diff --git a/test/algorithmid_test.c b/test/algorithmid_test.c
new file mode 100644
index 0000000000..b1b579b40d
--- /dev/null
+++ b/test/algorithmid_test.c
@@ -0,0 +1,338 @@
+/*
+ * Copyright 2021 The OpenSSL Project Authors. All Rights Reserved.
+ *
+ * Licensed under the Apache License 2.0 (the "License").  You may not use
+ * this file except in compliance with the License.  You can obtain a copy
+ * in the file LICENSE in the source distribution or at
+ * https://www.openssl.org/source/license.html
+ */
+
+#include <openssl/asn1.h>
+#include <openssl/pem.h>
+#include "internal/sizes.h"
+#include "crypto/evp.h"
+#include "testutil.h"
+
+/* Collected arguments */
+static const char *eecert_filename = NULL;  /* For test_x509_file() */
+static const char *cacert_filename = NULL;  /* For test_x509_file() */
+static const char *pubkey_filename = NULL;  /* For test_spki_file() */
+
+#define ALGORITHMID_NAME "algorithm-id"
+
+static int test_spki_aid(X509_PUBKEY *pubkey, const char *filename)
+{
+    const ASN1_OBJECT *oid;
+    X509_ALGOR *alg = NULL;
+    EVP_PKEY *pkey = NULL;
+    EVP_KEYMGMT *keymgmt = NULL;
+    void *keydata = NULL;
+    char name[OSSL_MAX_NAME_SIZE] = "";
+    unsigned char *algid_legacy = NULL;
+    int algid_legacy_len = 0;
+    static unsigned char algid_prov[OSSL_MAX_ALGORITHM_ID_SIZE];
+    size_t algid_prov_len = 0;
+    const OSSL_PARAM *gettable_params = NULL;
+    OSSL_PARAM params[] = {
+        OSSL_PARAM_octet_string(ALGORITHMID_NAME,
+                                &algid_prov, sizeof(algid_prov)),
+        OSSL_PARAM_END
+    };
+    int ret = 0;
+
+    if (!TEST_true(X509_PUBKEY_get0_param(NULL, NULL, NULL, &alg, pubkey))
+        || !TEST_ptr(pkey = X509_PUBKEY_get0(pubkey)))
+        goto end;
+
+    if (!TEST_int_ge(algid_legacy_len = i2d_X509_ALGOR(alg, &algid_legacy), 0))
+        goto end;
+
+    X509_ALGOR_get0(&oid, NULL, NULL, alg);
+    if (!TEST_true(OBJ_obj2txt(name, sizeof(name), oid, 0)))
+        goto end;
+
+    /*
+     * We use an internal functions to ensure we have a provided key.
+     * Note that |keydata| should not be freed, as it's cached in |pkey|.
+     * The |keymgmt|, however, should, as its reference count is incremented
+     * in this function.
+     */
+    if ((keydata = evp_pkey_export_to_provider(pkey, NULL,
+                                               &keymgmt, NULL)) == NULL) {
+        TEST_info("The public key found in '%s' doesn't have provider support."
+                  "  Skipping...",
+                  filename);
+        ret = 1;
+        goto end;
+    }
+
+    if (!TEST_true(EVP_KEYMGMT_is_a(keymgmt, name))) {
+        TEST_info("The AlgorithmID key type (%s) for the public key found in"
+                  " '%s' doesn't match the key type of the extracted public"
+                  " key.",
+                  name, filename);
+        ret = 1;
+        goto end;
+    }
+
+    if (!TEST_ptr(gettable_params = EVP_KEYMGMT_gettable_params(keymgmt))
+        || !TEST_ptr(OSSL_PARAM_locate_const(gettable_params, ALGORITHMID_NAME))) {
+        TEST_info("The %s provider keymgmt appears to lack support for algorithm-id."
