Improve optional 64-bit NIST-P224 implementation, and add NIST-P256 and

NIST-P521. (Now -DEC_NISTP_64_GCC_128 enables all three of these;
-DEC_NISTP224_64_GCC_128 no longer works.)

Submitted by: Google Inc.

1 files changed, 706 insertions, 548 deletions

diff --git a/crypto/ec/ecp_nistp224.c b/crypto/ec/ecp_nistp224.c
index 90c3589bdf..8b2c6d39c8 100644
--- a/crypto/ec/ecp_nistp224.c
+++ b/crypto/ec/ecp_nistp224.c
@@ -2,58 +2,20 @@
 /*
  * Written by Emilia Kasper (Google) for the OpenSSL project.
  */
-/* ====================================================================
- * Copyright (c) 2000-2010 The OpenSSL Project.  All rights reserved.
+/* Copyright 2011 Google Inc.
  *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
+ * Licensed under the Apache License, Version 2.0 (the "License");
  *
- * 1. Redistributions of source code must retain the above copyright
- *    notice, this list of conditions and the following disclaimer.
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
  *
- * 2. Redistributions in binary form must reproduce the above copyright
- *    notice, this list of conditions and the following disclaimer in
- *    the documentation and/or other materials provided with the
- *    distribution.
- *
- * 3. All advertising materials mentioning features or use of this
- *    software must display the following acknowledgment:
- *    "This product includes software developed by the OpenSSL Project
- *    for use in the OpenSSL Toolkit. (http://www.OpenSSL.org/)"
- *
- * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
- *    endorse or promote products derived from this software without
- *    prior written permission. For written permission, please contact
- *    licensing@OpenSSL.org.
- *
- * 5. Products derived from this software may not be called "OpenSSL"
- *    nor may "OpenSSL" appear in their names without prior written
- *    permission of the OpenSSL Project.
- *
- * 6. Redistributions of any form whatsoever must retain the following
- *    acknowledgment:
- *    "This product includes software developed by the OpenSSL Project
- *    for use in the OpenSSL Toolkit (http://www.OpenSSL.org/)"
- *
- * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
- * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
- * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
- * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
- * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
- * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
- * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
- * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
- * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
- * OF THE POSSIBILITY OF SUCH DAMAGE.
- * ====================================================================
- *
- * This product includes cryptographic software written by Eric Young
- * (eay@cryptsoft.com).  This product includes software written by Tim
- * Hudson (tjh@cryptsoft.com).
+ *     http://www.apache.org/licenses/LICENSE-2.0
  *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
  */
 
 /*
@@ -62,8 +24,7 @@
  * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
  * and Adam Langley's public domain 64-bit C implementation of curve25519
  */
-#include <openssl/opensslconf.h>
-#ifndef OPENSSL_NO_EC_NISTP224_64_GCC_128
+#ifdef EC_NISTP_64_GCC_128
 #include <stdint.h>
 #include <string.h>
 #include <openssl/err.h>
@@ -77,28 +38,39 @@
 #endif
 
 typedef uint8_t u8;
+typedef uint64_t u64;
+typedef int64_t s64;
 
 
 /******************************************************************************/
 /*		    INTERNAL REPRESENTATION OF FIELD ELEMENTS
  *
  * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
- * where each slice a_i is a 64-bit word, i.e., a field element is an fslice
- * array a with 4 elements, where a[i] = a_i.
- * Outputs from multiplications are represented as unreduced polynomials
+ * using 64-bit coefficients called 'limbs',
+ * and sometimes (for multiplication results) as
  * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
- * where each b_i is a 128-bit word. We ensure that inputs to each field
+ * using 128-bit coefficients called 'widelimbs'.
+ * A 4-limb representation is an 'felem';
+ * a 7-widelimb representation is a 'widefelem'.
+ * Even within felems, bits of adjacent limbs overlap, and we don't always
+ * reduce the representations: we ensure that inputs to each felem
  * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
  * and fit into a 128-bit word without overflow. The coefficients are then
- * again partially reduced to a_i < 2^57. We only reduce to the unique minimal
- * representation at the end of the computation.
- *
+ * again partially reduced to obtain an felem satisfying a_i < 2^57.
+ * We only reduce to the unique minimal representation at the end of the
+ * computation.
  */
 
