diff --git a/.gitignore b/.gitignore
index 5b2e4ac6..085bd54d 100644
--- a/.gitignore
+++ b/.gitignore
@@ -33,6 +33,8 @@ libtool
 *.lo
 *.o
 *~
+*.log
+*.trs
 src/libsecp256k1-config.h
 src/libsecp256k1-config.h.in
 src/ecmult_static_context.h
diff --git a/.travis.yml b/.travis.yml
index 2fbf8fcc..70d12b7e 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -17,19 +17,19 @@ compiler:
   - gcc
 env:
   global:
-    - FIELD=auto  BIGNUM=auto  SCALAR=auto  ENDOMORPHISM=no  STATICPRECOMPUTATION=yes  ECMULTGENPRECISION=auto  ASM=no  BUILD=check  EXTRAFLAGS=  HOST=  ECDH=no  RECOVERY=no  EXPERIMENTAL=no CTIMETEST=yes BENCH=yes ITERS=2 GENERATOR=no RANGEPROOF=no WHITELIST=no SCHNORRSIG=no MUSIG=no
+    - WIDEMUL=auto  BIGNUM=auto  ENDOMORPHISM=no  STATICPRECOMPUTATION=yes  ECMULTGENPRECISION=auto  ASM=no  BUILD=check  EXTRAFLAGS=  HOST=  ECDH=no  RECOVERY=no  EXPERIMENTAL=no CTIMETEST=yes BENCH=yes ITERS=2 GENERATOR=no RANGEPROOF=no WHITELIST=no SCHNORRSIG=no MUSIG=no
   matrix:
-    - SCALAR=32bit    FIELD=32bit       EXPERIMENTAL=yes RANGEPROOF=yes WHITELIST=yes GENERATOR=yes SCHNORRSIG=yes MUSIG=yes
-    - FIELD=64bit     EXPERIMENTAL=yes RANGEPROOF=yes WHITELIST=yes GENERATOR=yes  SCHNORRSIG=yes MUSIG=yes
-    - SCALAR=32bit    RECOVERY=yes
-    - SCALAR=32bit    FIELD=32bit       ECDH=yes  EXPERIMENTAL=yes
-    - SCALAR=64bit
-    - FIELD=64bit     RECOVERY=yes
-    - FIELD=64bit     ENDOMORPHISM=yes
-    - FIELD=64bit     ENDOMORPHISM=yes  ECDH=yes EXPERIMENTAL=yes    SCHNORRSIG=yes MUSIG=yes
-    - FIELD=64bit                       ASM=x86_64
-    - FIELD=64bit     ENDOMORPHISM=yes  ASM=x86_64
-    - FIELD=32bit     ENDOMORPHISM=yes
+    - WIDEMUL=int64   EXPERIMENTAL=yes RANGEPROOF=yes WHITELIST=yes GENERATOR=yes SCHNORRSIG=yes MUSIG=yes
+    - WIDEMUL=int128  EXPERIMENTAL=yes RANGEPROOF=yes WHITELIST=yes GENERATOR=yes  SCHNORRSIG=yes MUSIG=yes
+    - WIDEMUL=int64   RECOVERY=yes
+    - WIDEMUL=int64   ECDH=yes  EXPERIMENTAL=yes
+    - WIDEMUL=int64   ENDOMORPHISM=yes
+    - WIDEMUL=int128
+    - WIDEMUL=int128  RECOVERY=yes
+    - WIDEMUL=int128  ENDOMORPHISM=yes
+    - WIDEMUL=int128  ENDOMORPHISM=yes  ECDH=yes EXPERIMENTAL=yes    SCHNORRSIG=yes MUSIG=yes
+    - WIDEMUL=int128                    ASM=x86_64
+    - WIDEMUL=int128  ENDOMORPHISM=yes  ASM=x86_64
     - BIGNUM=no
     - BIGNUM=no       ENDOMORPHISM=yes RECOVERY=yes EXPERIMENTAL=yes SCHNORRSIG=yes MUSIG=yes
     - BIGNUM=no       STATICPRECOMPUTATION=no
@@ -85,6 +85,10 @@ matrix:
             - valgrind
             - libtool-bin
             - libc6-dbg:i386
+    # S390x build (big endian system)
+    - compiler: gcc
+      env: HOST=s390x-unknown-linux-gnu ECDH=yes RECOVERY=yes EXPERIMENTAL=yes CTIMETEST=
+      arch: s390x
 
 # We use this to install macOS dependencies instead of the built in `homebrew` plugin,
 # because in xcode earlier than 11 they have a bug requiring updating the system which overall takes ~8 minutes.
diff --git a/Makefile.am b/Makefile.am
index c29a9051..dcc8693f 100644
--- a/Makefile.am
+++ b/Makefile.am
@@ -34,9 +34,11 @@ noinst_HEADERS += src/field_5x52.h
 noinst_HEADERS += src/field_5x52_impl.h
 noinst_HEADERS += src/field_5x52_int128_impl.h
 noinst_HEADERS += src/field_5x52_asm_impl.h
+noinst_HEADERS += src/assumptions.h
 noinst_HEADERS += src/util.h
 noinst_HEADERS += src/scratch.h
 noinst_HEADERS += src/scratch_impl.h
+noinst_HEADERS += src/selftest.h
 noinst_HEADERS += src/testrand.h
 noinst_HEADERS += src/testrand_impl.h
 noinst_HEADERS += src/hash.h
@@ -99,7 +101,7 @@ if VALGRIND_ENABLED
 tests_CPPFLAGS += -DVALGRIND
 noinst_PROGRAMS += valgrind_ctime_test
 valgrind_ctime_test_SOURCES = src/valgrind_ctime_test.c
-valgrind_ctime_test_LDADD = libsecp256k1.la $(SECP_LIBS) $(SECP_TEST_LIBS) $(COMMON_LIB)
+valgrind_ctime_test_LDADD = libsecp256k1.la $(SECP_LIBS) $(SECP_LIBS) $(COMMON_LIB)
 endif
 if !ENABLE_COVERAGE
 tests_CPPFLAGS += -DVERIFY
diff --git a/TODO b/TODO
deleted file mode 100644
index a300e1c5..00000000
--- a/TODO
+++ /dev/null
@@ -1,3 +0,0 @@
-* Unit tests for fieldelem/groupelem, including ones intended to
-  trigger fieldelem's boundary cases.
-* Complete constant-time operations for signing/keygen
diff --git a/build-aux/m4/bitcoin_secp.m4 b/build-aux/m4/bitcoin_secp.m4
index 1b2b71e6..57595f44 100644
--- a/build-aux/m4/bitcoin_secp.m4
+++ b/build-aux/m4/bitcoin_secp.m4
@@ -1,8 +1,3 @@
-dnl libsecp25k1 helper checks
-AC_DEFUN([SECP_INT128_CHECK],[
-has_int128=$ac_cv_type___int128
-])
-
 dnl escape "$0x" below using the m4 quadrigaph @S|@, and escape it again with a \ for the shell.
 AC_DEFUN([SECP_64BIT_ASM_CHECK],[
 AC_MSG_CHECKING(for x86_64 assembly availability)
diff --git a/configure.ac b/configure.ac
index 811aea0d..ddfc76bd 100644
--- a/configure.ac
+++ b/configure.ac
@@ -176,15 +176,13 @@ AC_ARG_ENABLE(reduced_surjection_proof_size,
     [use_reduced_surjection_proof_size=$enableval],
     [use_reduced_surjection_proof_size=no])
 
