diff --git a/ChangeLog.d/makefile-python-windows.txt b/ChangeLog.d/makefile-python-windows.txt
new file mode 100644
index 000000000..57ccc1a39
--- /dev/null
+++ b/ChangeLog.d/makefile-python-windows.txt
@@ -0,0 +1,4 @@
+Bugfix
+   * The GNU makefiles invoke python3 in preference to python except on Windows.
+     The check was accidentally not performed when cross-compiling for Windows
+     on Linux. Fix this. Fixes #4774.
diff --git a/programs/Makefile b/programs/Makefile
index 997c19871..55ebd601a 100644
--- a/programs/Makefile
+++ b/programs/Makefile
@@ -43,11 +43,15 @@ LOCAL_LDFLAGS += -lws2_32
 ifdef SHARED
 SHARED_SUFFIX=.$(DLEXT)
 endif
-PYTHON ?= python
 else
 DLEXT ?= so
 EXEXT=
 SHARED_SUFFIX=
+endif
+
+ifdef WINDOWS
+PYTHON ?= python
+else
 PYTHON ?= $(shell if type python3 >/dev/null 2>/dev/null; then echo python3; else echo python; fi)
 endif
 
diff --git a/tests/Makefile b/tests/Makefile
index c1620d6e7..449fca274 100644
--- a/tests/Makefile
+++ b/tests/Makefile
@@ -50,31 +50,48 @@ LOCAL_LDFLAGS += -lws2_32
 ifdef SHARED
 SHARED_SUFFIX=.$(DLEXT)
 endif
-PYTHON ?= python
 else
 DLEXT ?= so
 EXEXT=
 SHARED_SUFFIX=
+endif
+
+ifdef WINDOWS
+PYTHON ?= python
+else
 PYTHON ?= $(shell if type python3 >/dev/null 2>/dev/null; then echo python3; else echo python; fi)
 endif
 
 .PHONY: generated_files
-GENERATED_DATA_FILES := $(patsubst tests/%,%,$(shell $(PYTHON) scripts/generate_psa_tests.py --list))
+GENERATED_DATA_FILES := $(patsubst tests/%,%,$(shell \
+	$(PYTHON) scripts/generate_psa_tests.py --list || \
+	echo FAILED \
+))
+ifeq ($(GENERATED_DATA_FILES),FAILED)
+$(error "$(PYTHON) scripts/generate_psa_tests.py --list" failed)
+endif
 GENERATED_FILES := $(GENERATED_DATA_FILES)
 generated_files: $(GENERATED_FILES)
 
-$(GENERATED_DATA_FILES): scripts/generate_psa_tests.py
+# generate_psa_tests.py spends more time analyzing inputs than generating
+# outputs. Its inputs are the same no matter which files are being generated.
+# It's rare not to want all the outputs. So always generate all of its outputs.
+# Use an intermediate phony dependency so that parallel builds don't run
+# a separate instance of the recipe for each output file.
+.SECONDARY: generated_psa_test_data
+$(GENERATED_DATA_FILES): generated_psa_test_data
+generated_psa_test_data: scripts/generate_psa_tests.py
 ## The generated file only depends on the options that are present in
 ## crypto_config.h, not on which options are set. To avoid regenerating this
 ## file all the time when switching between configurations, don't declare
 ## crypto_config.h as a dependency. Remove this file from your working tree
 ## if you've just added or removed an option in crypto_config.h.
-#$(GENERATED_DATA_FILES): ../include/psa/crypto_config.h
-$(GENERATED_DATA_FILES): ../include/psa/crypto_values.h
-$(GENERATED_DATA_FILES): ../include/psa/crypto_extra.h
-$(GENERATED_DATA_FILES): suites/test_suite_psa_crypto_metadata.data
-$(GENERATED_DATA_FILES):
-	echo "  Gen   $@ ..."
+#generated_psa_test_data: ../include/psa/crypto_config.h
+generated_psa_test_data: ../include/psa/crypto_values.h
+generated_psa_test_data: ../include/psa/crypto_extra.h
+generated_psa_test_data: suites/test_suite_psa_crypto_metadata.data
+generated_psa_test_data:
+	echo "  Gen   $(GENERATED_DATA_FILES) ..."
 	$(PYTHON) scripts/generate_psa_tests.py
 