-typedef uint64_t fslice;
+typedef uint64_t limb;
+typedef uint128_t widelimb;
+
+typedef limb felem[4];
+typedef widelimb widefelem[7];
 
 /* Field element represented as a byte arrary.
- * 28*8 = 224 bits is also the group order size for the elliptic curve.  */
+ * 28*8 = 224 bits is also the group order size for the elliptic curve,
+ * and we also use this type for scalars for point multiplication.
+  */
 typedef u8 felem_bytearray[28];
 
 static const felem_bytearray nistp224_curve_params[5] = {
@@ -120,72 +92,143 @@ static const felem_bytearray nistp224_curve_params[5] = {
 };
 
 /* Precomputed multiples of the standard generator
- * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for
- * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity,
- * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc.
- * Points are given in Jacobian projective coordinates: words 0-3 represent the
- * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the
- * Y-coordinate and words 8-11 represent the Z-coordinate. */
-static const fslice gmul[16][3][4] = {
-	{{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
-	 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
-	 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
-	 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
-	 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
-	 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
-	 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
-	 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
-	 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
-	 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
-	 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
-	 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
-	 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
-	 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
-	 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
-	 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
-	 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
-	{{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
-	 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
-	 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}
-};
+ * Points are given in coordinates (X, Y, Z) where Z normally is 1
+ * (0 for the point at infinity).
+ * For each field element, slice a_0 is word 0, etc.
+ *
+ * The table has 2 * 16 elements, starting with the following:
+ * index | bits    | point
+ * ------+---------+------------------------------
+ *     0 | 0 0 0 0 | 0G
+ *     1 | 0 0 0 1 | 1G
+ *     2 | 0 0 1 0 | 2^56G
+ *     3 | 0 0 1 1 | (2^56 + 1)G
+ *     4 | 0 1 0 0 | 2^112G
+ *     5 | 0 1 0 1 | (2^112 + 1)G
+ *     6 | 0 1 1 0 | (2^112 + 2^56)G
+ *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
+ *     8 | 1 0 0 0 | 2^168G
+ *     9 | 1 0 0 1 | (2^168 + 1)G
+ *    10 | 1 0 1 0 | (2^168 + 2^56)G
+ *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
+ *    12 | 1 1 0 0 | (2^168 + 2^112)G
+ *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
+ *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
+ *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
+ * followed by a copy of this with each element multiplied by 2^28.
+ *
+ * The reason for this is so that we can clock bits into four different
+ * locations when doing simple scalar multiplies against the base point,
+ * and then another four locations using the second 16 elements.
+ */
+static const felem gmul[2][16][3] =
+{{{{0, 0, 0, 0},
+   {0, 0, 0, 0},
+   {0, 0, 0, 0}},
+  {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
+   {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
+   {1, 0, 0, 0}},
+  {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
+   {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
+   {1, 0, 0, 0}},
+  {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
+   {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
+   {1, 0, 0, 0}},
+  {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
+   {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
+   {1, 0, 0, 0}},
+  {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
+   {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
+   {1, 0, 0, 0}},
+  {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
+   {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
+   {1, 0, 0, 0}},
+  {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
+   {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
+   {1, 0, 0, 0}},
+  {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
+   {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
+   {1, 0, 0, 0}},
+  {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
+   {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
+   {1, 0, 0, 0}},
+  {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
+   {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
+   {1, 0, 0, 0}},
+  {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
+   {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
+   {1, 0, 0, 0}},
+  {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
+   {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
+   {1, 0, 0, 0}},
+  {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
+   {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
+   {1, 0, 0, 0}},
+  {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
+   {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
+   {1, 0, 0, 0}},
+  {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
+   {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
+   {1, 0, 0, 0}}},
+ {{{0, 0, 0, 0},
+   {0, 0, 0, 0},
+   {0, 0, 0, 0}},
+  {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
+   {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
+   {1, 0, 0, 0}},
+  {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
+   {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
+   {1, 0, 0, 0}},
+  {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
+   {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
+   {1, 0, 0, 0}},
+  {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
+   {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
+   {1, 0, 0, 0}},
+  {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
+   {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
+   {1, 0, 0, 0}},
+  {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
+   {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
+   {1, 0, 0, 0}},
+  {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
+   {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
+   {1, 0, 0, 0}},
+  {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
+   {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
+   {1, 0, 0, 0}},
+  {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
+   {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
+   {1, 0, 0, 0}},
+  {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
+   {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
+   {1, 0, 0, 0}},
+  {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
+   {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
+   {1, 0, 0, 0}},
+  {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
+   {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
+   {1, 0, 0, 0}},
+  {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
+   {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
+   {1, 0, 0, 0}},
+  {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
+   {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
+   {1, 0, 0, 0}},
+  {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
+   {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
+   {1, 0, 0, 0}}}};
 