-AC_ARG_WITH([field], [AS_HELP_STRING([--with-field=64bit|32bit|auto],
-[finite field implementation to use [default=auto]])],[req_field=$withval], [req_field=auto])
+dnl Test-only override of the (autodetected by the C code) "widemul" setting.
+dnl Legal values are int64 (for [u]int64_t), int128 (for [unsigned] __int128), and auto (the default).
+AC_ARG_WITH([test-override-wide-multiply], [] ,[set_widemul=$withval], [set_widemul=auto])
 
 AC_ARG_WITH([bignum], [AS_HELP_STRING([--with-bignum=gmp|no|auto],
 [bignum implementation to use [default=auto]])],[req_bignum=$withval], [req_bignum=auto])
 
-AC_ARG_WITH([scalar], [AS_HELP_STRING([--with-scalar=64bit|32bit|auto],
-[scalar implementation to use [default=auto]])],[req_scalar=$withval], [req_scalar=auto])
-
 AC_ARG_WITH([asm], [AS_HELP_STRING([--with-asm=x86_64|arm|no|auto],
 [assembly optimizations to use (experimental: arm) [default=auto]])],[req_asm=$withval], [req_asm=auto])
 
@@ -205,8 +203,6 @@ AC_ARG_WITH([ecmult-gen-precision], [AS_HELP_STRING([--with-ecmult-gen-precision
 )],
 [req_ecmult_gen_precision=$withval], [req_ecmult_gen_precision=auto])
 
-AC_CHECK_TYPES([__int128])
-
 AC_CHECK_HEADER([valgrind/memcheck.h], [enable_valgrind=yes], [enable_valgrind=no], [])
 AM_CONDITIONAL([VALGRIND_ENABLED],[test "$enable_valgrind" = "yes"])
 
@@ -312,63 +308,6 @@ else
   esac
 fi
 
-if test x"$req_field" = x"auto"; then
-  if test x"set_asm" = x"x86_64"; then
-    set_field=64bit
-  fi
-  if test x"$set_field" = x; then
-    SECP_INT128_CHECK
-    if test x"$has_int128" = x"yes"; then
-      set_field=64bit
-    fi
-  fi
-  if test x"$set_field" = x; then
-    set_field=32bit
-  fi
-else
-  set_field=$req_field
-  case $set_field in
-  64bit)
-    if test x"$set_asm" != x"x86_64"; then
-      SECP_INT128_CHECK
-      if test x"$has_int128" != x"yes"; then
-        AC_MSG_ERROR([64bit field explicitly requested but neither __int128 support or x86_64 assembly available])
-      fi
-    fi
-    ;;
-  32bit)
-    ;;
-  *)
-    AC_MSG_ERROR([invalid field implementation selection])
-    ;;
-  esac
-fi
-
-if test x"$req_scalar" = x"auto"; then
-  SECP_INT128_CHECK
-  if test x"$has_int128" = x"yes"; then
-    set_scalar=64bit
-  fi
-  if test x"$set_scalar" = x; then
-    set_scalar=32bit
-  fi
-else
-  set_scalar=$req_scalar
-  case $set_scalar in
-  64bit)
-    SECP_INT128_CHECK
-    if test x"$has_int128" != x"yes"; then
-      AC_MSG_ERROR([64bit scalar explicitly requested but __int128 support not available])
-    fi
-    ;;
-  32bit)
-    ;;
-  *)
-    AC_MSG_ERROR([invalid scalar implementation selected])
-    ;;
-  esac
-fi
-
 if test x"$req_bignum" = x"auto"; then
   SECP_GMP_CHECK
   if test x"$has_gmp" = x"yes"; then
@@ -412,16 +351,18 @@ no)
   ;;
 esac
 
-# select field implementation
-case $set_field in
-64bit)
-  AC_DEFINE(USE_FIELD_5X52, 1, [Define this symbol to use the FIELD_5X52 implementation])
+# select wide multiplication implementation
+case $set_widemul in
+int128)
+  AC_DEFINE(USE_FORCE_WIDEMUL_INT128, 1, [Define this symbol to force the use of the (unsigned) __int128 based wide multiplication implementation])
   ;;
-32bit)
-  AC_DEFINE(USE_FIELD_10X26, 1, [Define this symbol to use the FIELD_10X26 implementation])
+int64)
+  AC_DEFINE(USE_FORCE_WIDEMUL_INT64, 1, [Define this symbol to force the use of the (u)int64_t based wide multiplication implementation])
+  ;;
+auto)
   ;;
 *)
-  AC_MSG_ERROR([invalid field implementation])
+  AC_MSG_ERROR([invalid wide multiplication implementation])
   ;;
 esac
 
@@ -443,19 +384,6 @@ no)
   ;;
 esac
 
-#select scalar implementation
-case $set_scalar in
-64bit)
-  AC_DEFINE(USE_SCALAR_4X64, 1, [Define this symbol to use the 4x64 scalar implementation])
-  ;;
-32bit)
-  AC_DEFINE(USE_SCALAR_8X32, 1, [Define this symbol to use the 8x32 scalar implementation])
-  ;;
-*)
-  AC_MSG_ERROR([invalid scalar implementation])
-  ;;
-esac
-
 #set ecmult window size
 if test x"$req_ecmult_window" = x"auto"; then
   set_ecmult_window=15
@@ -564,8 +492,6 @@ if test x"$enable_module_surjectionproof" = x"yes"; then
   AC_DEFINE(ENABLE_MODULE_SURJECTIONPROOF, 1, [Define this symbol to enable the surjection proof module])
 fi
 
-AC_C_BIGENDIAN()
-
 if test x"$use_external_asm" = x"yes"; then
   AC_DEFINE(USE_EXTERNAL_ASM, 1, [Define this symbol if an external (non-inline) assembly implementation is used])
 fi
@@ -682,10 +608,12 @@ echo "  module schnorrsig       = $enable_module_schnorrsig"
 echo
 echo "  asm                     = $set_asm"
 echo "  bignum                  = $set_bignum"
-echo "  field                   = $set_field"
-echo "  scalar                  = $set_scalar"
 echo "  ecmult window size      = $set_ecmult_window"
 echo "  ecmult gen prec. bits   = $set_ecmult_gen_precision"
+dnl Hide test-only options unless they're used.
+if test x"$set_widemul" != xauto; then
+echo "  wide multiplication     = $set_widemul"
+fi
 echo
 echo "  valgrind                = $enable_valgrind"
 echo "  CC                      = $CC"
diff --git a/contrib/travis.sh b/contrib/travis.sh
index 39d850e3..fad6bc18 100755
--- a/contrib/travis.sh
+++ b/contrib/travis.sh
@@ -14,7 +14,7 @@ fi
 
