xref: /linux/kernel/bpf/cfg.c (revision 5a8cd539ac19f7a68e68e1d25ef9ca2ff55b8500)

// SPDX-License-Identifier: GPL-2.0-only
/* Copyright (c) 2026 Meta Platforms, Inc. and affiliates. */
#include <linux/bpf.h>
#include <linux/bpf_verifier.h>
#include <linux/filter.h>
#include <linux/sort.h>

#include "diagnostics.h"

#define verbose(env, fmt, args...) bpf_verifier_log_write(env, fmt, ##args)

/* non-recursive DFS pseudo code
 * 1  procedure DFS-iterative(G,v):
 * 2      label v as discovered
 * 3      let S be a stack
 * 4      S.push(v)
 * 5      while S is not empty
 * 6            t <- S.peek()
 * 7            if t is what we're looking for:
 * 8                return t
 * 9            for all edges e in G.adjacentEdges(t) do
 * 10               if edge e is already labelled
 * 11                   continue with the next edge
 * 12               w <- G.adjacentVertex(t,e)
 * 13               if vertex w is not discovered and not explored
 * 14                   label e as tree-edge
 * 15                   label w as discovered
 * 16                   S.push(w)
 * 17                   continue at 5
 * 18               else if vertex w is discovered
 * 19                   label e as back-edge
 * 20               else
 * 21                   // vertex w is explored
 * 22                   label e as forward- or cross-edge
 * 23           label t as explored
 * 24           S.pop()
 *
 * convention:
 * 0x10 - discovered
 * 0x11 - discovered and fall-through edge labelled
 * 0x12 - discovered and fall-through and branch edges labelled
 * 0x20 - explored
 */

enum {
	DISCOVERED = 0x10,
	EXPLORED = 0x20,
	FALLTHROUGH = 1,
	BRANCH = 2,
};

static void mark_subprog_changes_pkt_data(struct bpf_verifier_env *env, int off)
{
	struct bpf_subprog_info *subprog;

	subprog = bpf_find_containing_subprog(env, off);
	subprog->changes_pkt_data = true;
}

static void mark_subprog_might_sleep(struct bpf_verifier_env *env, int off)
{
	struct bpf_subprog_info *subprog;

	subprog = bpf_find_containing_subprog(env, off);
	subprog->might_sleep = true;
}

static void mark_subprog_might_throw(struct bpf_verifier_env *env, int off)
{
	struct bpf_subprog_info *subprog;

	subprog = bpf_find_containing_subprog(env, off);
	subprog->might_throw = true;
}

/* 't' is an index of a call-site.
 * 'w' is a callee entry point.
 * Eventually this function would be called when env->cfg.insn_state[w] == EXPLORED.
 * Rely on DFS traversal order and absence of recursive calls to guarantee that
 * callee's effect marks would be correct at that moment.
 */
static void merge_callee_effects(struct bpf_verifier_env *env, int t, int w)
{
	struct bpf_subprog_info *caller, *callee;

	caller = bpf_find_containing_subprog(env, t);
	callee = bpf_find_containing_subprog(env, w);
	caller->changes_pkt_data |= callee->changes_pkt_data;
	caller->might_sleep |= callee->might_sleep;
	caller->might_throw |= callee->might_throw;
}

enum {
	DONE_EXPLORING = 0,
	KEEP_EXPLORING = 1,
};

/* t, w, e - match pseudo-code above:
 * t - index of current instruction
 * w - next instruction
 * e - edge
 */
static int push_insn(int t, int w, int e, struct bpf_verifier_env *env)
{
	int *insn_stack = env->cfg.insn_stack;
	int *insn_state = env->cfg.insn_state;

	if (e == FALLTHROUGH && insn_state[t] >= (DISCOVERED | FALLTHROUGH))
		return DONE_EXPLORING;

	if (e == BRANCH && insn_state[t] >= (DISCOVERED | BRANCH))
		return DONE_EXPLORING;

	if (w < 0 || w >= env->prog->len) {
		verbose_linfo(env, t, "%d: ", t);
		verbose(env, "jump out of range from insn %d to %d\n", t, w);
		bpf_diag_program_structure(
			env, t, "jump out of range", "Keep branch targets inside the program.",
			"Instruction %d jumps to instruction %d, but the program only contains instructions 0 through %d.",
			t, w, env->prog->len - 1);
		return -EINVAL;
	}