 /* Precomputation for the group generator. */
 typedef struct {
-	fslice g_pre_comp[16][3][4];
+	felem g_pre_comp[2][16][3];
 	int references;
 } NISTP224_PRE_COMP;
 
 const EC_METHOD *EC_GFp_nistp224_method(void)
 	{
 	static const EC_METHOD ret = {
+		EC_FLAGS_DEFAULT_OCT,
 		NID_X9_62_prime_field,
 		ec_GFp_nistp224_group_init,
 		ec_GFp_simple_group_finish,
@@ -204,9 +247,9 @@ const EC_METHOD *EC_GFp_nistp224_method(void)
 		ec_GFp_simple_get_Jprojective_coordinates_GFp,
 		ec_GFp_simple_point_set_affine_coordinates,
 		ec_GFp_nistp224_point_get_affine_coordinates,
-		ec_GFp_simple_set_compressed_coordinates,
-		ec_GFp_simple_point2oct,
-		ec_GFp_simple_oct2point,
+                0 /* point_set_compressed_coordinates */,
+                0 /* point2oct */,
+                0 /* oct2point */,
 		ec_GFp_simple_add,
 		ec_GFp_simple_dbl,
 		ec_GFp_simple_invert,
@@ -229,7 +272,7 @@ const EC_METHOD *EC_GFp_nistp224_method(void)
 	}
 
 /* Helper functions to convert field elements to/from internal representation */
-static void bin28_to_felem(fslice out[4], const u8 in[28])
+static void bin28_to_felem(felem out, const u8 in[28])
 	{
 	out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
 	out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
@@ -237,7 +280,7 @@ static void bin28_to_felem(fslice out[4], const u8 in[28])
 	out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
 	}
 
-static void felem_to_bin28(u8 out[28], const fslice in[4])
+static void felem_to_bin28(u8 out[28], const felem in)
 	{
 	unsigned i;
 	for (i = 0; i < 7; ++i)
@@ -258,9 +301,9 @@ static void flip_endian(u8 *out, const u8 *in, unsigned len)
 	}
 
 /* From OpenSSL BIGNUM to internal representation */
-static int BN_to_felem(fslice out[4], const BIGNUM *bn)
+static int BN_to_felem(felem out, const BIGNUM *bn)
 	{
-        felem_bytearray b_in;
+	felem_bytearray b_in;
 	felem_bytearray b_out;
 	unsigned num_bytes;
 
@@ -284,7 +327,7 @@ static int BN_to_felem(fslice out[4], const BIGNUM *bn)
 	}
 
 /* From internal representation to OpenSSL BIGNUM */
-static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4])
+static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
 	{
 	felem_bytearray b_in, b_out;
 	felem_to_bin28(b_in, in);
@@ -302,8 +345,24 @@ static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4])
  *
  */
 