 ./configure \
     --enable-experimental="$EXPERIMENTAL" --enable-endomorphism="$ENDOMORPHISM" \
-    --with-field="$FIELD" --with-bignum="$BIGNUM" --with-asm="$ASM" --with-scalar="$SCALAR" \
+    --with-test-override-wide-multiply="$WIDEMUL" --with-bignum="$BIGNUM" --with-asm="$ASM" \
     --enable-ecmult-static-precomputation="$STATICPRECOMPUTATION" --with-ecmult-gen-precision="$ECMULTGENPRECISION" \
     --enable-module-ecdh="$ECDH" --enable-module-recovery="$RECOVERY" \
     --enable-module-rangeproof="$RANGEPROOF" --enable-module-whitelist="$WHITELIST" --enable-module-generator="$GENERATOR" \
diff --git a/src/assumptions.h b/src/assumptions.h
new file mode 100644
index 00000000..f9d4e8e7
--- /dev/null
+++ b/src/assumptions.h
@@ -0,0 +1,74 @@
+/**********************************************************************
+ * Copyright (c) 2020 Pieter Wuille                                   *
+ * Distributed under the MIT software license, see the accompanying   *
+ * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
+ **********************************************************************/
+
+#ifndef SECP256K1_ASSUMPTIONS_H
+#define SECP256K1_ASSUMPTIONS_H
+
+#include "util.h"
+
+/* This library, like most software, relies on a number of compiler implementation defined (but not undefined)
+   behaviours. Although the behaviours we require are essentially universal we test them specifically here to
+   reduce the odds of experiencing an unwelcome surprise.
+*/
+
+struct secp256k1_assumption_checker {
+    /* This uses a trick to implement a static assertion in C89: a type with an array of negative size is not
+       allowed. */
+    int dummy_array[(
+        /* Bytes are 8 bits. */
+        CHAR_BIT == 8 &&
+
+        /* Conversions from unsigned to signed outside of the bounds of the signed type are
+           implementation-defined. Verify that they function as reinterpreting the lower
+           bits of the input in two's complement notation. Do this for conversions:
+           - from uint(N)_t to int(N)_t with negative result
+           - from uint(2N)_t to int(N)_t with negative result
+           - from int(2N)_t to int(N)_t with negative result
+           - from int(2N)_t to int(N)_t with positive result */
+
+        /* To int8_t. */
+        ((int8_t)(uint8_t)0xAB == (int8_t)-(int8_t)0x55) &&
+        ((int8_t)(uint16_t)0xABCD == (int8_t)-(int8_t)0x33) &&
+        ((int8_t)(int16_t)(uint16_t)0xCDEF == (int8_t)(uint8_t)0xEF) &&
+        ((int8_t)(int16_t)(uint16_t)0x9234 == (int8_t)(uint8_t)0x34) &&
+
+        /* To int16_t. */
+        ((int16_t)(uint16_t)0xBCDE == (int16_t)-(int16_t)0x4322) &&
+        ((int16_t)(uint32_t)0xA1B2C3D4 == (int16_t)-(int16_t)0x3C2C) &&
+        ((int16_t)(int32_t)(uint32_t)0xC1D2E3F4 == (int16_t)(uint16_t)0xE3F4) &&
+        ((int16_t)(int32_t)(uint32_t)0x92345678 == (int16_t)(uint16_t)0x5678) &&
+
+        /* To int32_t. */
+        ((int32_t)(uint32_t)0xB2C3D4E5 == (int32_t)-(int32_t)0x4D3C2B1B) &&
+        ((int32_t)(uint64_t)0xA123B456C789D012ULL == (int32_t)-(int32_t)0x38762FEE) &&
+        ((int32_t)(int64_t)(uint64_t)0xC1D2E3F4A5B6C7D8ULL == (int32_t)(uint32_t)0xA5B6C7D8) &&
+        ((int32_t)(int64_t)(uint64_t)0xABCDEF0123456789ULL == (int32_t)(uint32_t)0x23456789) &&
+
+        /* To int64_t. */
+        ((int64_t)(uint64_t)0xB123C456D789E012ULL == (int64_t)-(int64_t)0x4EDC3BA928761FEEULL) &&
+#if defined(SECP256K1_WIDEMUL_INT128)
+        ((int64_t)(((uint128_t)0xA1234567B8901234ULL << 64) + 0xC5678901D2345678ULL) == (int64_t)-(int64_t)0x3A9876FE2DCBA988ULL) &&
+        (((int64_t)(int128_t)(((uint128_t)0xB1C2D3E4F5A6B7C8ULL << 64) + 0xD9E0F1A2B3C4D5E6ULL)) == (int64_t)(uint64_t)0xD9E0F1A2B3C4D5E6ULL) &&
+        (((int64_t)(int128_t)(((uint128_t)0xABCDEF0123456789ULL << 64) + 0x0123456789ABCDEFULL)) == (int64_t)(uint64_t)0x0123456789ABCDEFULL) &&
+
+        /* To int128_t. */
+        ((int128_t)(((uint128_t)0xB1234567C8901234ULL << 64) + 0xD5678901E2345678ULL) == (int128_t)(-(int128_t)0x8E1648B3F50E80DCULL * 0x8E1648B3F50E80DDULL + 0x5EA688D5482F9464ULL)) &&
+#endif
+
+        /* Right shift on negative signed values is implementation defined. Verify that it
+           acts as a right shift in two's complement with sign extension (i.e duplicating
+           the top bit into newly added bits). */
+        ((((int8_t)0xE8) >> 2) == (int8_t)(uint8_t)0xFA) &&
+        ((((int16_t)0xE9AC) >> 4) == (int16_t)(uint16_t)0xFE9A) &&
+        ((((int32_t)0x937C918A) >> 9) == (int32_t)(uint32_t)0xFFC9BE48) &&
+        ((((int64_t)0xA8B72231DF9CF4B9ULL) >> 19) == (int64_t)(uint64_t)0xFFFFF516E4463BF3ULL) &&
+#if defined(SECP256K1_WIDEMUL_INT128)
+        ((((int128_t)(((uint128_t)0xCD833A65684A0DBCULL << 64) + 0xB349312F71EA7637ULL)) >> 39) == (int128_t)(((uint128_t)0xFFFFFFFFFF9B0674ULL << 64) + 0xCAD0941B79669262ULL)) &&
+#endif
+    1) * 2 - 1];
+};
+
+#endif /* SECP256K1_ASSUMPTIONS_H */
diff --git a/src/basic-config.h b/src/basic-config.h
index b5ccd839..83dbe6f2 100644
--- a/src/basic-config.h
+++ b/src/basic-config.h
@@ -14,24 +14,20 @@
 #undef USE_ENDOMORPHISM
 #undef USE_EXTERNAL_ASM
 #undef USE_EXTERNAL_DEFAULT_CALLBACKS
-#undef USE_FIELD_10X26
-#undef USE_FIELD_5X52
 #undef USE_FIELD_INV_BUILTIN
 #undef USE_FIELD_INV_NUM
 #undef USE_NUM_GMP
 #undef USE_NUM_NONE
-#undef USE_SCALAR_4X64
-#undef USE_SCALAR_8X32
 #undef USE_SCALAR_INV_BUILTIN
 #undef USE_SCALAR_INV_NUM
+#undef USE_FORCE_WIDEMUL_INT64
+#undef USE_FORCE_WIDEMUL_INT128
 #undef ECMULT_WINDOW_SIZE
-#undef HAVE___INT128 /* used in util.h */
 
 #define USE_NUM_NONE 1
 #define USE_FIELD_INV_BUILTIN 1
 #define USE_SCALAR_INV_BUILTIN 1
-#define USE_FIELD_10X26 1
-#define USE_SCALAR_8X32 1
+#define USE_WIDEMUL_64 1
 #define ECMULT_WINDOW_SIZE 15
 
 #endif /* USE_BASIC_CONFIG */
diff --git a/src/bench_internal.c b/src/bench_internal.c
index 20759127..9687fe44 100644
--- a/src/bench_internal.c
+++ b/src/bench_internal.c
@@ -7,6 +7,7 @@
 
 #include "include/secp256k1.h"
 
+#include "assumptions.h"
 #include "util.h"
 #include "hash_impl.h"
 #include "num_impl.h"
@@ -19,10 +20,10 @@
 #include "secp256k1.c"
 
 typedef struct {
-    secp256k1_scalar scalar_x, scalar_y;
-    secp256k1_fe fe_x, fe_y;
-    secp256k1_ge ge_x, ge_y;
-    secp256k1_gej gej_x, gej_y;
+    secp256k1_scalar scalar[2];
+    secp256k1_fe fe[4];
+    secp256k1_ge ge[2];
+    secp256k1_gej gej[2];
     unsigned char data[64];
     int wnaf[256];
 } bench_inv;
@@ -30,30 +31,53 @@ typedef struct {
 void bench_setup(void* arg) {
     bench_inv *data = (bench_inv*)arg;
 