	if (e == BRANCH) {
		/* mark branch target for state pruning */
		mark_prune_point(env, w);
		mark_jmp_point(env, w);
	}

	if (insn_state[w] == 0) {
		/* tree-edge */
		insn_state[t] = DISCOVERED | e;
		insn_state[w] = DISCOVERED;
		if (env->cfg.cur_stack >= env->prog->len)
			return -E2BIG;
		insn_stack[env->cfg.cur_stack++] = w;
		return KEEP_EXPLORING;
	} else if ((insn_state[w] & 0xF0) == DISCOVERED) {
		if (env->bpf_capable)
			return DONE_EXPLORING;
		verbose_linfo(env, t, "%d: ", t);
		verbose_linfo(env, w, "%d: ", w);
		verbose(env, "back-edge from insn %d to %d\n", t, w);
		bpf_diag_program_structure(
			env, t, "back-edge is not allowed",
			"Load with privileges that allow this back-edge, or rewrite the control flow so it does not branch backward.",
			"Instruction %d branches back to instruction %d. This program is being rejected without the privilege needed for this back-edge.",
			t, w);
		return -EINVAL;
	} else if (insn_state[w] == EXPLORED) {
		/* forward- or cross-edge */
		insn_state[t] = DISCOVERED | e;
	} else {
		verifier_bug(env, "insn state internal bug");
		return -EFAULT;
	}
	return DONE_EXPLORING;
}

static int visit_func_call_insn(int t, struct bpf_insn *insns,
				struct bpf_verifier_env *env,
				bool visit_callee)
{
	int ret, insn_sz;
	int w;

	insn_sz = bpf_is_ldimm64(&insns[t]) ? 2 : 1;
	ret = push_insn(t, t + insn_sz, FALLTHROUGH, env);
	if (ret)
		return ret;

	mark_prune_point(env, t + insn_sz);
	/* when we exit from subprog, we need to record non-linear history */
	mark_jmp_point(env, t + insn_sz);

	if (visit_callee) {
		w = t + insns[t].imm + 1;
		mark_prune_point(env, t);
		merge_callee_effects(env, t, w);
		ret = push_insn(t, w, BRANCH, env);
	}
	return ret;
}

struct bpf_iarray *bpf_iarray_realloc(struct bpf_iarray *old, size_t n_elem)
{
	size_t new_size = sizeof(struct bpf_iarray) + n_elem * sizeof(old->items[0]);
	struct bpf_iarray *new;

	new = kvrealloc(old, new_size, GFP_KERNEL_ACCOUNT);
	if (!new) {
		/* this is what callers always want, so simplify the call site */
		kvfree(old);
		return NULL;
	}

	new->cnt = n_elem;
	return new;
}

static int copy_insn_array(struct bpf_map *map, u32 start, u32 end, u32 *items)
{
	struct bpf_insn_array_value *value;
	u32 i;

	for (i = start; i <= end; i++) {
		value = map->ops->map_lookup_elem(map, &i);
		/*
		 * map_lookup_elem of an array map will never return an error,
		 * but not checking it makes some static analysers to worry
		 */
		if (IS_ERR(value))
			return PTR_ERR(value);
		else if (!value)
			return -EINVAL;
		items[i - start] = value->xlated_off;
	}
	return 0;
}

static int cmp_ptr_to_u32(const void *a, const void *b)
{
	return *(u32 *)a - *(u32 *)b;
}

static int sort_insn_array_uniq(u32 *items, int cnt)
{
	int unique = 1;
	int i;

	sort(items, cnt, sizeof(items[0]), cmp_ptr_to_u32, NULL);

	for (i = 1; i < cnt; i++)
		if (items[i] != items[unique - 1])
			items[unique++] = items[i];

	return unique;
}

/*
 * sort_unique({map[start], ..., map[end]}) into off
 */
int bpf_copy_insn_array_uniq(struct bpf_map *map, u32 start, u32 end, u32 *off)
{
	u32 n = end - start + 1;
	int err;

	err = copy_insn_array(map, start, end, off);
	if (err)
		return err;

	return sort_insn_array_uniq(off, n);
}

/*
 * Copy all unique offsets from the map
 */
static struct bpf_iarray *jt_from_map(struct bpf_map *map)
{
	struct bpf_iarray *jt;
	int err;
	int n;

