From patchwork Thu Aug 26 13:39:09 2021 Content-Type: text/plain; charset="utf-8" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit X-Patchwork-Submitter: Kumar Kartikeya Dwivedi X-Patchwork-Id: 12459851 X-Patchwork-Delegate: bpf@iogearbox.net Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.0 (2014-02-07) on aws-us-west-2-korg-lkml-1.web.codeaurora.org X-Spam-Level: X-Spam-Status: No, score=-12.9 required=3.0 tests=BAYES_00,DKIM_SIGNED, DKIM_VALID,DKIM_VALID_AU,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,INCLUDES_CR_TRAILER,INCLUDES_PATCH, MAILING_LIST_MULTI,SPF_HELO_NONE,SPF_PASS,UNWANTED_LANGUAGE_BODY, URIBL_BLOCKED,USER_AGENT_GIT autolearn=unavailable autolearn_force=no version=3.4.0 Received: from mail.kernel.org (mail.kernel.org [198.145.29.99]) by smtp.lore.kernel.org (Postfix) with ESMTP id BB858C4320A for ; Thu, 26 Aug 2021 13:39:26 +0000 (UTC) Received: from vger.kernel.org (vger.kernel.org [23.128.96.18]) by mail.kernel.org (Postfix) with ESMTP id 9C49560F14 for ; Thu, 26 Aug 2021 13:39:26 +0000 (UTC) Received: (majordomo@vger.kernel.org) by vger.kernel.org via listexpand id S242744AbhHZNkM (ORCPT ); Thu, 26 Aug 2021 09:40:12 -0400 Received: from lindbergh.monkeyblade.net ([23.128.96.19]:48410 "EHLO lindbergh.monkeyblade.net" rhost-flags-OK-OK-OK-OK) by vger.kernel.org with ESMTP id S242493AbhHZNkJ (ORCPT ); Thu, 26 Aug 2021 09:40:09 -0400 Received: from mail-pj1-x1043.google.com (mail-pj1-x1043.google.com [IPv6:2607:f8b0:4864:20::1043]) by lindbergh.monkeyblade.net (Postfix) with ESMTPS id 07EE0C061757; Thu, 26 Aug 2021 06:39:22 -0700 (PDT) Received: by mail-pj1-x1043.google.com with SMTP id n13-20020a17090a4e0d00b0017946980d8dso6595180pjh.5; Thu, 26 Aug 2021 06:39:22 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=from:to:cc:subject:date:message-id:in-reply-to:references :mime-version:content-transfer-encoding; bh=n+CSO3Tk5RJvAuQKM7dCZIYomNFSoJBC2AhA0046028=; b=CPp9Q6ESc2IiXi6+kdQCYoJj/fEJVhz0WCKhdxqz84GoKop3UdjZcpjxWU6tYAPpKm 2i3GWOc+lHZ5Zm27aAq7e89+CmY6DkRlDNhYD8xHNyzS0hCy++0xXFs0TB+mWJd5ODbg k/WxuDAKi+GUcwII8sBWClyPOPkldGjHbY//h2QRevS8ULbVd1QVqH3G6ZloLChbt3sD PNGKAV3WoioI2MMeuD7tltU+0PpmpHMQp3QxO1jshc+1UG7R2i17woK42eI5IkNmDvkV ak0QVKDMWYEud83gcPb5lW3E+9VoE1wEFgc7Jdv3UkarQyq+opsOyh2LFteC8Mzz5slU d3Yg== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=x-gm-message-state:from:to:cc:subject:date:message-id:in-reply-to :references:mime-version:content-transfer-encoding; bh=n+CSO3Tk5RJvAuQKM7dCZIYomNFSoJBC2AhA0046028=; b=PMmJhzchDgMeEhPH1h20EJOmz9DA1zpD2xsN3dHpLzhoY2Bi0KKUyJT79fCMTQwG6x Ml6tQvqzDq9hhTJ2gf3DzrAw5f/nXD+54YIezq6ktY6Ji982ocOk3jfTIBRzrC81bX/2 C1JK5Sz6ruPyUZRANzIlBi9aELVr1WEfIuGiF3+gAn5GnRoiVaRKOa7Kh6semzhSTGpr UgEyIiNp6a7zqJvsK80TcFO19jfR1DvHD7vEmk4H5zq7K2MXpMIcNAkorbDSozbtWBmh 5EArrBHIWsZ38xglZdWrMOEc+z8I0aqjWXOBxGS/vZQ6f4TCmB9Ha3moTCSRSx/KE2Wj WFUA== X-Gm-Message-State: AOAM533/blJhx0tcveWSVM/TniASWg7D2PFEEzR9x0z+4F5gSKP/qvmq UF1VuSZMTCNd9VgertURYjV5kQumnzw= X-Google-Smtp-Source: ABdhPJzuMFEdAhI+saEuELosBGgsVWQcefNgHOJhiv7743D8/LHSUqKXK7KGyoxVjSwolTQaKRUHCg== X-Received: by 2002:a17:902:7611:b029:12c:b602:5820 with SMTP id k17-20020a1709027611b029012cb6025820mr3849299pll.6.1629985161309; Thu, 26 Aug 2021 06:39:21 -0700 (PDT) Received: from localhost ([2405:201:6014:d820:9cc6:d37f:c2fd:dc6]) by smtp.gmail.com with ESMTPSA id b21sm3072294pfl.83.2021.08.26.06.39.20 (version=TLS1_3 cipher=TLS_AES_256_GCM_SHA384 bits=256/256); Thu, 26 Aug 2021 06:39:21 -0700 (PDT) From: Kumar Kartikeya Dwivedi To: bpf@vger.kernel.org Cc: Kumar Kartikeya Dwivedi , Alexei Starovoitov , Daniel Borkmann , Andrii Nakryiko , Martin KaFai Lau , KP Singh , Spencer Baugh , Pavel Emelyanov , Alexander Mihalicyn , Andrei Vagin , linux-security-module@vger.kernel.org Subject: [PATCH bpf-next v2 1/5] bpf: Implement file local storage Date: Thu, 26 Aug 2021 19:09:09 +0530 Message-Id: <20210826133913.627361-2-memxor@gmail.com> X-Mailer: git-send-email 2.33.0 In-Reply-To: <20210826133913.627361-1-memxor@gmail.com> References: <20210826133913.627361-1-memxor@gmail.com> MIME-Version: 1.0 Precedence: bulk List-ID: X-Mailing-List: bpf@vger.kernel.org X-Patchwork-Delegate: bpf@iogearbox.net This map is useful in general to tie data associated with a open file (not fd) from eBPF programs, such that data is released when the file goes away (e.g. checkpoint/restore usecase). Another usecase is implementing Capsicum [0] style capability sandbox in userspace using eBPF LSM, enforcing rights at the file level using this mechanism. [0]: https://www.usenix.org/legacy/event/sec10/tech/full_papers/Watson.pdf Signed-off-by: Kumar Kartikeya Dwivedi --- include/linux/bpf_lsm.h | 21 +++ include/linux/bpf_types.h | 1 + include/uapi/linux/bpf.h | 39 ++++++ kernel/bpf/Makefile | 2 +- kernel/bpf/bpf_file_storage.c | 244 ++++++++++++++++++++++++++++++++++ kernel/bpf/bpf_lsm.c | 4 + kernel/bpf/syscall.c | 3 +- kernel/bpf/verifier.c | 10 ++ security/bpf/hooks.c | 2 + 9 files changed, 324 insertions(+), 2 deletions(-) create mode 100644 kernel/bpf/bpf_file_storage.c diff --git a/include/linux/bpf_lsm.h b/include/linux/bpf_lsm.h index 479c101546ad..5901a39cd5ac 100644 --- a/include/linux/bpf_lsm.h +++ b/include/linux/bpf_lsm.h @@ -42,6 +42,18 @@ extern const struct bpf_func_proto bpf_inode_storage_get_proto; extern const struct bpf_func_proto bpf_inode_storage_delete_proto; void bpf_inode_storage_free(struct inode *inode); +static inline struct bpf_storage_blob *bpf_file(const struct file *file) +{ + if (unlikely(!file->f_security)) + return NULL; + + return file->f_security + bpf_lsm_blob_sizes.lbs_file; +} + +extern const struct bpf_func_proto bpf_file_storage_get_proto; +extern const struct bpf_func_proto bpf_file_storage_delete_proto; +void bpf_file_storage_free(struct file *file); + #else /* !CONFIG_BPF_LSM */ static inline bool bpf_lsm_is_sleepable_hook(u32 btf_id) @@ -65,6 +77,15 @@ static inline void bpf_inode_storage_free(struct inode *inode) { } +static inline struct bpf_storage_blob *bpf_file(const struct file *file) +{ + return NULL; +} + +static inline void bpf_file_storage_free(struct file *file) +{ +} + #endif /* CONFIG_BPF_LSM */ #endif /* _LINUX_BPF_LSM_H */ diff --git a/include/linux/bpf_types.h b/include/linux/bpf_types.h index 9c81724e4b98..c68cc6d9e7da 100644 --- a/include/linux/bpf_types.h +++ b/include/linux/bpf_types.h @@ -107,6 +107,7 @@ BPF_MAP_TYPE(BPF_MAP_TYPE_DEVMAP_HASH, dev_map_hash_ops) BPF_MAP_TYPE(BPF_MAP_TYPE_SK_STORAGE, sk_storage_map_ops) #ifdef CONFIG_BPF_LSM BPF_MAP_TYPE(BPF_MAP_TYPE_INODE_STORAGE, inode_storage_map_ops) +BPF_MAP_TYPE(BPF_MAP_TYPE_FILE_STORAGE, file_storage_map_ops) #endif BPF_MAP_TYPE(BPF_MAP_TYPE_TASK_STORAGE, task_storage_map_ops) BPF_MAP_TYPE(BPF_MAP_TYPE_CPUMAP, cpu_map_ops) diff --git a/include/uapi/linux/bpf.h b/include/uapi/linux/bpf.h index 791f31dd0abe..62aa1ff2dcfb 100644 --- a/include/uapi/linux/bpf.h +++ b/include/uapi/linux/bpf.h @@ -906,6 +906,7 @@ enum bpf_map_type { BPF_MAP_TYPE_RINGBUF, BPF_MAP_TYPE_INODE_STORAGE, BPF_MAP_TYPE_TASK_STORAGE, + BPF_MAP_TYPE_FILE_STORAGE, }; /* Note that tracing related programs such as @@ -4877,6 +4878,42 @@ union bpf_attr { * Get the struct pt_regs associated with **task**. * Return * A pointer to struct pt_regs. + * + * void *bpf_file_storage_get(struct bpf_map *map, void *file, void *value, u64 flags) + * Description + * Get a bpf_local_storage from a *file*. + * + * Logically, it could be thought of as getting the value from + * a *map* with *file* as the **key**. From this + * perspective, the usage is not much different from + * **bpf_map_lookup_elem**\ (*map*, **&**\ *file*) except this + * helper enforces the key must be an file and the map must also + * be a **BPF_MAP_TYPE_FILE_STORAGE**. + * + * Underneath, the value is stored locally at *file* instead of + * the *map*. The *map* is used as the bpf-local-storage + * "type". The bpf-local-storage "type" (i.e. the *map*) is + * searched against all bpf_local_storage residing at *file*. + * + * An optional *flags* (**BPF_LOCAL_STORAGE_GET_F_CREATE**) can be + * used such that a new bpf_local_storage will be + * created if one does not exist. *value* can be used + * together with **BPF_LOCAL_STORAGE_GET_F_CREATE** to specify + * the initial value of a bpf_local_storage. If *value* is + * **NULL**, the new bpf_local_storage will be zero initialized. + * Return + * A bpf_local_storage pointer is returned on success. + * + * **NULL** if not found or there was an error in adding + * a new bpf_local_storage. + * + * int bpf_file_storage_delete(struct bpf_map *map, void *file) + * Description + * Delete a bpf_local_storage from a *file*. + * Return + * 0 on success. + * + * **-ENOENT** if the bpf_local_storage cannot be found. */ #define __BPF_FUNC_MAPPER(FN) \ FN(unspec), \ @@ -5055,6 +5092,8 @@ union bpf_attr { FN(get_func_ip), \ FN(get_attach_cookie), \ FN(task_pt_regs), \ + FN(file_storage_get), \ + FN(file_storage_delete), \ /* */ /* integer value in 'imm' field of BPF_CALL instruction selects which helper diff --git a/kernel/bpf/Makefile b/kernel/bpf/Makefile index 7f33098ca63f..98a18e402a0a 100644 --- a/kernel/bpf/Makefile +++ b/kernel/bpf/Makefile @@ -10,7 +10,7 @@ obj-$(CONFIG_BPF_SYSCALL) += syscall.o