-    static const unsigned char init_x[32] = {
-        0x02, 0x03, 0x05, 0x07, 0x0b, 0x0d, 0x11, 0x13,
-        0x17, 0x1d, 0x1f, 0x25, 0x29, 0x2b, 0x2f, 0x35,
-        0x3b, 0x3d, 0x43, 0x47, 0x49, 0x4f, 0x53, 0x59,
-        0x61, 0x65, 0x67, 0x6b, 0x6d, 0x71, 0x7f, 0x83
+    static const unsigned char init[4][32] = {
+        /* Initializer for scalar[0], fe[0], first half of data, the X coordinate of ge[0],
+           and the (implied affine) X coordinate of gej[0]. */
+        {
+            0x02, 0x03, 0x05, 0x07, 0x0b, 0x0d, 0x11, 0x13,
+            0x17, 0x1d, 0x1f, 0x25, 0x29, 0x2b, 0x2f, 0x35,
+            0x3b, 0x3d, 0x43, 0x47, 0x49, 0x4f, 0x53, 0x59,
+            0x61, 0x65, 0x67, 0x6b, 0x6d, 0x71, 0x7f, 0x83
+        },
+        /* Initializer for scalar[1], fe[1], first half of data, the X coordinate of ge[1],
+           and the (implied affine) X coordinate of gej[1]. */
+        {
+            0x82, 0x83, 0x85, 0x87, 0x8b, 0x8d, 0x81, 0x83,
+            0x97, 0xad, 0xaf, 0xb5, 0xb9, 0xbb, 0xbf, 0xc5,
+            0xdb, 0xdd, 0xe3, 0xe7, 0xe9, 0xef, 0xf3, 0xf9,
+            0x11, 0x15, 0x17, 0x1b, 0x1d, 0xb1, 0xbf, 0xd3
+        },
+        /* Initializer for fe[2] and the Z coordinate of gej[0]. */
+        {
+            0x3d, 0x2d, 0xef, 0xf4, 0x25, 0x98, 0x4f, 0x5d,
+            0xe2, 0xca, 0x5f, 0x41, 0x3f, 0x3f, 0xce, 0x44,
+            0xaa, 0x2c, 0x53, 0x8a, 0xc6, 0x59, 0x1f, 0x38,
+            0x38, 0x23, 0xe4, 0x11, 0x27, 0xc6, 0xa0, 0xe7
+        },
+        /* Initializer for fe[3] and the Z coordinate of gej[1]. */
+        {
+            0xbd, 0x21, 0xa5, 0xe1, 0x13, 0x50, 0x73, 0x2e,
+            0x52, 0x98, 0xc8, 0x9e, 0xab, 0x00, 0xa2, 0x68,
+            0x43, 0xf5, 0xd7, 0x49, 0x80, 0x72, 0xa7, 0xf3,
+            0xd7, 0x60, 0xe6, 0xab, 0x90, 0x92, 0xdf, 0xc5
+        }
     };
 
-    static const unsigned char init_y[32] = {
-        0x82, 0x83, 0x85, 0x87, 0x8b, 0x8d, 0x81, 0x83,
-        0x97, 0xad, 0xaf, 0xb5, 0xb9, 0xbb, 0xbf, 0xc5,
-        0xdb, 0xdd, 0xe3, 0xe7, 0xe9, 0xef, 0xf3, 0xf9,
-        0x11, 0x15, 0x17, 0x1b, 0x1d, 0xb1, 0xbf, 0xd3
-    };
-
-    secp256k1_scalar_set_b32(&data->scalar_x, init_x, NULL);
-    secp256k1_scalar_set_b32(&data->scalar_y, init_y, NULL);
-    secp256k1_fe_set_b32(&data->fe_x, init_x);
-    secp256k1_fe_set_b32(&data->fe_y, init_y);
-    CHECK(secp256k1_ge_set_xo_var(&data->ge_x, &data->fe_x, 0));
-    CHECK(secp256k1_ge_set_xo_var(&data->ge_y, &data->fe_y, 1));
-    secp256k1_gej_set_ge(&data->gej_x, &data->ge_x);
-    secp256k1_gej_set_ge(&data->gej_y, &data->ge_y);
-    memcpy(data->data, init_x, 32);
-    memcpy(data->data + 32, init_y, 32);
+    secp256k1_scalar_set_b32(&data->scalar[0], init[0], NULL);
+    secp256k1_scalar_set_b32(&data->scalar[1], init[1], NULL);
+    secp256k1_fe_set_b32(&data->fe[0], init[0]);
+    secp256k1_fe_set_b32(&data->fe[1], init[1]);
+    secp256k1_fe_set_b32(&data->fe[2], init[2]);
+    secp256k1_fe_set_b32(&data->fe[3], init[3]);
+    CHECK(secp256k1_ge_set_xo_var(&data->ge[0], &data->fe[0], 0));
+    CHECK(secp256k1_ge_set_xo_var(&data->ge[1], &data->fe[1], 1));
+    secp256k1_gej_set_ge(&data->gej[0], &data->ge[0]);
+    secp256k1_gej_rescale(&data->gej[0], &data->fe[2]);
+    secp256k1_gej_set_ge(&data->gej[1], &data->ge[1]);
+    secp256k1_gej_rescale(&data->gej[1], &data->fe[3]);
+    memcpy(data->data, init[0], 32);
+    memcpy(data->data + 32, init[1], 32);
 }
 
 void bench_scalar_add(void* arg, int iters) {
@@ -61,7 +85,7 @@ void bench_scalar_add(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        j += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        j += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(j <= iters);
 }
@@ -71,7 +95,7 @@ void bench_scalar_negate(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_negate(&data->scalar_x, &data->scalar_x);
+        secp256k1_scalar_negate(&data->scalar[0], &data->scalar[0]);
     }
 }
 
@@ -80,7 +104,7 @@ void bench_scalar_sqr(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_sqr(&data->scalar_x, &data->scalar_x);
+        secp256k1_scalar_sqr(&data->scalar[0], &data->scalar[0]);
     }
 }
 
@@ -89,7 +113,7 @@ void bench_scalar_mul(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_mul(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        secp256k1_scalar_mul(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
 }
 
@@ -99,8 +123,8 @@ void bench_scalar_split(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_split_lambda(&data->scalar_x, &data->scalar_y, &data->scalar_x);
-        j += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        secp256k1_scalar_split_lambda(&data->scalar[0], &data->scalar[1], &data->scalar[0]);
+        j += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(j <= iters);
 }
@@ -111,8 +135,8 @@ void bench_scalar_inverse(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_inverse(&data->scalar_x, &data->scalar_x);
-        j += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        secp256k1_scalar_inverse(&data->scalar[0], &data->scalar[0]);
+        j += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(j <= iters);
 }
@@ -122,8 +146,8 @@ void bench_scalar_inverse_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_scalar_inverse_var(&data->scalar_x, &data->scalar_x);
-        j += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        secp256k1_scalar_inverse_var(&data->scalar[0], &data->scalar[0]);
+        j += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(j <= iters);
 }
@@ -133,7 +157,7 @@ void bench_field_normalize(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_normalize(&data->fe_x);
+        secp256k1_fe_normalize(&data->fe[0]);
     }
 }
 
@@ -142,7 +166,7 @@ void bench_field_normalize_weak(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_normalize_weak(&data->fe_x);
+        secp256k1_fe_normalize_weak(&data->fe[0]);
     }
 }
 
@@ -151,7 +175,7 @@ void bench_field_mul(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_mul(&data->fe_x, &data->fe_x, &data->fe_y);
+        secp256k1_fe_mul(&data->fe[0], &data->fe[0], &data->fe[1]);
     }
 }
 