+                  "  Skipping...",
+                  name);
+        ret = 1;
+        goto end;
+    }
+
+    algid_prov[0] = '\0';
+    if (!TEST_true(evp_keymgmt_get_params(keymgmt, keydata, params)))
+        goto end;
+    algid_prov_len = params[0].return_size;
+
+    /* We now have all the algorithm IDs we need, let's compare them */
+    if (TEST_mem_eq(algid_legacy, algid_legacy_len,
+                    algid_prov, algid_prov_len))
+        ret = 1;
+
+ end:
+    EVP_KEYMGMT_free(keymgmt);
+    OPENSSL_free(algid_legacy);
+    return ret;
+}
+
+static int test_x509_spki_aid(X509 *cert, const char *filename)
+{
+    X509_PUBKEY *pubkey = X509_get_X509_PUBKEY(cert);
+
+    return test_spki_aid(pubkey, filename);
+}
+
+/*
+ * TODO
+ * When we gain the ability to get an EVP_SIGNATURE with a complete signature
+ * algorithm name (like "sha1WithRSAEncryption" or its corresponding OID in
+ * text form, "1.2.840.113549.1.1.2"), we won't have to limit this test to
+ * what we have in libcrypto's cross-reference db, i.e. won't have to call
+ * OBJ_find_sigid_algs() to find out the EVP_PKEY_METHOD NID any more.
+ * All we'd have to do is used OBJ_obj2txt() on an ASN1_OBJECT and pass the
+ * result.
+ */
+static int test_x509_sig_aid(X509 *eecert, const char *ee_filename,
+                             X509 *cacert, const char *ca_filename)
+{
+    const ASN1_OBJECT *sig_oid = NULL;
+    const X509_ALGOR *alg = NULL;
+    int sig_nid = NID_undef, dig_nid = NID_undef, pkey_nid = NID_undef;
+    EVP_MD_CTX *mdctx = NULL;
+    EVP_PKEY_CTX *pctx = NULL;
+    EVP_PKEY *pkey = NULL;
+    unsigned char *algid_legacy = NULL;
+    int algid_legacy_len = 0;
+    static unsigned char algid_prov[OSSL_MAX_ALGORITHM_ID_SIZE];
+    size_t algid_prov_len = 0;
+    const OSSL_PARAM *gettable_params = NULL;
+    OSSL_PARAM params[] = {
+        OSSL_PARAM_octet_string("algorithm-id",
+                                &algid_prov, sizeof(algid_prov)),
+        OSSL_PARAM_END
+    };
+    int ret = 0;
+
+    X509_get0_signature(NULL, &alg, eecert);
+    X509_ALGOR_get0(&sig_oid, NULL, NULL, alg);
+    if (!TEST_int_eq(X509_ALGOR_cmp(alg, X509_get0_tbs_sigalg(eecert)), 0))
+        goto end;
+    if (!TEST_int_ne(sig_nid = OBJ_obj2nid(sig_oid), NID_undef)
+        || !TEST_true(OBJ_find_sigid_algs(sig_nid, &dig_nid, &pkey_nid))
+        || !TEST_ptr(pkey = X509_get0_pubkey(cacert)))
+        goto end;
+
+    if (!TEST_true(EVP_PKEY_is_a(pkey, OBJ_nid2sn(pkey_nid)))) {
+        TEST_info("The '%s' pubkey can't be used to verify the '%s' signature",
+                  ca_filename, ee_filename);
+        TEST_info("Signature algorithm is %s (pkey type %s, hash type %s)",
+                  OBJ_nid2sn(sig_nid), OBJ_nid2sn(pkey_nid), OBJ_nid2sn(dig_nid));
+        TEST_info("Pkey key type is %s", EVP_PKEY_get0_first_alg_name(pkey));
+        goto end;
+    }
+
+    if (!TEST_int_ge(algid_legacy_len = i2d_X509_ALGOR(alg, &algid_legacy), 0))
+        goto end;
+
+    if (!TEST_ptr(mdctx = EVP_MD_CTX_new())
+        || !TEST_true(EVP_DigestVerifyInit_ex(mdctx, &pctx,
+                                              OBJ_nid2sn(dig_nid),
+                                              NULL, NULL, pkey))) {
+        TEST_info("Couldn't initialize a DigestVerify operation with "
+                  "pkey type %s and hash type %s",
+                  OBJ_nid2sn(pkey_nid), OBJ_nid2sn(dig_nid));
+        goto end;
+    }
+
+    if (!TEST_ptr(gettable_params = EVP_PKEY_CTX_gettable_params(pctx))
+        || !TEST_ptr(OSSL_PARAM_locate_const(gettable_params, ALGORITHMID_NAME))) {
