diff --git a/ssl/internal.h b/ssl/internal.h
index e307cb9..eb2af49 100644
--- a/ssl/internal.h
+++ b/ssl/internal.h
@@ -1061,22 +1061,22 @@
 int tls13_prepare_finished(SSL_HANDSHAKE *hs);
 int tls13_process_new_session_ticket(SSL *ssl);
 
-int ssl_ext_key_share_parse_serverhello(SSL *ssl, uint8_t **out_secret,
+int ssl_ext_key_share_parse_serverhello(SSL_HANDSHAKE *hs, uint8_t **out_secret,
                                         size_t *out_secret_len,
                                         uint8_t *out_alert, CBS *contents);
-int ssl_ext_key_share_parse_clienthello(SSL *ssl, int *out_found,
+int ssl_ext_key_share_parse_clienthello(SSL_HANDSHAKE *hs, int *out_found,
                                         uint8_t **out_secret,
                                         size_t *out_secret_len,
                                         uint8_t *out_alert, CBS *contents);
-int ssl_ext_key_share_add_serverhello(SSL *ssl, CBB *out);
+int ssl_ext_key_share_add_serverhello(SSL_HANDSHAKE *hs, CBB *out);
 
-int ssl_ext_pre_shared_key_parse_serverhello(SSL *ssl, uint8_t *out_alert,
-                                             CBS *contents);
-int ssl_ext_pre_shared_key_parse_clienthello(SSL *ssl,
+int ssl_ext_pre_shared_key_parse_serverhello(SSL_HANDSHAKE *hs,
+                                             uint8_t *out_alert, CBS *contents);
+int ssl_ext_pre_shared_key_parse_clienthello(SSL_HANDSHAKE *hs,
                                              SSL_SESSION **out_session,
                                              CBS *out_binders,
                                              uint8_t *out_alert, CBS *contents);
-int ssl_ext_pre_shared_key_add_serverhello(SSL *ssl, CBB *out);
+int ssl_ext_pre_shared_key_add_serverhello(SSL_HANDSHAKE *hs, CBB *out);
 
 /* ssl_is_sct_list_valid does a shallow parse of the SCT list in |contents| and
  * returns one iff it's valid. */
diff --git a/ssl/t1_lib.c b/ssl/t1_lib.c
index a87bee1..0b27165 100644
--- a/ssl/t1_lib.c
+++ b/ssl/t1_lib.c
@@ -1990,7 +1990,8 @@
   return CBB_flush(out);
 }
 
-int ssl_ext_pre_shared_key_parse_serverhello(SSL *ssl, uint8_t *out_alert,
+int ssl_ext_pre_shared_key_parse_serverhello(SSL_HANDSHAKE *hs,
+                                             uint8_t *out_alert,
                                              CBS *contents) {
   uint16_t psk_id;
   if (!CBS_get_u16(contents, &psk_id) ||
@@ -2010,11 +2011,12 @@
   return 1;
 }
 
-int ssl_ext_pre_shared_key_parse_clienthello(SSL *ssl,
+int ssl_ext_pre_shared_key_parse_clienthello(SSL_HANDSHAKE *hs,
                                              SSL_SESSION **out_session,
                                              CBS *out_binders,
                                              uint8_t *out_alert,
                                              CBS *contents) {
+  SSL *const ssl = hs->ssl;
   /* We only process the first PSK identity since we don't support pure PSK. */
   uint32_t obfuscated_ticket_age;
   CBS identities, ticket, binders;
@@ -2081,8 +2083,8 @@
   return 1;
 }
 
-int ssl_ext_pre_shared_key_add_serverhello(SSL *ssl, CBB *out) {
-  if (!ssl->s3->session_reused) {
+int ssl_ext_pre_shared_key_add_serverhello(SSL_HANDSHAKE *hs, CBB *out) {
+  if (!hs->ssl->s3->session_reused) {
     return 1;
   }
 