+static void felem_one(felem out)
+	{
+	out[0] = 1;
+	out[1] = 0;
+	out[2] = 0;
+	out[3] = 0;
+	}
+
+static void felem_assign(felem out, const felem in)
+	{
+	out[0] = in[0];
+	out[1] = in[1];
+	out[2] = in[2];
+	out[3] = in[3];
+	}
+
 /* Sum two field elements: out += in */
-static void felem_sum64(fslice out[4], const fslice in[4])
+static void felem_sum(felem out, const felem in)
 	{
 	out[0] += in[0];
 	out[1] += in[1];
@@ -311,14 +370,30 @@ static void felem_sum64(fslice out[4], const fslice in[4])
 	out[3] += in[3];
 	}
 
+/* Get negative value: out = -in */
+/* Assumes in[i] < 2^57 */
+static void felem_neg(felem out, const felem in)
+	{
+	static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
+	static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
+	static const limb two58m42m2 = (((limb) 1) << 58) -
+	    (((limb) 1) << 42) - (((limb) 1) << 2);
+
+	/* Set to 0 mod 2^224-2^96+1 to ensure out > in */
+	out[0] = two58p2 - in[0];
+	out[1] = two58m42m2 - in[1];
+	out[2] = two58m2 - in[2];
+	out[3] = two58m2 - in[3];
+	}
+
 /* Subtract field elements: out -= in */
 /* Assumes in[i] < 2^57 */
-static void felem_diff64(fslice out[4], const fslice in[4])
+static void felem_diff(felem out, const felem in)
 	{
-	static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2);
-	static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2);
-	static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) -
-	    (((uint64_t) 1) << 42) - (((uint64_t) 1) << 2);
+	static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
+	static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
+	static const limb two58m42m2 = (((limb) 1) << 58) -
+	    (((limb) 1) << 42) - (((limb) 1) << 2);
 
 	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
 	out[0] += two58p2;
@@ -332,15 +407,15 @@ static void felem_diff64(fslice out[4], const fslice in[4])
 	out[3] -= in[3];
 	}
 
-/* Subtract in unreduced 128-bit mode: out128 -= in128 */
+/* Subtract in unreduced 128-bit mode: out -= in */
 /* Assumes in[i] < 2^119 */
-static void felem_diff128(uint128_t out[7], const uint128_t in[4])
+static void widefelem_diff(widefelem out, const widefelem in)
 	{
-	static const uint128_t two120 = ((uint128_t) 1) << 120;
-	static const uint128_t two120m64 = (((uint128_t) 1) << 120) -
-		(((uint128_t) 1) << 64);
-	static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) -
-		(((uint128_t) 1) << 104) - (((uint128_t) 1) << 64);
+	static const widelimb two120 = ((widelimb) 1) << 120;
+	static const widelimb two120m64 = (((widelimb) 1) << 120) -
+		(((widelimb) 1) << 64);
+	static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
+		(((widelimb) 1) << 104) - (((widelimb) 1) << 64);
 
 	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
 	out[0] += two120;
@@ -362,14 +437,14 @@ static void felem_diff128(uint128_t out[7], const uint128_t in[4])
 
 /* Subtract in mixed mode: out128 -= in64 */
 /* in[i] < 2^63 */
-static void felem_diff_128_64(uint128_t out[7], const fslice in[4])
+static void felem_diff_128_64(widefelem out, const felem in)
 	{
-	static const uint128_t two64p8 = (((uint128_t) 1) << 64) +
-		(((uint128_t) 1) << 8);
-	static const uint128_t two64m8 = (((uint128_t) 1) << 64) -
-		(((uint128_t) 1) << 8);
-	static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) -
-		(((uint128_t) 1) << 48) - (((uint128_t) 1) << 8);
+	static const widelimb two64p8 = (((widelimb) 1) << 64) +
+		(((widelimb) 1) << 8);
+	static const widelimb two64m8 = (((widelimb) 1) << 64) -
+		(((widelimb) 1) << 8);
+	static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
+		(((widelimb) 1) << 48) - (((widelimb) 1) << 8);
 