	jt = bpf_iarray_realloc(NULL, map->max_entries);
	if (!jt)
		return ERR_PTR(-ENOMEM);

	n = bpf_copy_insn_array_uniq(map, 0, map->max_entries - 1, jt->items);
	if (n < 0) {
		err = n;
		goto err_free;
	}
	if (n == 0) {
		err = -EINVAL;
		goto err_free;
	}
	jt->cnt = n;
	return jt;

err_free:
	kvfree(jt);
	return ERR_PTR(err);
}

/*
 * Find and collect all maps which fit in the subprog. Return the result as one
 * combined jump table in jt->items (allocated with kvcalloc)
 */
static struct bpf_iarray *jt_from_subprog(struct bpf_verifier_env *env,
					  int subprog_start, int subprog_end)
{
	struct bpf_iarray *jt = NULL;
	struct bpf_map *map;
	struct bpf_iarray *jt_cur;
	int i;

	for (i = 0; i < env->insn_array_map_cnt; i++) {
		/*
		 * TODO (when needed): collect only jump tables, not static keys
		 * or maps for indirect calls
		 */
		map = env->insn_array_maps[i];

		jt_cur = jt_from_map(map);
		if (IS_ERR(jt_cur)) {
			kvfree(jt);
			return jt_cur;
		}

		/*
		 * This is enough to check one element. The full table is
		 * checked to fit inside the subprog later in create_jt()
		 */
		if (jt_cur->items[0] >= subprog_start && jt_cur->items[0] < subprog_end) {
			u32 old_cnt = jt ? jt->cnt : 0;
			jt = bpf_iarray_realloc(jt, old_cnt + jt_cur->cnt);
			if (!jt) {
				kvfree(jt_cur);
				return ERR_PTR(-ENOMEM);
			}
			memcpy(jt->items + old_cnt, jt_cur->items, jt_cur->cnt << 2);
		}

		kvfree(jt_cur);
	}

	if (!jt) {
		verbose(env, "no jump tables found for subprog starting at %u\n", subprog_start);
		bpf_diag_program_structure(
			env, subprog_start, "missing jump table",
			"Make sure subprograms containing gotox instructions are accompanied by jump tables referencing these subprograms.",
			"No jump table was found for the subprogram that starts at instruction %u.",
			subprog_start);
		return ERR_PTR(-EINVAL);
	}

	jt->cnt = sort_insn_array_uniq(jt->items, jt->cnt);
	return jt;
}

static struct bpf_iarray *
create_jt(int t, struct bpf_verifier_env *env)
{
	struct bpf_subprog_info *subprog;
	int subprog_start, subprog_end;
	struct bpf_iarray *jt;
	int i;

	subprog = bpf_find_containing_subprog(env, t);
	subprog_start = subprog->start;
	subprog_end = (subprog + 1)->start;
	jt = jt_from_subprog(env, subprog_start, subprog_end);
	if (IS_ERR(jt))
		return jt;

	/* Check that the every element of the jump table fits within the given subprogram */
	for (i = 0; i < jt->cnt; i++) {
		if (jt->items[i] < subprog_start || jt->items[i] >= subprog_end) {
			verbose(env, "jump table for insn %d points outside of the subprog [%u,%u]\n",
					t, subprog_start, subprog_end);
			bpf_diag_program_structure(
				env, t, "jump table target out of range",
				"Keep every jump-table target inside the same subprogram.",
				"The jump table for instruction %d points outside subprogram range [%u,%u).",
				t, subprog_start, subprog_end);
			kvfree(jt);
			return ERR_PTR(-EINVAL);
		}
	}

	return jt;
}

/* "conditional jump with N edges" */
static int visit_gotox_insn(int t, struct bpf_verifier_env *env)
{
	int *insn_stack = env->cfg.insn_stack;
	int *insn_state = env->cfg.insn_state;
	bool keep_exploring = false;
	struct bpf_iarray *jt;
	int i, w;

	jt = env->insn_aux_data[t].jt;
	if (!jt) {
		jt = create_jt(t, env);
		if (IS_ERR(jt))
			return PTR_ERR(jt);

		env->insn_aux_data[t].jt = jt;
	}

	mark_prune_point(env, t);
	for (i = 0; i < jt->cnt; i++) {
		w = jt->items[i];
		if (w < 0 || w >= env->prog->len) {
			verbose(env, "indirect jump out of range from insn %d to %d\n", t, w);
			bpf_diag_program_structure(
				env, t, "indirect jump out of range",
				"Keep indirect jump targets inside the program.",
				"Instruction %d can jump indirectly to instruction %d, but the program only contains instructions 0 through %d.",
				t, w, env->prog->len - 1);
			return -EINVAL;
		}

		mark_jmp_point(env, w);