@@ -160,7 +184,7 @@ void bench_field_sqr(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_sqr(&data->fe_x, &data->fe_x);
+        secp256k1_fe_sqr(&data->fe[0], &data->fe[0]);
     }
 }
 
@@ -169,8 +193,8 @@ void bench_field_inverse(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_inv(&data->fe_x, &data->fe_x);
-        secp256k1_fe_add(&data->fe_x, &data->fe_y);
+        secp256k1_fe_inv(&data->fe[0], &data->fe[0]);
+        secp256k1_fe_add(&data->fe[0], &data->fe[1]);
     }
 }
 
@@ -179,8 +203,8 @@ void bench_field_inverse_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_fe_inv_var(&data->fe_x, &data->fe_x);
-        secp256k1_fe_add(&data->fe_x, &data->fe_y);
+        secp256k1_fe_inv_var(&data->fe[0], &data->fe[0]);
+        secp256k1_fe_add(&data->fe[0], &data->fe[1]);
     }
 }
 
@@ -190,9 +214,9 @@ void bench_field_sqrt(void* arg, int iters) {
     secp256k1_fe t;
 
     for (i = 0; i < iters; i++) {
-        t = data->fe_x;
-        j += secp256k1_fe_sqrt(&data->fe_x, &t);
-        secp256k1_fe_add(&data->fe_x, &data->fe_y);
+        t = data->fe[0];
+        j += secp256k1_fe_sqrt(&data->fe[0], &t);
+        secp256k1_fe_add(&data->fe[0], &data->fe[1]);
     }
     CHECK(j <= iters);
 }
@@ -202,7 +226,7 @@ void bench_group_double_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_gej_double_var(&data->gej_x, &data->gej_x, NULL);
+        secp256k1_gej_double_var(&data->gej[0], &data->gej[0], NULL);
     }
 }
 
@@ -211,7 +235,7 @@ void bench_group_add_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_gej_add_var(&data->gej_x, &data->gej_x, &data->gej_y, NULL);
+        secp256k1_gej_add_var(&data->gej[0], &data->gej[0], &data->gej[1], NULL);
     }
 }
 
@@ -220,7 +244,7 @@ void bench_group_add_affine(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_gej_add_ge(&data->gej_x, &data->gej_x, &data->ge_y);
+        secp256k1_gej_add_ge(&data->gej[0], &data->gej[0], &data->ge[1]);
     }
 }
 
@@ -229,7 +253,7 @@ void bench_group_add_affine_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        secp256k1_gej_add_ge_var(&data->gej_x, &data->gej_x, &data->ge_y, NULL);
+        secp256k1_gej_add_ge_var(&data->gej[0], &data->gej[0], &data->ge[1], NULL);
     }
 }
 
@@ -238,9 +262,37 @@ void bench_group_jacobi_var(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        j += secp256k1_gej_has_quad_y_var(&data->gej_x);
+        j += secp256k1_gej_has_quad_y_var(&data->gej[0]);
+        /* Vary the Y and Z coordinates of the input (the X coordinate doesn't matter to
+           secp256k1_gej_has_quad_y_var). Note that the resulting coordinates will
+           generally not correspond to a point on the curve, but this is not a problem
+           for the code being benchmarked here. Adding and normalizing have less
+           overhead than EC operations (which could guarantee the point remains on the
+           curve). */
+        secp256k1_fe_add(&data->gej[0].y, &data->fe[1]);
+        secp256k1_fe_add(&data->gej[0].z, &data->fe[2]);
+        secp256k1_fe_normalize_var(&data->gej[0].y);
+        secp256k1_fe_normalize_var(&data->gej[0].z);
+    }
+    CHECK(j <= iters);
+}
+
+void bench_group_to_affine_var(void* arg, int iters) {
+    int i;
+    bench_inv *data = (bench_inv*)arg;
+
+    for (i = 0; i < iters; ++i) {
+        secp256k1_ge_set_gej_var(&data->ge[1], &data->gej[0]);
+        /* Use the output affine X/Y coordinates to vary the input X/Y/Z coordinates.
+           Similar to bench_group_jacobi_var, this approach does not result in
+           coordinates of points on the curve. */
+        secp256k1_fe_add(&data->gej[0].x, &data->ge[1].y);
+        secp256k1_fe_add(&data->gej[0].y, &data->fe[2]);
+        secp256k1_fe_add(&data->gej[0].z, &data->ge[1].x);
+        secp256k1_fe_normalize_var(&data->gej[0].x);
+        secp256k1_fe_normalize_var(&data->gej[0].y);
+        secp256k1_fe_normalize_var(&data->gej[0].z);
     }
-    CHECK(j == iters);
 }
 
 void bench_ecmult_wnaf(void* arg, int iters) {
@@ -248,8 +300,8 @@ void bench_ecmult_wnaf(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        bits += secp256k1_ecmult_wnaf(data->wnaf, 256, &data->scalar_x, WINDOW_A);
-        overflow += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        bits += secp256k1_ecmult_wnaf(data->wnaf, 256, &data->scalar[0], WINDOW_A);
+        overflow += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(overflow >= 0);
     CHECK(bits <= 256*iters);
@@ -260,8 +312,8 @@ void bench_wnaf_const(void* arg, int iters) {
     bench_inv *data = (bench_inv*)arg;
 
     for (i = 0; i < iters; i++) {
-        bits += secp256k1_wnaf_const(data->wnaf, &data->scalar_x, WINDOW_A, 256);
-        overflow += secp256k1_scalar_add(&data->scalar_x, &data->scalar_x, &data->scalar_y);
+        bits += secp256k1_wnaf_const(data->wnaf, &data->scalar[0], WINDOW_A, 256);
+        overflow += secp256k1_scalar_add(&data->scalar[0], &data->scalar[0], &data->scalar[1]);
     }
     CHECK(overflow >= 0);
     CHECK(bits <= 256*iters);
@@ -323,14 +375,15 @@ void bench_context_sign(void* arg, int iters) {
 void bench_num_jacobi(void* arg, int iters) {
     int i, j = 0;
     bench_inv *data = (bench_inv*)arg;
-    secp256k1_num nx, norder;
+    secp256k1_num nx, na, norder;
 
-    secp256k1_scalar_get_num(&nx, &data->scalar_x);
+    secp256k1_scalar_get_num(&nx, &data->scalar[0]);
     secp256k1_scalar_order_get_num(&norder);
-    secp256k1_scalar_get_num(&norder, &data->scalar_y);
+    secp256k1_scalar_get_num(&na, &data->scalar[1]);
 
     for (i = 0; i < iters; i++) {
         j += secp256k1_num_jacobi(&nx, &norder);
+        secp256k1_num_add(&nx, &nx, &na);
     }
     CHECK(j <= iters);
 }
@@ -363,6 +416,7 @@ int main(int argc, char **argv) {
     if (have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine", bench_group_add_affine, bench_setup, NULL, &data, 10, iters*10);
     if (have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine_var", bench_group_add_affine_var, bench_setup, NULL, &data, 10, iters*10);
     if (have_flag(argc, argv, "group") || have_flag(argc, argv, "jacobi")) run_benchmark("group_jacobi_var", bench_group_jacobi_var, bench_setup, NULL, &data, 10, iters);
+    if (have_flag(argc, argv, "group") || have_flag(argc, argv, "to_affine")) run_benchmark("group_to_affine_var", bench_group_to_affine_var, bench_setup, NULL, &data, 10, iters);
 
     if (have_flag(argc, argv, "ecmult") || have_flag(argc, argv, "wnaf")) run_benchmark("wnaf_const", bench_wnaf_const, bench_setup, NULL, &data, 10, iters);
     if (have_flag(argc, argv, "ecmult") || have_flag(argc, argv, "wnaf")) run_benchmark("ecmult_wnaf", bench_ecmult_wnaf, bench_setup, NULL, &data, 10, iters);
diff --git a/src/field.h b/src/field.h
index 7993a1f1..aca1fb72 100644
--- a/src/field.h
+++ b/src/field.h
@@ -22,16 +22,16 @@
 #include "libsecp256k1-config.h"
 #endif
 