 	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
 	out[0] += two64p8;
@@ -383,9 +458,9 @@ static void felem_diff_128_64(uint128_t out[7], const fslice in[4])
 	out[3] -= in[3];
 	}
 
-/* Multiply a field element by a scalar: out64 = out64 * scalar
+/* Multiply a field element by a scalar: out = out * scalar
  * The scalars we actually use are small, so results fit without overflow */
-static void felem_scalar64(fslice out[4], const fslice scalar)
+static void felem_scalar(felem out, const limb scalar)
 	{
 	out[0] *= scalar;
 	out[1] *= scalar;
@@ -393,9 +468,9 @@ static void felem_scalar64(fslice out[4], const fslice scalar)
 	out[3] *= scalar;
 	}
 
-/* Multiply an unreduced field element by a scalar: out128 = out128 * scalar
+/* Multiply an unreduced field element by a scalar: out = out * scalar
  * The scalars we actually use are small, so results fit without overflow */
-static void felem_scalar128(uint128_t out[7], const uint128_t scalar)
+static void widefelem_scalar(widefelem out, const widelimb scalar)
 	{
 	out[0] *= scalar;
 	out[1] *= scalar;
@@ -407,44 +482,47 @@ static void felem_scalar128(uint128_t out[7], const uint128_t scalar)
 	}
 
 /* Square a field element: out = in^2 */
-static void felem_square(uint128_t out[7], const fslice in[4])
+static void felem_square(widefelem out, const felem in)
 	{
-	out[0] = ((uint128_t) in[0]) * in[0];
-	out[1] = ((uint128_t) in[0]) * in[1] * 2;
-	out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1];
-	out[3] = ((uint128_t) in[0]) * in[3] * 2 +
-		((uint128_t) in[1]) * in[2] * 2;
-	out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2];
-	out[5] = ((uint128_t) in[2]) * in[3] * 2;
-	out[6] = ((uint128_t) in[3]) * in[3];
+	limb tmp0, tmp1, tmp2;
+	tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2];
+	out[0] = ((widelimb) in[0]) * in[0];
+	out[1] = ((widelimb) in[0]) * tmp1;
+	out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
+	out[3] = ((widelimb) in[3]) * tmp0 +
+		((widelimb) in[1]) * tmp2;
+	out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
+	out[5] = ((widelimb) in[3]) * tmp2;
+	out[6] = ((widelimb) in[3]) * in[3];
 	}
 
 /* Multiply two field elements: out = in1 * in2 */
-static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4])
+static void felem_mul(widefelem out, const felem in1, const felem in2)
 	{
-	out[0] = ((uint128_t) in1[0]) * in2[0];
-	out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0];
-	out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] +
-		((uint128_t) in1[2]) * in2[0];
-	out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] +
-		((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0];
-	out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] +
-		((uint128_t) in1[3]) * in2[1];
-	out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2];
-	out[6] = ((uint128_t) in1[3]) * in2[3];
+	out[0] = ((widelimb) in1[0]) * in2[0];
+	out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
+	out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
+		((widelimb) in1[2]) * in2[0];
+	out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
+		((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
+	out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
+		((widelimb) in1[3]) * in2[1];
+	out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
+	out[6] = ((widelimb) in1[3]) * in2[3];
 	}
 
