ static void test_add_neg_y_diff_x(void) {
      * which this test is a regression test for.
      *
      * These points were generated in sage as
-     * # secp256k1 params
-     * F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
-     * C = EllipticCurve ([F (0), F (7)])
-     * G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
-     * N = FiniteField(G.order())
      *
-     * # endomorphism values (lambda is 1^{1/3} in N, beta is 1^{1/3} in F)
-     * x = polygen(N)
-     * lam  = (1 - x^3).roots()[1][0]
+     * load("secp256k1_params.sage")
      *
      * # random "bad pair"
      * P = C.random_element()
-     * Q = -int(lam) * P
-     * print "    P: %x %x" % P.xy()
-     * print "    Q: %x %x" % Q.xy()
-     * print "P + Q: %x %x" % (P + Q).xy()
+     * Q = -int(LAMBDA) * P
+     * print("    P: %x %x" % P.xy())
+     * print("    Q: %x %x" % Q.xy())
+     * print("P + Q: %x %x" % (P + Q).xy())
      */
     secp256k1_gej aj = SECP256K1_GEJ_CONST(
         0x8d24cd95, 0x0a355af1, 0x3c543505, 0x44238d30,