diff --git a/tests/scripts/curves.pl b/tests/scripts/curves.pl
index cd6ea0d9a5..8db44303cf 100755
--- a/tests/scripts/curves.pl
+++ b/tests/scripts/curves.pl
@@ -2,7 +2,7 @@
 
 # curves.pl
 #
-# Copyright (c) 2014-2016, ARM Limited, All Rights Reserved
+# Copyright (c) 2014-2020, ARM Limited, All Rights Reserved
 # SPDX-License-Identifier: Apache-2.0
 #
 # Licensed under the Apache License, Version 2.0 (the "License"); you may
@@ -21,21 +21,25 @@
 #
 # Purpose
 #
-# To test the code dependencies on individual curves in each test suite. This
-# is a verification step to ensure we don't ship test suites that do not work
-# for some build options.
+# The purpose of this test script is to validate that the library works
+# with any combination of elliptic curves. To this effect, build the library
+# and run the test suite with each tested combination of elliptic curves.
 #
-# The process is:
-#       for each possible curve
-#           build the library and test suites with the curve disabled
-#           execute the test suites
-#
-# And any test suite with the wrong dependencies will fail.
+# Testing all 2^n combinations would be too much, so we only test 2*n:
 #
+# 1. Test with a single curve, for each curve. This validates that the
+#    library works with any curve, and in particular that curve-specific
+#    code is guarded by the proper preprocessor conditionals.
+# 2. Test with all curves except one, for each curve. This validates that
+#    the test cases have correct dependencies. Testing with a single curve
+#    doesn't validate this for tests that require more than one curve.
+
 # Usage: tests/scripts/curves.pl
 #
 # This script should be executed from the root of the project directory.
 #
+# Only curves that are enabled in config.h will be tested.
+#
 # For best effect, run either with cmake disabled, or cmake enabled in a mode
 # that includes -Werror.
 
@@ -48,6 +52,25 @@ my $sed_cmd = 's/^#define \(MBEDTLS_ECP_DP.*_ENABLED\)/\1/p';
 my $config_h = 'include/mbedtls/config.h';
 my @curves = split( /\s+/, `sed -n -e '$sed_cmd' $config_h` );
 
+# Determine which curves support ECDSA by checking the dependencies of
+# ECDSA in check_config.h.
+my %curve_supports_ecdsa = ();
+{
+    local $/ = "";
+    local *CHECK_CONFIG;
+    open(CHECK_CONFIG, '<', 'include/mbedtls/check_config.h')
+        or die "open include/mbedtls/check_config.h: $!";
+    while (my $stanza = <CHECK_CONFIG>) {
+        if ($stanza =~ /\A#if defined\(MBEDTLS_ECDSA_C\)/) {
+            for my $curve ($stanza =~ /(?<=\()MBEDTLS_ECP_DP_\w+_ENABLED(?=\))/g) {
+                $curve_supports_ecdsa{$curve} = 1;
+            }
+            last;
+        }
+    }
+    close(CHECK_CONFIG);
+}
+
 system( "cp $config_h $config_h.bak" ) and die;
 sub abort {
     system( "mv $config_h.bak $config_h" ) and warn "$config_h not restored\n";
@@ -56,6 +79,46 @@ sub abort {
     exit 1;
 }
 
+# Disable all the curves. We'll then re-enable them one by one.
+for my $curve (@curves) {
+    system( "scripts/config.pl unset $curve" )
+        and abort "Failed to disable $curve\n";
+}
+# Depends on a specific curve. Also, ignore error if it wasn't enabled.
+system( "scripts/config.pl unset MBEDTLS_KEY_EXCHANGE_ECJPAKE_ENABLED" );
+
+# Test with only $curve enabled, for each $curve.
+for my $curve (@curves) {
+    system( "make clean" ) and die;
+
+    print "\n******************************************\n";
+    print "* Testing with only curve: $curve\n";
+    print "******************************************\n";
+    $ENV{MBEDTLS_TEST_CONFIGURATION} = "$curve";
+
+    system( "scripts/config.pl set $curve" )
+        and abort "Failed to enable $curve\n";
+
+    my $ecdsa = $curve_supports_ecdsa{$curve} ? "set" : "unset";
+    for my $dep (qw(MBEDTLS_ECDSA_C
+                    MBEDTLS_KEY_EXCHANGE_ECDH_ECDSA_ENABLED
+                    MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED)) {
+        system( "scripts/config.pl $ecdsa $dep" )
+            and abort "Failed to $ecdsa $dep\n";
+    }
+
+    system( "CFLAGS='-Werror -Wall -Wextra' make" )
+        and abort "Failed to build: only $curve\n";
+    system( "make test" )
+        and abort "Failed test suite: only $curve\n";
+
+    system( "scripts/config.pl unset $curve" )
+        and abort "Failed to disable $curve\n";
+}
+
+system( "cp $config_h.bak $config_h" ) and die "$config_h not restored\n";
+
+# Test with $curve disabled but the others enabled, for each $curve.
 for my $curve (@curves) {
     system( "cp $config_h.bak $config_h" ) and die "$config_h not restored\n";
     system( "make clean" ) and die;
@@ -71,10 +134,10 @@ for my $curve (@curves) {
     system( "scripts/config.py unset $curve" )
         and abort "Failed to disable $curve\n";
 
-    system( "CFLAGS='-Werror -Wall -Wextra' make lib" )
-        and abort "Failed to build lib: $curve\n";
-    system( "make" ) and abort "Failed to build tests: $curve\n";
-    system( "make test" ) and abort "Failed test suite: $curve\n";
+    system( "CFLAGS='-Werror -Wall -Wextra' make" )
+        and abort "Failed to build: all but $curve\n";
+    system( "make test" )
+        and abort "Failed test suite: all but $curve\n";
 
 }