-/* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126,
- * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */
-static void felem_reduce(fslice out[4], const uint128_t in[7])
+/* Reduce seven 128-bit coefficients to four 64-bit coefficients.
+ * Requires in[i] < 2^126,
+ * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
+static void felem_reduce(felem out, const widefelem in)
 	{
-	static const uint128_t two127p15 = (((uint128_t) 1) << 127) +
-		(((uint128_t) 1) << 15);
-	static const uint128_t two127m71 = (((uint128_t) 1) << 127) -
-		(((uint128_t) 1) << 71);
-	static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) -
-		(((uint128_t) 1) << 71) - (((uint128_t) 1) << 55);
-	uint128_t output[5];
+	static const widelimb two127p15 = (((widelimb) 1) << 127) +
+		(((widelimb) 1) << 15);
+	static const widelimb two127m71 = (((widelimb) 1) << 127) -
+		(((widelimb) 1) << 71);
+	static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
+		(((widelimb) 1) << 71) - (((widelimb) 1) << 55);
+	widelimb output[5];
 
 	/* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
 	output[0] = in[0] + two127p15;
@@ -455,30 +533,30 @@ static void felem_reduce(fslice out[4], const uint128_t in[7])
 
 	/* Eliminate in[4], in[5], in[6] */
 	output[4] += in[6] >> 16;
-	output[3] += (in[6]&0xffff) << 40;
+	output[3] += (in[6] & 0xffff) << 40;
 	output[2] -= in[6];
 
 	output[3] += in[5] >> 16;
-	output[2] += (in[5]&0xffff) << 40;
+	output[2] += (in[5] & 0xffff) << 40;
 	output[1] -= in[5];
 
 	output[2] += output[4] >> 16;
-	output[1] += (output[4]&0xffff) << 40;
+	output[1] += (output[4] & 0xffff) << 40;
 	output[0] -= output[4];
-	output[4] = 0;
 
 	/* Carry 2 -> 3 -> 4 */
 	output[3] += output[2] >> 56;
 	output[2] &= 0x00ffffffffffffff;
 
-	output[4] += output[3] >> 56;
+	output[4] = output[3] >> 56;
 	output[3] &= 0x00ffffffffffffff;
 
-	/* Now output[2] < 2^56, output[3] < 2^56 */
+	/* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
 
 	/* Eliminate output[4] */
 	output[2] += output[4] >> 16;
-	output[1] += (output[4]&0xffff) << 40;
+	/* output[2] < 2^56 + 2^56 = 2^57 */
+	output[1] += (output[4] & 0xffff) << 40;
 	output[0] -= output[4];
 
 	/* Carry 0 -> 1 -> 2 -> 3 */
@@ -486,76 +564,68 @@ static void felem_reduce(fslice out[4], const uint128_t in[7])
 	out[0] = output[0] & 0x00ffffffffffffff;
 
 	output[2] += output[1] >> 56;
+	/* output[2] < 2^57 + 2^72 */
 	out[1] = output[1] & 0x00ffffffffffffff;
 	output[3] += output[2] >> 56;
+	/* output[3] <= 2^56 + 2^16 */
 	out[2] = output[2] & 0x00ffffffffffffff;
 
 	/* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
-	 * out[3] < 2^57 (due to final carry) */
+	 * out[3] <= 2^56 + 2^16 (due to final carry),
+	 * so out < 2*p */
 	out[3] = output[3];
 	}
 
-/* Reduce to unique minimal representation */
-static void felem_contract(fslice out[4], const fslice in[4])
+static void felem_square_reduce(felem out, const felem in)
 	{
-	static const int64_t two56 = ((uint64_t) 1) << 56;
-	/* 0 <= in < 2^225 */
-	/* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
-	int64_t tmp[4], a;
-	tmp[0] = (int64_t) in[0] - (in[3] >> 56);
-	tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000);
-	tmp[2] = (int64_t) in[2];
-	tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff;
-
-	/* eliminate negative coefficients */
-	a = tmp[0] >> 63;
-	tmp[0] += two56 & a;
-	tmp[1] -= 1 & a;
-
-	a = tmp[1] >> 63;
-	tmp[1] += two56 & a;
-	tmp[2] -= 1 & a;
-
-	a = tmp[2] >> 63;
-	tmp[2] += two56 & a;
-	tmp[3] -= 1 & a;
-
-	a = tmp[3] >> 63;
-	tmp[3] += two56 & a;
-	tmp[0] += 1 & a;
-	tmp[1] -= (1 & a) << 40;
-
-	/* carry 1 -> 2 -> 3 */
-	tmp[2] += tmp[1] >> 56;
-	tmp[1] &= 0x00ffffffffffffff;
+	widefelem tmp;
+	felem_square(tmp, in);
+	felem_reduce(out, tmp);
+	}
 