-#if defined(USE_FIELD_10X26)
-#include "field_10x26.h"
-#elif defined(USE_FIELD_5X52)
-#include "field_5x52.h"
-#else
-#error "Please select field implementation"
-#endif
-
 #include "util.h"
 
+#if defined(SECP256K1_WIDEMUL_INT128)
+#include "field_5x52.h"
+#elif defined(SECP256K1_WIDEMUL_INT64)
+#include "field_10x26.h"
+#else
+#error "Please select wide multiplication implementation"
+#endif
+
 /** Normalize a field element. This brings the field element to a canonical representation, reduces
  *  its magnitude to 1, and reduces it modulo field size `p`.
  */
diff --git a/src/field_5x52.h b/src/field_5x52.h
index fc5bfe35..6a068484 100644
--- a/src/field_5x52.h
+++ b/src/field_5x52.h
@@ -46,4 +46,10 @@ typedef struct {
     (d6) | (((uint64_t)(d7)) << 32) \
 }}
 
+#define SECP256K1_FE_STORAGE_CONST_GET(d) \
+    (uint32_t)(d.n[3] >> 32), (uint32_t)d.n[3], \
+    (uint32_t)(d.n[2] >> 32), (uint32_t)d.n[2], \
+    (uint32_t)(d.n[1] >> 32), (uint32_t)d.n[1], \
+    (uint32_t)(d.n[0] >> 32), (uint32_t)d.n[0]
+
 #endif /* SECP256K1_FIELD_REPR_H */
diff --git a/src/field_impl.h b/src/field_impl.h
index 485921a6..18e4d2f3 100644
--- a/src/field_impl.h
+++ b/src/field_impl.h
@@ -14,12 +14,12 @@
 #include "util.h"
 #include "num.h"
 
-#if defined(USE_FIELD_10X26)
-#include "field_10x26_impl.h"
-#elif defined(USE_FIELD_5X52)
+#if defined(SECP256K1_WIDEMUL_INT128)
 #include "field_5x52_impl.h"
+#elif defined(SECP256K1_WIDEMUL_INT64)
+#include "field_10x26_impl.h"
 #else
-#error "Please select field implementation"
+#error "Please select wide multiplication implementation"
 #endif
 
 SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b) {
diff --git a/src/gen_context.c b/src/gen_context.c
index 539f574b..8b7729ae 100644
--- a/src/gen_context.c
+++ b/src/gen_context.c
@@ -13,6 +13,7 @@
 #include "basic-config.h"
 
 #include "include/secp256k1.h"
+#include "assumptions.h"
 #include "util.h"
 #include "field_impl.h"
 #include "scalar_impl.h"
diff --git a/src/group_impl.h b/src/group_impl.h
index fbfd3489..ccd93d34 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -399,7 +399,7 @@ static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, cons
             if (rzr != NULL) {
                 secp256k1_fe_set_int(rzr, 0);
             }
-            r->infinity = 1;
+            secp256k1_gej_set_infinity(r);
         }
         return;
     }
@@ -449,7 +449,7 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, c
             if (rzr != NULL) {
                 secp256k1_fe_set_int(rzr, 0);
             }
-            r->infinity = 1;
+            secp256k1_gej_set_infinity(r);
         }
         return;
     }
@@ -508,7 +508,7 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
             secp256k1_gej_double_var(r, a, NULL);
         } else {
-            r->infinity = 1;
+            secp256k1_gej_set_infinity(r);
         }
         return;
     }
diff --git a/src/hash_impl.h b/src/hash_impl.h
index 782f9721..1985a078 100644
--- a/src/hash_impl.h
+++ b/src/hash_impl.h
@@ -8,6 +8,7 @@
 #define SECP256K1_HASH_IMPL_H
 
 #include "hash.h"
+#include "util.h"
 
 #include <stdlib.h>
 #include <stdint.h>
@@ -27,9 +28,9 @@
     (h) = t1 + t2; \
 } while(0)
 
-#ifdef WORDS_BIGENDIAN
+#if defined(SECP256K1_BIG_ENDIAN)
 #define BE32(x) (x)
-#else
+#elif defined(SECP256K1_LITTLE_ENDIAN)
 #define BE32(p) ((((p) & 0xFF) << 24) | (((p) & 0xFF00) << 8) | (((p) & 0xFF0000) >> 8) | (((p) & 0xFF000000) >> 24))
 #endif
 
diff --git a/src/modules/rangeproof/borromean_impl.h b/src/modules/rangeproof/borromean_impl.h
index 3a82f096..e11db181 100644
--- a/src/modules/rangeproof/borromean_impl.h
+++ b/src/modules/rangeproof/borromean_impl.h
@@ -20,9 +20,9 @@
 #include <limits.h>
 #include <string.h>
 
-#ifdef WORDS_BIGENDIAN
+#if defined(SECP256K1_BIG_ENDIAN)
 #define BE32(x) (x)
-#else
+#elif defined(SECP256K1_LITTLE_ENDIAN)
 #define BE32(p) ((((p) & 0xFF) << 24) | (((p) & 0xFF00) << 8) | (((p) & 0xFF0000) >> 8) | (((p) & 0xFF000000) >> 24))
 #endif
 
diff --git a/src/scalar.h b/src/scalar.h
index bf609a1b..d190e019 100644
--- a/src/scalar.h
+++ b/src/scalar.h
@@ -8,6 +8,7 @@
 #define SECP256K1_SCALAR_H
 
 #include "num.h"
+#include "util.h"
 
 #if defined HAVE_CONFIG_H
 #include "libsecp256k1-config.h"
@@ -15,12 +16,12 @@
 
 #if defined(EXHAUSTIVE_TEST_ORDER)
 #include "scalar_low.h"
-#elif defined(USE_SCALAR_4X64)
+#elif defined(SECP256K1_WIDEMUL_INT128)
 #include "scalar_4x64.h"
-#elif defined(USE_SCALAR_8X32)
+#elif defined(SECP256K1_WIDEMUL_INT64)
 #include "scalar_8x32.h"
 #else
-#error "Please select scalar implementation"
+#error "Please select wide multiplication implementation"
 #endif
 
 /** Clear a scalar to prevent the leak of sensitive data. */
diff --git a/src/scalar_4x64_impl.h b/src/scalar_4x64_impl.h
index 294413bd..2cd0f9bc 100644
--- a/src/scalar_4x64_impl.h
+++ b/src/scalar_4x64_impl.h
@@ -974,9 +974,9 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se
   a += b; d = ROTL32(d ^ a, 8); \
   c += d; b = ROTL32(b ^ c, 7);
 
-#ifdef WORDS_BIGENDIAN
+#if defined(SECP256K1_BIG_ENDIAN)
 #define LE32(p) ((((p) & 0xFF) << 24) | (((p) & 0xFF00) << 8) | (((p) & 0xFF0000) >> 8) | (((p) & 0xFF000000) >> 24))
-#else
+#elif defined(SECP256K1_LITTLE_ENDIAN)
 #define LE32(p) (p)
 #endif
 
diff --git a/src/scalar_8x32_impl.h b/src/scalar_8x32_impl.h
index c12b06e6..9afb9e95 100644
--- a/src/scalar_8x32_impl.h
+++ b/src/scalar_8x32_impl.h
@@ -753,9 +753,9 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se
   a += b; d = ROTL32(d ^ a, 8); \
   c += d; b = ROTL32(b ^ c, 7);
 