@@ -2260,9 +2262,10 @@
   return CBB_flush(out);
 }
 
-int ssl_ext_key_share_parse_serverhello(SSL *ssl, uint8_t **out_secret,
+int ssl_ext_key_share_parse_serverhello(SSL_HANDSHAKE *hs, uint8_t **out_secret,
                                         size_t *out_secret_len,
                                         uint8_t *out_alert, CBS *contents) {
+  SSL *const ssl = hs->ssl;
   CBS peer_key;
   uint16_t group_id;
   if (!CBS_get_u16(contents, &group_id) ||
@@ -2272,28 +2275,28 @@
     return 0;
   }
 
-  if (SSL_ECDH_CTX_get_id(&ssl->s3->hs->ecdh_ctx) != group_id) {
+  if (SSL_ECDH_CTX_get_id(&hs->ecdh_ctx) != group_id) {
     *out_alert = SSL_AD_ILLEGAL_PARAMETER;
     OPENSSL_PUT_ERROR(SSL, SSL_R_WRONG_CURVE);
     return 0;
   }
 
-  if (!SSL_ECDH_CTX_finish(&ssl->s3->hs->ecdh_ctx, out_secret, out_secret_len,
-                           out_alert, CBS_data(&peer_key),
-                           CBS_len(&peer_key))) {
+  if (!SSL_ECDH_CTX_finish(&hs->ecdh_ctx, out_secret, out_secret_len, out_alert,
+                           CBS_data(&peer_key), CBS_len(&peer_key))) {
     *out_alert = SSL_AD_INTERNAL_ERROR;
     return 0;
   }
 
   ssl->s3->new_session->key_exchange_info = group_id;
-  SSL_ECDH_CTX_cleanup(&ssl->s3->hs->ecdh_ctx);
+  SSL_ECDH_CTX_cleanup(&hs->ecdh_ctx);
   return 1;
 }
 
-int ssl_ext_key_share_parse_clienthello(SSL *ssl, int *out_found,
+int ssl_ext_key_share_parse_clienthello(SSL_HANDSHAKE *hs, int *out_found,
                                         uint8_t **out_secret,
                                         size_t *out_secret_len,
                                         uint8_t *out_alert, CBS *contents) {
+  SSL *const ssl = hs->ssl;
   uint16_t group_id;
   CBS key_shares;
   if (!tls1_get_shared_group(ssl, &group_id)) {
@@ -2348,11 +2351,9 @@
   CBB public_key;
   if (!CBB_init(&public_key, 32) ||
       !SSL_ECDH_CTX_init(&group, group_id) ||
-      !SSL_ECDH_CTX_accept(&group, &public_key, &secret, &secret_len,
-                           out_alert, CBS_data(&peer_key),
-                           CBS_len(&peer_key)) ||
-      !CBB_finish(&public_key, &ssl->s3->hs->public_key,
-                  &ssl->s3->hs->public_key_len)) {
+      !SSL_ECDH_CTX_accept(&group, &public_key, &secret, &secret_len, out_alert,
+                           CBS_data(&peer_key), CBS_len(&peer_key)) ||
+      !CBB_finish(&public_key, &hs->public_key, &hs->public_key_len)) {
     OPENSSL_free(secret);
     SSL_ECDH_CTX_cleanup(&group);
     CBB_cleanup(&public_key);
@@ -2368,7 +2369,8 @@
   return 1;
 }
 
-int ssl_ext_key_share_add_serverhello(SSL *ssl, CBB *out) {
+int ssl_ext_key_share_add_serverhello(SSL_HANDSHAKE *hs, CBB *out) {
+  SSL *const ssl = hs->ssl;
   uint16_t group_id;
   CBB kse_bytes, public_key;
   if (!tls1_get_shared_group(ssl, &group_id) ||
@@ -2376,15 +2378,14 @@
       !CBB_add_u16_length_prefixed(out, &kse_bytes) ||
       !CBB_add_u16(&kse_bytes, group_id) ||
       !CBB_add_u16_length_prefixed(&kse_bytes, &public_key) ||
-      !CBB_add_bytes(&public_key, ssl->s3->hs->public_key,
-                     ssl->s3->hs->public_key_len) ||
+      !CBB_add_bytes(&public_key, hs->public_key, hs->public_key_len) ||
       !CBB_flush(out)) {
     return 0;
   }
 