verifier.o inode.o helpers.o tnum.o bpf_i obj-$(CONFIG_BPF_SYSCALL) += hashtab.o arraymap.o percpu_freelist.o bpf_lru_list.o lpm_trie.o map_in_map.o obj-$(CONFIG_BPF_SYSCALL) += local_storage.o queue_stack_maps.o ringbuf.o obj-$(CONFIG_BPF_SYSCALL) += bpf_local_storage.o bpf_task_storage.o -obj-${CONFIG_BPF_LSM} += bpf_inode_storage.o +obj-${CONFIG_BPF_LSM} += bpf_inode_storage.o bpf_file_storage.o obj-$(CONFIG_BPF_SYSCALL) += disasm.o obj-$(CONFIG_BPF_JIT) += trampoline.o obj-$(CONFIG_BPF_SYSCALL) += btf.o diff --git a/kernel/bpf/bpf_file_storage.c b/kernel/bpf/bpf_file_storage.c new file mode 100644 index 000000000000..c826bc0405c4 --- /dev/null +++ b/kernel/bpf/bpf_file_storage.c @@ -0,0 +1,244 @@ +// SPDX-License-Identifier: GPL-2.0 + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +DEFINE_BPF_STORAGE_CACHE(file_cache); + +static struct bpf_local_storage __rcu **file_storage_ptr(void *owner) +{ + struct bpf_storage_blob *bsb; + struct file *file = owner; + + bsb = bpf_file(file); + if (!bsb) + return NULL; + return &bsb->storage; +} + +static struct bpf_local_storage_data * +file_storage_lookup(struct file *file, struct bpf_map *map, bool cacheit_lockit) +{ + struct bpf_local_storage *file_storage; + struct bpf_local_storage_map *smap; + struct bpf_storage_blob *bsb; + + bsb = bpf_file(file); + if (!bsb) + return NULL; + + file_storage = rcu_dereference(bsb->storage); + if (!file_storage) + return NULL; + + smap = (struct bpf_local_storage_map *)map; + return bpf_local_storage_lookup(file_storage, smap, cacheit_lockit); +} + +void bpf_file_storage_free(struct file *file) +{ + struct bpf_local_storage *local_storage; + struct bpf_local_storage_elem *selem; + bool free_file_storage = false; + struct bpf_storage_blob *bsb; + struct hlist_node *n; + + bsb = bpf_file(file); + if (!bsb) + return; + + rcu_read_lock(); + + local_storage = rcu_dereference(bsb->storage); + if (!local_storage) { + rcu_read_unlock(); + return; + } + + raw_spin_lock_bh(&local_storage->lock); + hlist_for_each_entry_safe(selem, n, &local_storage->list, snode) { + bpf_selem_unlink_map(selem); + free_file_storage = bpf_selem_unlink_storage_nolock(local_storage, + selem, false); + } + raw_spin_unlock_bh(&local_storage->lock); + rcu_read_unlock(); + + if (free_file_storage) + kfree_rcu(local_storage, rcu); +} + +static void *bpf_fd_file_storage_lookup_elem(struct bpf_map *map, void *key) +{ + struct bpf_local_storage_data *sdata; + struct file *file; + int fd; + + fd = *(int *)key; + file = fget_raw(fd); + if (!file) + return ERR_PTR(-EBADF); + + sdata = file_storage_lookup(file, map, true); + fput(file); + return sdata ? sdata->data : NULL; +} + +static int bpf_fd_file_storage_update_elem(struct bpf_map *map, void *key, + void *value, u64 map_flags) +{ + struct bpf_local_storage_data *sdata; + struct file *file; + int fd; + + fd = *(int *)key; + file = fget_raw(fd); + if (!file) + return -EBADF; + if (!file_storage_ptr(file)) { + fput(file); + return -EBADF; + } + + sdata = bpf_local_storage_update(file, + (struct bpf_local_storage_map *)map, + value, map_flags); + fput(file); + return PTR_ERR_OR_ZERO(sdata); +} + +static int file_storage_delete(struct file *file, struct bpf_map *map) +{ + struct bpf_local_storage_data *sdata; + + sdata = file_storage_lookup(file, map, false); + if (!sdata) + return -ENOENT; + + bpf_selem_unlink(SELEM(sdata)); + + return 0; +} + +static int bpf_fd_file_storage_delete_elem(struct bpf_map *map, void *key) +{ + struct file *file; + int fd, err; + + fd = *(int *)key; + file = fget_raw(fd); + if (!file) + return -EBADF; + + err = file_storage_delete(file, map); + fput(file); + return err; +} + +BPF_CALL_4(bpf_file_storage_get, struct bpf_map *, map, struct file *, file, + void *, value, u64, flags) +{ + struct bpf_local_storage_data *sdata; + + if (flags & ~(BPF_LOCAL_STORAGE_GET_F_CREATE)) + return (unsigned long)NULL; + + if (!file || !file_storage_ptr(file)) + return (unsigned long)NULL; + + sdata = file_storage_lookup(file, map, true); + if (sdata) + return (unsigned long)sdata->data; + + if (flags & BPF_LOCAL_STORAGE_GET_F_CREATE) { + sdata = bpf_local_storage_update( + file, (struct bpf_local_storage_map *)map, value, + BPF_NOEXIST); + return IS_ERR(sdata) ? (unsigned long)NULL : + (unsigned long)sdata->data; + } + + return (unsigned long)NULL; +} + +BPF_CALL_2(bpf_file_storage_delete, struct bpf_map *, map, struct file *, file) +{ + if (!file) + return -EINVAL; + + return file_storage_delete(file, map); +} + +static int notsupp_get_next_key(struct bpf_map *map, void *key, void *next_key) +{ + return -ENOTSUPP; +} + +static struct bpf_map *file_storage_map_alloc(union bpf_attr *attr) +{ + struct bpf_local_storage_map *smap; + + smap = bpf_local_storage_map_alloc(attr); + if (IS_ERR(smap)) + return ERR_CAST(smap); + + smap->cache_idx = bpf_local_storage_cache_idx_get(&file_cache); + return &smap->map; +} + +static void file_storage_map_free(struct bpf_map *map) +{ + struct bpf_local_storage_map *smap; + + smap = (struct bpf_local_storage_map *)map; + bpf_local_storage_cache_idx_free(&file_cache, smap->cache_idx); + bpf_local_storage_map_free(smap, NULL); +} + +static int file_storage_map_btf_id; + +const struct bpf_map_ops file_storage_map_ops = { + .map_meta_equal = bpf_map_meta_equal, + .map_alloc_check = bpf_local_storage_map_alloc_check, + .map_alloc = file_storage_map_alloc, + .map_free = file_storage_map_free, + .map_get_next_key = notsupp_get_next_key, + .map_lookup_elem = bpf_fd_file_storage_lookup_elem, + .map_update_elem = bpf_fd_file_storage_update_elem, + .map_delete_elem = bpf_fd_file_storage_delete_elem, + .map_check_btf = bpf_local_storage_map_check_btf, + .map_btf_name = "bpf_local_storage_map", + .map_btf_id = &file_storage_map_btf_id, + .map_owner_storage_ptr = file_storage_ptr, +}; + +BTF_ID_LIST_SINGLE(bpf_file_storage_btf_ids, struct, file) + +const struct bpf_func_proto bpf_file_storage_get_proto = { + .func = bpf_file_storage_get, + .gpl_only = false, + .ret_type = RET_PTR_TO_MAP_VALUE_OR_NULL, + .arg1_type = ARG_CONST_MAP_PTR, + .arg2_type = ARG_PTR_TO_BTF_ID, + .arg2_btf_id = &bpf_file_storage_btf_ids[0], + .arg3_type = ARG_PTR_TO_MAP_VALUE_OR_NULL, + .arg4_type = ARG_ANYTHING, +}; + +const struct bpf_func_proto bpf_file_storage_delete_proto = { + .func = bpf_file_storage_delete, + .gpl_only = false, + .ret_type = RET_INTEGER, + .arg1_type = ARG_CONST_MAP_PTR, + .arg2_type = ARG_PTR_TO_BTF_ID, + .arg2_btf_id = &bpf_file_storage_btf_ids[0], +}; diff --git a/kernel/bpf/bpf_lsm.c b/kernel/bpf/bpf_lsm.c index 06062370c3b8..48c2022fd958 100644 --- a/kernel/bpf/bpf_lsm.c +++ b/kernel/bpf/bpf_lsm.c @@ -121,6 +121,10 @@ bpf_lsm_func_proto(enum bpf_func_id func_id, const struct bpf_prog *prog) return &bpf_bprm_opts_set_proto; case BPF_FUNC_ima_inode_hash: return prog->aux->sleepable ? &bpf_ima_inode_hash_proto : NULL; + case BPF_FUNC_file_storage_get: + return &bpf_file_storage_get_proto; + case BPF_FUNC_file_storage_delete: + return &bpf_file_storage_delete_proto; default: return tracing_prog_func_proto(func_id, prog); } diff --git a/kernel/bpf/syscall.c b/kernel/bpf/syscall.c index 4e50c0bfdb7d..946a85945776 100644 --- a/kernel/bpf/syscall.c +++ b/kernel/bpf/syscall.c @@ -783,7 +783,8 @@ static int map_check_btf(struct bpf_map *map, const struct btf *btf, map->map_type != BPF_MAP_TYPE_CGROUP_STORAGE && map->map_type != BPF_MAP_TYPE_SK_STORAGE && map->map_type != BPF_MAP_TYPE_INODE_STORAGE && - map->map_type != BPF_MAP_TYPE_TASK_STORAGE) + map->map_type != BPF_MAP_TYPE_TASK_STORAGE && + map->map_type != BPF_MAP_TYPE_FILE_STORAGE) return -ENOTSUPP; if (map->spin_lock_off + sizeof(struct bpf_spin_lock) > map->value_size) { diff --git a/kernel/bpf/verifier.c b/kernel/bpf/verifier.c index 206c221453cf..c703d58681a8 100644 --- a/kernel/bpf/verifier.c +++ b/kernel/bpf/verifier.c @@ -5390,6 +5390,11 @@ static int check_map_func_compatibility(struct bpf_verifier_env *env, func_id != BPF_FUNC_task_storage_delete) goto error; break; + case BPF_MAP_TYPE_FILE_STORAGE: + if (func_id != BPF_FUNC_file_storage_get && + func_id != BPF_FUNC_file_storage_delete) + goto error; + break; default: break; } @@ -5473,6 +5478,11 @@ static int check_map_func_compatibility(struct bpf_verifier_env *env, if (map->map_type != BPF_MAP_TYPE_TASK_STORAGE) goto error; break; + case BPF_FUNC_file_storage_get: + case BPF_FUNC_file_storage_delete: + if (map->map_type != BPF_MAP_TYPE_FILE_STORAGE) + goto error; + break; default: break; } diff --git a/security/bpf/hooks.c b/security/bpf/hooks.c index e5971fa74fd7..faa70467db4d 100644 --- a/security/bpf/hooks.c +++ b/security/bpf/hooks.c @@ -13,6 +13,7 @@ static struct security_hook_list bpf_lsm_hooks[] __lsm_ro_after_init = { #undef LSM_HOOK LSM_HOOK_INIT(inode_free_security, bpf_inode_storage_free), LSM_HOOK_INIT(task_free, bpf_task_storage_free), + LSM_HOOK_INIT(file_free_security, bpf_file_storage_free), }; static int __init bpf_lsm_init(void) @@ -25,6 +26,7 @@ static int __init bpf_lsm_init(void) struct lsm_blob_sizes bpf_lsm_blob_sizes __lsm_ro_after_init = { .lbs_inode = sizeof(struct bpf_storage_blob), .lbs_task = sizeof(struct bpf_storage_blob), + .lbs_file = sizeof(struct bpf_storage_blob), }; DEFINE_LSM(bpf) = {