-#ifdef WORDS_BIGENDIAN
+#if defined(SECP256K1_BIG_ENDIAN)
 #define LE32(p) ((((p) & 0xFF) << 24) | (((p) & 0xFF00) << 8) | (((p) & 0xFF0000) >> 8) | (((p) & 0xFF000000) >> 24))
-#else
+#elif defined(SECP256K1_LITTLE_ENDIAN)
 #define LE32(p) (p)
 #endif
 
diff --git a/src/scalar_impl.h b/src/scalar_impl.h
index 70cd73db..2ec04b1a 100644
--- a/src/scalar_impl.h
+++ b/src/scalar_impl.h
@@ -16,12 +16,12 @@
 
 #if defined(EXHAUSTIVE_TEST_ORDER)
 #include "scalar_low_impl.h"
-#elif defined(USE_SCALAR_4X64)
+#elif defined(SECP256K1_WIDEMUL_INT128)
 #include "scalar_4x64_impl.h"
-#elif defined(USE_SCALAR_8X32)
+#elif defined(SECP256K1_WIDEMUL_INT64)
 #include "scalar_8x32_impl.h"
 #else
-#error "Please select scalar implementation"
+#error "Please select wide multiplication implementation"
 #endif
 
 static const secp256k1_scalar secp256k1_scalar_one = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1);
diff --git a/src/scratch_impl.h b/src/scratch_impl.h
index 4cee7000..b2056202 100644
--- a/src/scratch_impl.h
+++ b/src/scratch_impl.h
@@ -11,7 +11,7 @@
 #include "scratch.h"
 
 static secp256k1_scratch* secp256k1_scratch_create(const secp256k1_callback* error_callback, size_t size) {
-    const size_t base_alloc = ((sizeof(secp256k1_scratch) + ALIGNMENT - 1) / ALIGNMENT) * ALIGNMENT;
+    const size_t base_alloc = ROUND_TO_ALIGN(sizeof(secp256k1_scratch));
     void *alloc = checked_malloc(error_callback, base_alloc + size);
     secp256k1_scratch* ret = (secp256k1_scratch *)alloc;
     if (ret != NULL) {
@@ -60,6 +60,10 @@ static size_t secp256k1_scratch_max_allocation(const secp256k1_callback* error_c
         secp256k1_callback_call(error_callback, "invalid scratch space");
         return 0;
     }
+    /* Ensure that multiplication will not wrap around */
+    if (ALIGNMENT > 1 && objects > SIZE_MAX/(ALIGNMENT - 1)) {
+        return 0;
+    }
     if (scratch->max_size - scratch->alloc_size <= objects * (ALIGNMENT - 1)) {
         return 0;
     }
@@ -68,7 +72,14 @@ static size_t secp256k1_scratch_max_allocation(const secp256k1_callback* error_c
 
 static void *secp256k1_scratch_alloc(const secp256k1_callback* error_callback, secp256k1_scratch* scratch, size_t size) {
     void *ret;
-    size = ROUND_TO_ALIGN(size);
+    size_t rounded_size;
+
+    rounded_size = ROUND_TO_ALIGN(size);
+    /* Check that rounding did not wrap around */
+    if (rounded_size < size) {
+        return NULL;
+    }
+    size = rounded_size;
 
     if (memcmp(scratch->magic, "scratch", 8) != 0) {
         secp256k1_callback_call(error_callback, "invalid scratch space");
diff --git a/src/secp256k1.c b/src/secp256k1.c
index c2fe1c63..0c3a09e3 100644
--- a/src/secp256k1.c
+++ b/src/secp256k1.c
@@ -7,6 +7,7 @@
 #include "include/secp256k1.h"
 #include "include/secp256k1_preallocated.h"
 
+#include "assumptions.h"
 #include "util.h"
 #include "num_impl.h"
 #include "field_impl.h"
@@ -19,6 +20,7 @@
 #include "eckey_impl.h"
 #include "hash_impl.h"
 #include "scratch_impl.h"
+#include "selftest.h"
 
 #if defined(VALGRIND)
 # include <valgrind/memcheck.h>
@@ -127,6 +129,9 @@ secp256k1_context* secp256k1_context_preallocated_create(void* prealloc, unsigne
     size_t prealloc_size;
     secp256k1_context* ret;
 
+    if (!secp256k1_selftest()) {
+        secp256k1_callback_call(&default_error_callback, "self test failed");
+    }
     VERIFY_CHECK(prealloc != NULL);
     prealloc_size = secp256k1_context_preallocated_size(flags);
     ret = (secp256k1_context*)manual_alloc(&prealloc, sizeof(secp256k1_context), base, prealloc_size);
diff --git a/src/selftest.h b/src/selftest.h
new file mode 100644
index 00000000..885983aa
--- /dev/null
+++ b/src/selftest.h
@@ -0,0 +1,32 @@
+/**********************************************************************
+ * Copyright (c) 2020 Pieter Wuille                                   *
+ * Distributed under the MIT software license, see the accompanying   *
+ * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
+ **********************************************************************/
+
+#ifndef SECP256K1_SELFTEST_H
+#define SECP256K1_SELFTEST_H
+
+#include "hash.h"
+
+#include <string.h>
+
+static int secp256k1_selftest_sha256(void) {
+    static const char *input63 = "For this sample, this 63-byte string will be used as input data";
+    static const unsigned char output32[32] = {
+        0xf0, 0x8a, 0x78, 0xcb, 0xba, 0xee, 0x08, 0x2b, 0x05, 0x2a, 0xe0, 0x70, 0x8f, 0x32, 0xfa, 0x1e,
+        0x50, 0xc5, 0xc4, 0x21, 0xaa, 0x77, 0x2b, 0xa5, 0xdb, 0xb4, 0x06, 0xa2, 0xea, 0x6b, 0xe3, 0x42,
+    };
+    unsigned char out[32];
+    secp256k1_sha256 hasher;
+    secp256k1_sha256_initialize(&hasher);
+    secp256k1_sha256_write(&hasher, (const unsigned char*)input63, 63);
+    secp256k1_sha256_finalize(&hasher, out);
+    return memcmp(out, output32, 32) == 0;
+}
+
+static int secp256k1_selftest(void) {
+    return secp256k1_selftest_sha256();
+}
+
+#endif /* SECP256K1_SELFTEST_H */
diff --git a/src/tests.c b/src/tests.c
index c994ab6b..c594dee9 100644
--- a/src/tests.c
+++ b/src/tests.c
@@ -410,8 +410,8 @@ void run_scratch_tests(void) {
     CHECK(scratch->alloc_size != 0);
     CHECK(scratch->alloc_size % ALIGNMENT == 0);
 
-    /* Allocating another 500 bytes fails */
-    CHECK(secp256k1_scratch_alloc(&none->error_callback, scratch, 500) == NULL);
+    /* Allocating another 501 bytes fails */
+    CHECK(secp256k1_scratch_alloc(&none->error_callback, scratch, 501) == NULL);
     CHECK(secp256k1_scratch_max_allocation(&none->error_callback, scratch, 0) == 1000 - adj_alloc);
     CHECK(secp256k1_scratch_max_allocation(&none->error_callback, scratch, 1) == 1000 - adj_alloc - (ALIGNMENT - 1));
     CHECK(scratch->alloc_size != 0);
@@ -444,6 +444,18 @@ void run_scratch_tests(void) {
     secp256k1_scratch_space_destroy(none, scratch);
     CHECK(ecount == 5);
 