+        TEST_info("The %s provider keymgmt appears to lack support for algorithm-id"
+                  "  Skipping...",
+                  OBJ_nid2sn(pkey_nid));
+        ret = 1;
+        goto end;
+    }
+
+    algid_prov[0] = '\0';
+    if (!TEST_true(EVP_PKEY_CTX_get_params(pctx, params)))
+        goto end;
+    algid_prov_len = params[0].return_size;
+
+    /* We now have all the algorithm IDs we need, let's compare them */
+    if (TEST_mem_eq(algid_legacy, algid_legacy_len,
+                    algid_prov, algid_prov_len))
+        ret = 1;
+
+ end:
+    EVP_MD_CTX_free(mdctx);
+    /* pctx is free by EVP_MD_CTX_free() */
+    OPENSSL_free(algid_legacy);
+    return ret;
+}
+
+static int test_spki_file(void)
+{
+    X509_PUBKEY *pubkey = NULL;
+    BIO *b = BIO_new_file(pubkey_filename, "r");
+    int ret = 0;
+
+    if (b == NULL) {
+        TEST_error("Couldn't open '%s' for reading\n", pubkey_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+
+    if ((pubkey = PEM_read_bio_X509_PUBKEY(b, NULL, NULL, NULL)) == NULL) {
+        TEST_error("'%s' doesn't appear to be a SubjectPublicKeyInfo in PEM format\n",
+                   pubkey_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+
+    ret = test_spki_aid(pubkey, pubkey_filename);
+ end:
+    BIO_free(b);
+    X509_PUBKEY_free(pubkey);
+    return ret;
+}
+
+static int test_x509_files(void)
+{
+    X509 *eecert = NULL, *cacert = NULL;
+    BIO *bee = NULL, *bca = NULL;
+    int ret = 0;
+
+    if ((bee = BIO_new_file(eecert_filename, "r")) == NULL) {
+        TEST_error("Couldn't open '%s' for reading\n", eecert_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+    if ((bca = BIO_new_file(cacert_filename, "r")) == NULL) {
+        TEST_error("Couldn't open '%s' for reading\n", cacert_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+
+    if ((eecert = PEM_read_bio_X509(bee, NULL, NULL, NULL)) == NULL) {
+        TEST_error("'%s' doesn't appear to be a X.509 certificate in PEM format\n",
+                   eecert_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+    if ((cacert = PEM_read_bio_X509(bca, NULL, NULL, NULL)) == NULL) {
+        TEST_error("'%s' doesn't appear to be a X.509 certificate in PEM format\n",
+                   cacert_filename);
+        TEST_openssl_errors();
+        goto end;
+    }
+
+    ret = test_x509_sig_aid(eecert, eecert_filename, cacert, cacert_filename)
+        & test_x509_spki_aid(eecert, eecert_filename)
+        & test_x509_spki_aid(cacert, cacert_filename);
+ end:
+    BIO_free(bee);
+    BIO_free(bca);
+    X509_free(eecert);
+    X509_free(cacert);
+    return ret;
+}
+
+typedef enum OPTION_choice {
+    OPT_ERR = -1,
+    OPT_EOF = 0,
+    OPT_X509,
+    OPT_SPKI,
+    OPT_TEST_ENUM
+} OPTION_CHOICE;
+
+const OPTIONS *test_get_options(void)
+{
+    static const OPTIONS test_options[] = {
+        OPT_TEST_OPTIONS_WITH_EXTRA_USAGE("file...\n"),
+        { "x509", OPT_X509, '-', "Test X.509 certificates.  Requires two files" },
+        { "spki", OPT_SPKI, '-', "Test public keys in SubjectPublicKeyInfo form.  Requires one file" },
+        { OPT_HELP_STR, 1, '-',
+          "file...\tFile(s) to run tests on.  All files must be PEM encoded.\n" },
+        { NULL }
+    };
+    return test_options;
+}
+
+int setup_tests(void)
+{
+    OPTION_CHOICE o;
+    int n, x509 = 0, spki = 0, testcount = 0;
+
+    while ((o = opt_next()) != OPT_EOF) {
+        switch (o) {
+        case OPT_X509:
+            x509 = 1;
+            break;
+        case OPT_SPKI:
+            spki = 1;
+            break;
+        case OPT_TEST_CASES:
+           break;
+        default:
+        case OPT_ERR:
+            return 0;
+        }
+    }
+
+    /* |testcount| adds all the given test types together */
+    testcount = x509 + spki;
+
+    if (testcount < 1)
+        BIO_printf(bio_err, "No test type given\n");
+    else if (testcount > 1)
+        BIO_printf(bio_err, "Only one test type may be given\n");
+    if (testcount != 1)
+        return 0;
+
+    n = test_get_argument_count();
+    if (spki && n == 1) {
+        pubkey_filename = test_get_argument(0);
+    } else if (x509 && n == 2) {
+        eecert_filename = test_get_argument(0);
+        cacert_filename = test_get_argument(1);
+    }
+
+    if (spki && pubkey_filename == NULL) {
+        BIO_printf(bio_err, "Missing -spki argument\n");
+        return 0;
+    } else if (x509 && (eecert_filename == NULL || cacert_filename == NULL)) {
+        BIO_printf(bio_err, "Missing -x509 argument(s)\n");
+        return 0;
+    }
+
+    if (x509)
+        ADD_TEST(test_x509_files);
+    if (spki)
+        ADD_TEST(test_spki_file);
+    return 1;
+}
diff --git a/test/build.info b/test/build.info
index b9e6d4f3b1..159f7146e3 100644
--- a/test/build.info
+++ b/test/build.info
@@ -707,6 +707,11 @@ IF[{- !$disabled{tests} -}]
     SOURCE[ssl_old_test]=ssl_old_test.c helpers/predefined_dhparams.c
     INCLUDE[ssl_old_test]=.. ../include ../apps/include
     DEPEND[ssl_old_test]=../libcrypto.a ../libssl.a libtestutil.a
+
+    PROGRAMS{noinst}=algorithmid_test
+    SOURCE[algorithmid_test]=algorithmid_test.c
+    INCLUDE[algorithmid_test]=../include ../apps/include
+    DEPEND[algorithmid_test]=../libcrypto.a libtestutil.a
   ENDIF
 
   PROGRAMS{noinst}=asn1_time_test
diff --git a/test/recipes/06-test_algorithmid.t b/test/recipes/06-test_algorithmid.t
new file mode 100644
index 0000000000..7d56c09150
--- /dev/null
+++ b/test/recipes/06-test_algorithmid.t
@@ -0,0 +1,60 @@
+#! /usr/bin/perl
+
+# Copyright 2018 The OpenSSL Project Authors. All Rights Reserved.
+# 
+# Licensed under the Apache License 2.0 (the "License").  You may not use
+# this file except in compliance with the License.  You can obtain a copy
+# in the file LICENSE in the source distribution or at
+# https://www.openssl.org/source/license.html
+
+use strict;
+
+use OpenSSL::Test qw(:DEFAULT srctop_file);
+use OpenSSL::Test::Utils;
+
+setup("test_algorithmid");
+
+# eecert => cacert
+my %certs_info =
+    (
+     'ee-cert' => 'ca-cert',
+     'ee-cert2' => 'ca-cert2',
+
+     # 'ee-pss-sha1-cert' => 'ca-cert',
+     # 'ee-pss-sha256-cert' => 'ca-cert',
+     # 'ee-pss-cert' => 'ca-pss-cert',
+     # 'server-pss-restrict-cert' => 'rootcert',
+
+     (
+      disabled('ec')
+      ? ()
+      : (
+         'ee-cert-ec-explicit' => 'ca-cert-ec-named',
+         'ee-cert-ec-named-explicit' => 'ca-cert-ec-explicit',
+         'ee-cert-ec-named-named' => 'ca-cert-ec-named',
+         # 'server-ed448-cert' => 'root-ed448-cert'
+         'server-ecdsa-brainpoolP256r1-cert' => 'rootcert',
+        )
+     )
+    );
+my @pubkeys =
+    (
+     'testrsapub',
+     disabled('dsa') ? () : 'testdsapub',
+     disabled('ec') ? () : qw(testecpub-p256 tested25519pub tested448pub)
+    );
+my @certs = sort keys %certs_info;
+
+plan tests =>
+    scalar @certs
+    + scalar @pubkeys;
+
+foreach (@certs) {
+    ok(run(test(['algorithmid_test', '-x509',
+                 srctop_file('test', 'certs', "$_.pem"),
+                 srctop_file('test', 'certs', "$certs_info{$_}.pem")])));
+}
+
+foreach (sort @pubkeys) {
+    ok(run(test(['algorithmid_test', '-spki', srctop_file('test', "$_.pem")])));
+}