 # A test application is built for each suites/test_suite_*.data file.
diff --git a/tests/ssl-opt.sh b/tests/ssl-opt.sh
index 584202daa..4f0a60ea1 100755
--- a/tests/ssl-opt.sh
+++ b/tests/ssl-opt.sh
@@ -6784,7 +6784,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: both (MTU=512)" \
@@ -6815,7 +6815,7 @@ not_with_valgrind
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: proxy MTU: auto-reduction (not valgrind)" \
@@ -6839,7 +6839,7 @@ only_with_valgrind
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: proxy MTU: auto-reduction (with valgrind)" \
@@ -6891,7 +6891,7 @@ not_with_valgrind # spurious autoreduction due to timeout
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: proxy MTU, simple handshake (MTU=512)" \
@@ -6940,7 +6940,7 @@ not_with_valgrind # spurious autoreduction due to timeout
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: proxy MTU, simple handshake, nbio (MTU=512)" \
@@ -6976,7 +6976,7 @@ not_with_valgrind # spurious autoreduction due to timeout
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 run_test    "DTLS fragmenting: proxy MTU, resumed handshake" \
@@ -7005,7 +7005,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_SSL_RENEGOTIATION
 requires_config_enabled MBEDTLS_CHACHAPOLY_C
 run_test    "DTLS fragmenting: proxy MTU, ChachaPoly renego" \
@@ -7036,7 +7036,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_SSL_RENEGOTIATION
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
@@ -7068,7 +7068,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_SSL_RENEGOTIATION
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_CCM_C
@@ -7100,7 +7100,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_SSL_RENEGOTIATION
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_CIPHER_MODE_CBC
@@ -7133,7 +7133,7 @@ requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
 requires_config_enabled MBEDTLS_SHA256_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_SSL_RENEGOTIATION
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_CIPHER_MODE_CBC
@@ -7162,7 +7162,7 @@ run_test    "DTLS fragmenting: proxy MTU, AES-CBC non-EtM renego" \
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 client_needs_more_time 2
@@ -7186,7 +7186,7 @@ run_test    "DTLS fragmenting: proxy MTU + 3d" \
 requires_config_enabled MBEDTLS_SSL_PROTO_DTLS
 requires_config_enabled MBEDTLS_RSA_C
 requires_config_enabled MBEDTLS_ECDSA_C
-requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA
+requires_config_enabled MBEDTLS_KEY_EXCHANGE_ECDHE_ECDSA_ENABLED
 requires_config_enabled MBEDTLS_AES_C
 requires_config_enabled MBEDTLS_GCM_C
 client_needs_more_time 2
diff --git a/tests/suites/test_suite_psa_crypto_storage_format.misc.data b/tests/suites/test_suite_psa_crypto_storage_format.misc.data
index 114c402a2..48e3804b4 100644
--- a/tests/suites/test_suite_psa_crypto_storage_format.misc.data
+++ b/tests/suites/test_suite_psa_crypto_storage_format.misc.data
@@ -3,11 +3,9 @@
 # debugging changes to the test code or to the test case generation.
 
 PSA storage read: AES-GCM+CTR
-#depends_on:PSA_WANT_KEY_TYPE_AES:PSA_WANT_ALG_GCM:PSA_WANT_ALG_CTR
-depends_on:MBEDTLS_AES_C:MBEDTLS_GCM_C:MBEDTLS_CTR_C
+depends_on:PSA_WANT_KEY_TYPE_AES:PSA_WANT_ALG_GCM:PSA_WANT_ALG_CTR
 key_storage_read:PSA_KEY_LIFETIME_PERSISTENT:PSA_KEY_TYPE_AES:128:PSA_KEY_USAGE_EXPORT | PSA_KEY_USAGE_ENCRYPT:PSA_ALG_GCM:PSA_ALG_CTR:"404142434445464748494a4b4c4d4e4f":"505341004b45590000000000010000000024800001010000000250050010c00410000000404142434445464748494a4b4c4d4e4f":1
 
 PSA storage save: AES-GCM+CTR
-#depends_on:PSA_WANT_KEY_TYPE_AES
-depends_on:MBEDTLS_AES_C
+depends_on:PSA_WANT_KEY_TYPE_AES
 key_storage_save:PSA_KEY_LIFETIME_PERSISTENT:PSA_KEY_TYPE_AES:128:PSA_KEY_USAGE_EXPORT | PSA_KEY_USAGE_ENCRYPT:PSA_ALG_GCM:PSA_ALG_CTR:"404142434445464748494a4b4c4d4e4f":"505341004b45590000000000010000000024800001010000000250050010c00410000000404142434445464748494a4b4c4d4e4f"