-	tmp[3] += tmp[2] >> 56;
-	tmp[2] &= 0x00ffffffffffffff;
+static void felem_mul_reduce(felem out, const felem in1, const felem in2)
+	{
+	widefelem tmp;
+	felem_mul(tmp, in1, in2);
+	felem_reduce(out, tmp);
+	}
 
-	/* 0 <= in < 2^224 + 2^96 - 1 */
-	/* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
-	tmp[0] -= (tmp[3] >> 56);
-	tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000);
+/* Reduce to unique minimal representation.
+ * Requires 0 <= in < 2*p (always call felem_reduce first) */
+static void felem_contract(felem out, const felem in)
+	{
+	static const int64_t two56 = ((limb) 1) << 56;
+	/* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
+	/* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
+	int64_t tmp[4], a;
+	tmp[0] = in[0];
+	tmp[1] = in[1];
+	tmp[2] = in[2];
+	tmp[3] = in[3];
+	/* Case 1: a = 1 iff in >= 2^224 */
+	a = (in[3] >> 56);
+	tmp[0] -= a;
+	tmp[1] += a << 40;
 	tmp[3] &= 0x00ffffffffffffff;
+	/* Case 2: a = 0 iff p <= in < 2^224, i.e.,
+	 * the high 128 bits are all 1 and the lower part is non-zero */
+	a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
+		(((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
+	a &= 0x00ffffffffffffff;
+	/* turn a into an all-one mask (if a = 0) or an all-zero mask */
+	a = (a - 1) >> 63;
+	/* subtract 2^224 - 2^96 + 1 if a is all-one*/
+	tmp[3] &= a ^ 0xffffffffffffffff;
+	tmp[2] &= a ^ 0xffffffffffffffff;
+	tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
+	tmp[0] -= 1 & a;
 
-	/* eliminate negative coefficients */
+	/* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
+	 * be non-zero, so we only need one step */
 	a = tmp[0] >> 63;
 	tmp[0] += two56 & a;
 	tmp[1] -= 1 & a;
 
-	a = tmp[1] >> 63;
-	tmp[1] += two56 & a;
-	tmp[2] -= 1 & a;
-
-	a = tmp[2] >> 63;
-	tmp[2] += two56 & a;
-	tmp[3] -= 1 & a;
-
-	a = tmp[3] >> 63;
-	tmp[3] += two56 & a;
-	tmp[0] += 1 & a;
-	tmp[1] -= (1 & a) << 40;
-
 	/* carry 1 -> 2 -> 3 */
 	tmp[2] += tmp[1] >> 56;
 	tmp[1] &= 0x00ffffffffffffff;
@@ -563,27 +633,7 @@ static void felem_contract(fslice out[4], const fslice in[4])
 	tmp[3] += tmp[2] >> 56;
 	tmp[2] &= 0x00ffffffffffffff;
 
