diff --git a/tests/suites/test_suite_x509parse.data b/tests/suites/test_suite_x509parse.data
index 2829cca42..90d6f6172 100644
--- a/tests/suites/test_suite_x509parse.data
+++ b/tests/suites/test_suite_x509parse.data
@@ -1083,40 +1083,45 @@ x509parse_crt:"301c3007a0030201028285300d06092a864886f70d01010b0500030200ff":"":
 X509 Certificate ASN1 (TBSCertificate, invalid serial, length out of bounds)
 x509parse_crt:"301c3007a0030201028201300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_SERIAL + MBEDTLS_ERR_ASN1_OUT_OF_DATA
 
-X509 Certificate ASN1 (TBSCertificate, correct serial, no alg)
+X509 Certificate ASN1 (TBSCertificate, AlgorithmIdentifier missing)
 x509parse_crt:"3020300ba0030201028204deadbeef300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
 
-X509 Certificate ASN1 (TBSCertificate, correct serial, no alg oid tag)
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, tag wrong)
 x509parse_crt:"3021300ca0030201028204deadbeef00300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_UNEXPECTED_TAG
 
-X509 Certificate ASN1 (TBSCertificate, alg oid no data in sequence)
-x509parse_crt:"300f300da0030201028204deadbeef3000":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID missing)
+x509parse_crt:"308189308173a0030201008204deadbeef3000300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
 
-X509 Certificate ASN1 (TBSCertificate, alg with params)
-x509parse_crt:"30163014a0030201028204deadbeef30070604cafed00d01":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID tag wrong)
+x509parse_crt:"30818b308175a0030201008204deadbeef30020500300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_UNEXPECTED_TAG
 
-X509 Certificate ASN1 (TBSCertificate, correct alg data, no params unknown version)
-x509parse_crt:"30153013a0030201028204deadbeef30060604cafed00d":"":MBEDTLS_ERR_X509_UNKNOWN_VERSION
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID invalid length encoding)
+x509parse_crt:"30818b308175a0030201008204deadbeef30020685300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_INVALID_LENGTH
 
-X509 Certificate ASN1 (TBSCertificate, correct alg data, unknown version)
-x509parse_crt:"30173015a0030201028204deadbeef30080604cafed00d0500":"":MBEDTLS_ERR_X509_UNKNOWN_VERSION
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID length out of bounds)
+x509parse_crt:"30818b308175a0030201008204deadbeef30020601300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
 
-X509 Certificate ASN1 (TBSCertificate, correct alg data, length mismatch)
-x509parse_crt:"30183016a0030201028204deadbeef30090604cafed00d050000":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_LENGTH_MISMATCH
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID empty)
+x509parse_crt:"30818b308175a0030201008204deadbeef30020600300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_UNKNOWN_SIG_ALG + MBEDTLS_ERR_OID_NOT_FOUND
 
-X509 Certificate ASN1 (TBSCertificate, correct alg, unknown alg_id)
-x509parse_crt:"30173015a0030201028204deadbeef30080604cafed00d0500":"":MBEDTLS_ERR_X509_UNKNOWN_SIG_ALG + MBEDTLS_ERR_OID_NOT_FOUND
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, OID unknown)
+x509parse_crt:"30818f308179a0030201008204deadbeef30060604deadbeef300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_UNKNOWN_SIG_ALG + MBEDTLS_ERR_OID_NOT_FOUND
 
-X509 Certificate ASN1 (TBSCertificate, correct alg, specific alg_id)
-depends_on:MBEDTLS_RSA_C:MBEDTLS_SHA256_C
-x509parse_crt:"301c301aa0030201028204deadbeef300d06092a864886f70d01010b0500":"":MBEDTLS_ERR_X509_INVALID_FORMAT + MBEDTLS_ERR_ASN1_OUT_OF_DATA
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, param invalid length encoding)
+x509parse_crt:"308196308180a0030201008204deadbeef300d06092a864886f70d01010b0685300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_INVALID_LENGTH
 
-X509 Certificate ASN1 (TBSCertificate, correct alg, unknown specific alg_id)
-x509parse_crt:"301c301aa0030201028204deadbeef300d06092a864886f70d0101010500":"":MBEDTLS_ERR_X509_UNKNOWN_SIG_ALG + MBEDTLS_ERR_OID_NOT_FOUND
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, param length out of bounds)
+x509parse_crt:"308196308180a0030201008204deadbeef300d06092a864886f70d01010b0601300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_OUT_OF_DATA
 
-X509 Certificate ASN1 (TBSCertificate, correct alg, bad RSASSA-PSS params)
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, param length mismatch)
+x509parse_crt:"308198308182a0030201008204deadbeef300f06092a864886f70d01010b06010000300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_LENGTH_MISMATCH
+
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, params present but empty)
+x509parse_crt:"308196308180a0030201008204deadbeef300d06092a864886f70d01010b0600300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG
+
+X509 Certificate ASN1 (TBSCertificate, invalid AlgorithmIdentifier, bad RSASSA-PSS params)
 depends_on:MBEDTLS_X509_RSASSA_PSS_SUPPORT
-x509parse_crt:"30193017a003020102020118300d06092a864886f70d01010a3100":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_UNEXPECTED_TAG
+x509parse_crt:"308196308180a0030201008204deadbeef300d06092a864886f70d01010a3100300c310a30080600130454657374301c170c303930313031303030303030170c303931323331323335393539300c310a30080600130454657374302a300d06092a864886f70d010101050003190030160210ffffffffffffffffffffffffffffffff0202ffff300d06092a864886f70d01010b0500030200ff":"":MBEDTLS_ERR_X509_INVALID_ALG + MBEDTLS_ERR_ASN1_UNEXPECTED_TAG
 
 X509 Certificate ASN1 (TBSCertificate, issuer no set data)
 depends_on:MBEDTLS_RSA_C:MBEDTLS_SHA256_C