-  OPENSSL_free(ssl->s3->hs->public_key);
-  ssl->s3->hs->public_key = NULL;
-  ssl->s3->hs->public_key_len = 0;
+  OPENSSL_free(hs->public_key);
+  hs->public_key = NULL;
+  hs->public_key_len = 0;
 
   ssl->s3->new_session->key_exchange_info = group_id;
   return 1;
diff --git a/ssl/tls13_client.c b/ssl/tls13_client.c
index 6992b52..7319ca1 100644
--- a/ssl/tls13_client.c
+++ b/ssl/tls13_client.c
@@ -230,7 +230,7 @@
       return ssl_hs_error;
     }
 
-    if (!ssl_ext_pre_shared_key_parse_serverhello(ssl, &alert,
+    if (!ssl_ext_pre_shared_key_parse_serverhello(hs, &alert,
                                                   &pre_shared_key)) {
       ssl3_send_alert(ssl, SSL3_AL_FATAL, alert);
       return ssl_hs_error;
@@ -293,7 +293,7 @@
   /* Resolve ECDHE and incorporate it into the secret. */
   uint8_t *dhe_secret;
   size_t dhe_secret_len;
-  if (!ssl_ext_key_share_parse_serverhello(ssl, &dhe_secret, &dhe_secret_len,
+  if (!ssl_ext_key_share_parse_serverhello(hs, &dhe_secret, &dhe_secret_len,
                                            &alert, &key_share)) {
     ssl3_send_alert(ssl, SSL3_AL_FATAL, alert);
     return ssl_hs_error;
diff --git a/ssl/tls13_server.c b/ssl/tls13_server.c
index c1af999..9778610 100644
--- a/ssl/tls13_server.c
+++ b/ssl/tls13_server.c
@@ -71,7 +71,7 @@
   uint8_t *dhe_secret;
   size_t dhe_secret_len;
   uint8_t alert = SSL_AD_DECODE_ERROR;
-  if (!ssl_ext_key_share_parse_clienthello(ssl, &found_key_share, &dhe_secret,
+  if (!ssl_ext_key_share_parse_clienthello(hs, &found_key_share, &dhe_secret,
                                            &dhe_secret_len, &alert,
                                            &key_share)) {
     ssl3_send_alert(ssl, SSL3_AL_FATAL, alert);
@@ -223,7 +223,7 @@
       return ssl_hs_error;
     }
 
-    if (!ssl_ext_pre_shared_key_parse_clienthello(ssl, &session, &binders,
+    if (!ssl_ext_pre_shared_key_parse_clienthello(hs, &session, &binders,
                                                   &alert, &pre_shared_key)) {
       ssl3_send_alert(ssl, SSL3_AL_FATAL, alert);
       return ssl_hs_error;
@@ -388,8 +388,8 @@
       !CBB_add_bytes(&body, ssl->s3->server_random, SSL3_RANDOM_SIZE) ||
       !CBB_add_u16(&body, ssl_cipher_get_value(ssl->s3->tmp.new_cipher)) ||
       !CBB_add_u16_length_prefixed(&body, &extensions) ||
-      !ssl_ext_pre_shared_key_add_serverhello(ssl, &extensions) ||
-      !ssl_ext_key_share_add_serverhello(ssl, &extensions) ||
+      !ssl_ext_pre_shared_key_add_serverhello(hs, &extensions) ||
+      !ssl_ext_key_share_add_serverhello(hs, &extensions) ||
       !ssl_complete_message(ssl, &cbb)) {
     goto err;
   }