+    /* Test that large integers do not wrap around in a bad way */
+    scratch = secp256k1_scratch_space_create(none, 1000);
+    /* Try max allocation with a large number of objects. Only makes sense if
+     * ALIGNMENT is greater than 1 because otherwise the objects take no extra
+     * space. */
+    CHECK(ALIGNMENT <= 1 || !secp256k1_scratch_max_allocation(&none->error_callback, scratch, (SIZE_MAX / (ALIGNMENT - 1)) + 1));
+    /* Try allocating SIZE_MAX to test wrap around which only happens if
+     * ALIGNMENT > 1, otherwise it returns NULL anyway because the scratch
+     * space is too small. */
+    CHECK(secp256k1_scratch_alloc(&none->error_callback, scratch, SIZE_MAX) == NULL);
+    secp256k1_scratch_space_destroy(none, scratch);
+
     /* cleanup */
     secp256k1_scratch_space_destroy(none, NULL); /* no-op */
     secp256k1_context_destroy(none);
@@ -2462,6 +2474,39 @@ void test_ge(void) {
     free(zinv);
 }
 
+
+void test_intialized_inf(void) {
+    secp256k1_ge p;
+    secp256k1_gej pj, npj, infj1, infj2, infj3;
+    secp256k1_fe zinv;
+
+    /* Test that adding P+(-P) results in a fully initalized infinity*/
+    random_group_element_test(&p);
+    secp256k1_gej_set_ge(&pj, &p);
+    secp256k1_gej_neg(&npj, &pj);
+
+    secp256k1_gej_add_var(&infj1, &pj, &npj, NULL);
+    CHECK(secp256k1_gej_is_infinity(&infj1));
+    CHECK(secp256k1_fe_is_zero(&infj1.x));
+    CHECK(secp256k1_fe_is_zero(&infj1.y));
+    CHECK(secp256k1_fe_is_zero(&infj1.z));
+
+    secp256k1_gej_add_ge_var(&infj2, &npj, &p, NULL);
+    CHECK(secp256k1_gej_is_infinity(&infj2));
+    CHECK(secp256k1_fe_is_zero(&infj2.x));
+    CHECK(secp256k1_fe_is_zero(&infj2.y));
+    CHECK(secp256k1_fe_is_zero(&infj2.z));
+
+    secp256k1_fe_set_int(&zinv, 1);
+    secp256k1_gej_add_zinv_var(&infj3, &npj, &p, &zinv);
+    CHECK(secp256k1_gej_is_infinity(&infj3));
+    CHECK(secp256k1_fe_is_zero(&infj3.x));
+    CHECK(secp256k1_fe_is_zero(&infj3.y));
+    CHECK(secp256k1_fe_is_zero(&infj3.z));
+
+
+}
+
 void test_add_neg_y_diff_x(void) {
     /* The point of this test is to check that we can add two points
      * whose y-coordinates are negatives of each other but whose x
@@ -2535,6 +2580,7 @@ void run_ge(void) {
         test_ge();
     }
     test_add_neg_y_diff_x();
+    test_intialized_inf();
 }
 
 void test_ec_combine(void) {
diff --git a/src/tests_exhaustive.c b/src/tests_exhaustive.c
index 8f346701..681ed80b 100644
--- a/src/tests_exhaustive.c
+++ b/src/tests_exhaustive.c
@@ -22,6 +22,7 @@
 #endif
 
 #include "include/secp256k1.h"
+#include "assumptions.h"
 #include "group.h"
 #include "secp256k1.c"
 #include "testrand_impl.h"
diff --git a/src/util.h b/src/util.h
index 944180ce..f62c41d4 100644
--- a/src/util.h
+++ b/src/util.h
@@ -196,13 +196,35 @@ SECP256K1_INLINE static int secp256k1_clz64_var(uint64_t x) {
 # define I64uFORMAT "llu"
 #endif
 
-#if defined(HAVE___INT128)
-# if defined(__GNUC__)
-#  define SECP256K1_GNUC_EXT __extension__
-# else
-#  define SECP256K1_GNUC_EXT
+#if defined(__GNUC__)
+# define SECP256K1_GNUC_EXT __extension__
+#else
+# define SECP256K1_GNUC_EXT
+#endif
+
+/* If SECP256K1_{LITTLE,BIG}_ENDIAN is not explicitly provided, infer from various other system macros. */
+#if !defined(SECP256K1_LITTLE_ENDIAN) && !defined(SECP256K1_BIG_ENDIAN)
+/* Inspired by https://github.com/rofl0r/endianness.h/blob/9853923246b065a3b52d2c43835f3819a62c7199/endianness.h#L52L73 */
+# if (defined(__BYTE_ORDER__) && defined(__ORDER_LITTLE_ENDIAN__) && __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__) || \
+     defined(_X86_) || defined(__x86_64__) || defined(__i386__) || \
+     defined(__i486__) || defined(__i586__) || defined(__i686__) || \
+     defined(__MIPSEL) || defined(_MIPSEL) || defined(MIPSEL) || \
+     defined(__ARMEL__) || defined(__AARCH64EL__) || \
+     (defined(__LITTLE_ENDIAN__) && __LITTLE_ENDIAN__ == 1) || \
+     (defined(_LITTLE_ENDIAN) && _LITTLE_ENDIAN == 1) || \
+     defined(_M_IX86) || defined(_M_AMD64) || defined(_M_ARM) /* MSVC */
+#  define SECP256K1_LITTLE_ENDIAN
 # endif
-SECP256K1_GNUC_EXT typedef unsigned __int128 uint128_t;
+# if (defined(__BYTE_ORDER__) && defined(__ORDER_BIG_ENDIAN__) && __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__) || \
+     defined(__MIPSEB) || defined(_MIPSEB) || defined(MIPSEB) || \
+     defined(__MICROBLAZEEB__) || defined(__ARMEB__) || defined(__AARCH64EB__) || \
+     (defined(__BIG_ENDIAN__) && __BIG_ENDIAN__ == 1) || \
+     (defined(_BIG_ENDIAN) && _BIG_ENDIAN == 1)
+#  define SECP256K1_BIG_ENDIAN
+# endif
+#endif
+#if defined(SECP256K1_LITTLE_ENDIAN) == defined(SECP256K1_BIG_ENDIAN)
+# error Please make sure that either SECP256K1_LITTLE_ENDIAN or SECP256K1_BIG_ENDIAN is set, see src/util.h.
 #endif
 
 /* Zero memory if flag == 1. Flag must be 0 or 1. Constant time. */
@@ -239,4 +261,21 @@ static SECP256K1_INLINE void secp256k1_int_cmov(int *r, const int *a, int flag)
     *r = (int)(r_masked | a_masked);
 }
 
+/* If USE_FORCE_WIDEMUL_{INT128,INT64} is set, use that wide multiplication implementation.
+ * Otherwise use the presence of __SIZEOF_INT128__ to decide.
+ */
+#if defined(USE_FORCE_WIDEMUL_INT128)
+# define SECP256K1_WIDEMUL_INT128 1
+#elif defined(USE_FORCE_WIDEMUL_INT64)
+# define SECP256K1_WIDEMUL_INT64 1
+#elif defined(__SIZEOF_INT128__)
+# define SECP256K1_WIDEMUL_INT128 1
+#else
+# define SECP256K1_WIDEMUL_INT64 1
+#endif
+#if defined(SECP256K1_WIDEMUL_INT128)
+SECP256K1_GNUC_EXT typedef unsigned __int128 uint128_t;
+SECP256K1_GNUC_EXT typedef __int128 int128_t;
+#endif
+
 #endif /* SECP256K1_UTIL_H */
diff --git a/src/valgrind_ctime_test.c b/src/valgrind_ctime_test.c
index 60a82d59..090fde2b 100644
--- a/src/valgrind_ctime_test.c
+++ b/src/valgrind_ctime_test.c
@@ -6,6 +6,7 @@
 
 #include <valgrind/memcheck.h>
 #include "include/secp256k1.h"
+#include "assumptions.h"
 #include "util.h"
 
 #if ENABLE_MODULE_ECDH