diff --git a/ChangeLog.d/sha256-armv8-crypto-pragma.txt b/ChangeLog.d/sha256-armv8-crypto-pragma.txt
new file mode 100644
index 0000000000..8b55f4542b
--- /dev/null
+++ b/ChangeLog.d/sha256-armv8-crypto-pragma.txt
@@ -0,0 +1,7 @@
+Bugfix
+   * Fix a build failure with GCC on AArch64 when
+     MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_IF_PRESENT or
+     MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_ONLY is enabled and the target CPU has
+     extensions beyond Armv8.0-A, such as -mcpu=cortex-a53 with CRC. The
+     sha256 module then failed to inline always_inline functions such as a
+     fortified memset().
diff --git a/ChangeLog.d/sha512-a64-crypto-pragma.txt b/ChangeLog.d/sha512-a64-crypto-pragma.txt
new file mode 100644
index 0000000000..470ef1476a
--- /dev/null
+++ b/ChangeLog.d/sha512-a64-crypto-pragma.txt
@@ -0,0 +1,7 @@
+Bugfix
+   * Fix a build failure with GCC 13 or later on AArch64 when
+     MBEDTLS_SHA512_USE_A64_CRYPTO_IF_PRESENT or
+     MBEDTLS_SHA512_USE_A64_CRYPTO_ONLY is enabled and the target CPU has
+     extensions that Armv8.2-A does not include, such as -mcpu=cortex-a76.
+     The sha512 module then failed to inline always_inline functions such as
+     a fortified memset().
diff --git a/library/sha256.c b/library/sha256.c
index 06d2f53bc3..69381dfe86 100644
--- a/library/sha256.c
+++ b/library/sha256.c
@@ -102,7 +102,11 @@
 #          error "A more recent GCC is required for MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_*"
 #        else
 #          pragma GCC push_options
-#          pragma GCC target ("arch=armv8-a+crypto")
+#          if defined(MBEDTLS_ARCH_IS_ARM64)
+#            pragma GCC target ("+crypto")
+#          else
+#            pragma GCC target ("arch=armv8-a+crypto")
+#          endif
 #          define MBEDTLS_POP_TARGET_PRAGMA
 #        endif
 #      else
diff --git a/library/sha512.c b/library/sha512.c
index 286cbdf9f0..99ab8ac409 100644
--- a/library/sha512.c
+++ b/library/sha512.c
@@ -91,7 +91,12 @@
 #          error "A more recent GCC is required for MBEDTLS_SHA512_USE_A64_CRYPTO_*"
 #        else
 #          pragma GCC push_options
-#          pragma GCC target ("arch=armv8.2-a+sha3")
+/* Before GCC 13, the SHA-512 intrinsics require Armv8.2-A. */
+#          if __GNUC__ < 13
+#            pragma GCC target ("arch=armv8.2-a+sha3")
+#          else
+#            pragma GCC target ("+sha3")
+#          endif
 #          define MBEDTLS_POP_TARGET_PRAGMA
 #        endif
 #      else
diff --git a/tests/scripts/components-platform.sh b/tests/scripts/components-platform.sh
index 9a1d42e95d..f83f15cb13 100644
--- a/tests/scripts/components-platform.sh
+++ b/tests/scripts/components-platform.sh
@@ -276,6 +276,40 @@ component_test_m32_no_asm () {
     make test
 }
 
+component_build_sha_armce_gcc () {
+    # The SHA-256 crypto extensions must be added to the target of the build,
+    # not replace it: with -mcpu=cortex-a53 (which has CRC) GCC otherwise
+    # refuses to inline always_inline functions such as a fortified memset().
+    scripts/config.py unset MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_IF_PRESENT
+    for opt in MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_ONLY MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_IF_PRESENT; do
+        scripts/config.py set $opt
+            msg "$opt ${AARCH64_LINUX_GNU_GCC_PREFIX}gcc, -mcpu=cortex-a53 -D_FORTIFY_SOURCE=2"
+            make -B library/sha256.o library/sha256.s CC="${AARCH64_LINUX_GNU_GCC_PREFIX}gcc" CFLAGS="-std=c99 -Werror -O2 -mcpu=cortex-a53 -D_FORTIFY_SOURCE=2"
+            msg "$opt ${AARCH64_LINUX_GNU_GCC_PREFIX}gcc, test aarch64 crypto instructions built"
+            grep -E 'sha256[a-z0-9]+\s+[qv]' library/sha256.s
+        scripts/config.py unset $opt
+    done
+
+    # The same for the SHA-512 extensions with -mcpu=cortex-a76, which has
+    # extensions beyond Armv8.2-A. Before GCC 13, the SHA-512 intrinsics
+    # require Armv8.2-A, so this only builds with GCC 13 or later.
+    if [ "$(gcc_version "${AARCH64_LINUX_GNU_GCC_PREFIX}gcc")" -ge 13 ]; then
+        for opt in MBEDTLS_SHA512_USE_A64_CRYPTO_ONLY MBEDTLS_SHA512_USE_A64_CRYPTO_IF_PRESENT; do
+            scripts/config.py set $opt
+                msg "$opt ${AARCH64_LINUX_GNU_GCC_PREFIX}gcc, -mcpu=cortex-a76 -D_FORTIFY_SOURCE=2"
+                make -B library/sha512.o library/sha512.s CC="${AARCH64_LINUX_GNU_GCC_PREFIX}gcc" CFLAGS="-std=c99 -Werror -O2 -mcpu=cortex-a76 -D_FORTIFY_SOURCE=2"
+                msg "$opt ${AARCH64_LINUX_GNU_GCC_PREFIX}gcc, test aarch64 crypto instructions built"
+                grep -E 'sha512[a-z0-9]+\s+[qv]' library/sha512.s
+            scripts/config.py unset $opt
+        done
+    fi
+}
+
+support_build_sha_armce_gcc () {
+    # Minimum version of GCC for MBEDTLS_SHA256_USE_ARMV8_A_CRYPTO_* is 6.0
+    [ "$(gcc_version "${AARCH64_LINUX_GNU_GCC_PREFIX}gcc")" -ge 6 ]
+}
+
 support_test_m32_no_asm () {
     case $(uname -m) in
         amd64|x86_64) true;;