-	/* Now 0 <= in < 2^224 */
-
-	/* if in > 2^224 - 2^96, reduce */
-	/* a = 0 iff in > 2^224 - 2^96, i.e.,
-	 * the high 128 bits are all 1 and the lower part is non-zero */
-	a = (tmp[3] + 1) | (tmp[2] + 1) |
-		((tmp[1] | 0x000000ffffffffff) + 1) |
-		((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63));
-	/* turn a into an all-one mask (if a = 0) or an all-zero mask */
-	a = ((a & 0x00ffffffffffffff) - 1) >> 63;
-	/* subtract 2^224 - 2^96 + 1 if a is all-one*/
-	tmp[3] &= a ^ 0xffffffffffffffff;
-	tmp[2] &= a ^ 0xffffffffffffffff;
-	tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
-	tmp[0] -= 1 & a;
-	/* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
-	 * non-zero, so we only need one step */
-	a = tmp[0] >> 63;
-	tmp[0] += two56 & a;
-	tmp[1] -= 1 & a;
-
+	/* Now 0 <= out < p */
 	out[0] = tmp[0];
 	out[1] = tmp[1];
 	out[2] = tmp[2];
@@ -594,9 +644,9 @@ static void felem_contract(fslice out[4], const fslice in[4])
  * We know that field elements are reduced to in < 2^225,
  * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
  * and 2^225 - 2^97 + 2 */
-static fslice felem_is_zero(const fslice in[4])
+static limb felem_is_zero(const felem in)
 	{
-	fslice zero, two224m96p1, two225m97p2;
+	limb zero, two224m96p1, two225m97p2;
 
 	zero = in[0] | in[1] | in[2] | in[3];
 	zero = (((int64_t)(zero) - 1) >> 63) & 1;
@@ -609,12 +659,17 @@ static fslice felem_is_zero(const fslice in[4])
 	return (zero | two224m96p1 | two225m97p2);
 	}
 
+static limb felem_is_zero_int(const felem in)
+	{
+	return (int) (felem_is_zero(in) & ((limb)1));
+	}
+
 /* Invert a field element */
 /* Computation chain copied from djb's code */
-static void felem_inv(fslice out[4], const fslice in[4])
+static void felem_inv(felem out, const felem in)
 	{
-	fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4];
-	uint128_t tmp[7];
+	felem ftmp, ftmp2, ftmp3, ftmp4;
+	widefelem tmp;
 	unsigned i;
 
 	felem_square(tmp, in); felem_reduce(ftmp, tmp);		/* 2 */
@@ -673,34 +728,18 @@ static void felem_inv(fslice out[4], const fslice in[4])
  * if icopy == 1, copy in to out,
  * if icopy == 0, copy out to itself. */
 static void
-copy_conditional(fslice *out, const fslice *in, unsigned len, fslice icopy)
+copy_conditional(felem out, const felem in, limb icopy)
 	{
 	unsigned i;
 	/* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
-	const fslice copy = -icopy;
-	for (i = 0; i < len; ++i)
+	const limb copy = -icopy;
+	for (i = 0; i < 4; ++i)
 		{
-		const fslice tmp = copy & (in[i] ^ out[i]);
+		const limb tmp = copy & (in[i] ^ out[i]);
 		out[i] ^= tmp;
 		}
 	}
 
-/* Copy in constant time:
- * if isel == 1, copy in2 to out,
- * if isel == 0, copy in1 to out. */
-static void select_conditional(fslice *out, const fslice *in1, const fslice *in2,
-	unsigned len, fslice isel)
-	{
-	unsigned i;
-	/* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */
-	const fslice sel = -isel;
-	for (i = 0; i < len; ++i)
-		{
-		const fslice tmp = sel & (in1[i] ^ in2[i]);
-		out[i] = in1[i] ^ tmp;
-		}
-}
-
 /******************************************************************************/
 /*			 ELLIPTIC CURVE POINT OPERATIONS
  *
@@ -718,17 +757,14 @@ static void select_conditional(fslice *out, const fslice *in1, const fslice *in2
  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
  * while x_out == y_in is not (maybe this works, but it's not tested). */
 static void
-point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4],
-	     const fslice x_in[4], const fslice y_in[4], const fslice z_in[4])
+point_double(felem x_out, felem y_out, felem z_out,
+             const felem x_in, const felem y_in, const felem z_in)
 	{
-	uint128_t tmp[7], tmp2[7];
-	fslice delta[4];
-	fslice gamma[4];
-	fslice beta[4];
-	fslice alpha[4];