		/* EXPLORED || DISCOVERED */
		if (insn_state[w])
			continue;

		if (env->cfg.cur_stack >= env->prog->len)
			return -E2BIG;

		insn_stack[env->cfg.cur_stack++] = w;
		insn_state[w] |= DISCOVERED;
		keep_exploring = true;
	}

	return keep_exploring ? KEEP_EXPLORING : DONE_EXPLORING;
}

/*
 * Instructions that can abnormally return from a subprog (tail_call
 * upon success, ld_{abs,ind} upon load failure) have a hidden exit
 * that the verifier must account for.
 */
static int visit_abnormal_return_insn(struct bpf_verifier_env *env, int t)
{
	struct bpf_subprog_info *subprog;
	struct bpf_iarray *jt;

	if (env->insn_aux_data[t].jt)
		return 0;

	jt = bpf_iarray_realloc(NULL, 2);
	if (!jt)
		return -ENOMEM;

	subprog = bpf_find_containing_subprog(env, t);
	jt->items[0] = t + 1;
	jt->items[1] = subprog->exit_idx;
	env->insn_aux_data[t].jt = jt;
	return 0;
}

/* Visits the instruction at index t and returns one of the following:
 *  < 0 - an error occurred
 *  DONE_EXPLORING - the instruction was fully explored
 *  KEEP_EXPLORING - there is still work to be done before it is fully explored
 */
static int visit_insn(int t, struct bpf_verifier_env *env)
{
	struct bpf_insn *insns = env->prog->insnsi, *insn = &insns[t];
	int ret, off, insn_sz;

	if (bpf_pseudo_func(insn))
		return visit_func_call_insn(t, insns, env, true);

	/* All non-branch instructions have a single fall-through edge. */
	if (BPF_CLASS(insn->code) != BPF_JMP &&
	    BPF_CLASS(insn->code) != BPF_JMP32) {
		if (BPF_CLASS(insn->code) == BPF_LD &&
		    (BPF_MODE(insn->code) == BPF_ABS ||
		     BPF_MODE(insn->code) == BPF_IND)) {
			ret = visit_abnormal_return_insn(env, t);
			if (ret)
				return ret;
		}
		insn_sz = bpf_is_ldimm64(insn) ? 2 : 1;
		return push_insn(t, t + insn_sz, FALLTHROUGH, env);
	}

	switch (BPF_OP(insn->code)) {
	case BPF_EXIT:
		return DONE_EXPLORING;

	case BPF_CALL:
		if (bpf_is_async_callback_calling_insn(insn))
			/* Mark this call insn as a prune point to trigger
			 * is_state_visited() check before call itself is
			 * processed by __check_func_call(). Otherwise new
			 * async state will be pushed for further exploration.
			 */
			mark_prune_point(env, t);
		/* For functions that invoke callbacks it is not known how many times
		 * callback would be called. Verifier models callback calling functions
		 * by repeatedly visiting callback bodies and returning to origin call
		 * instruction.
		 * In order to stop such iteration verifier needs to identify when a
		 * state identical some state from a previous iteration is reached.
		 * Check below forces creation of checkpoint before callback calling
		 * instruction to allow search for such identical states.
		 */
		if (bpf_is_sync_callback_calling_insn(insn)) {
			mark_calls_callback(env, t);
			mark_force_checkpoint(env, t);
			mark_prune_point(env, t);
			mark_jmp_point(env, t);
		}
		if (bpf_helper_call(insn)) {
			const struct bpf_func_proto *fp;

			ret = bpf_get_helper_proto(env, insn->imm, &fp);
			/* If called in a non-sleepable context program will be
			 * rejected anyway, so we should end up with precise
			 * sleepable marks on subprogs, except for dead code
			 * elimination.
			 */
			if (ret == 0 && fp->might_sleep)
				mark_subprog_might_sleep(env, t);
			if (bpf_helper_changes_pkt_data(insn->imm))
				mark_subprog_changes_pkt_data(env, t);
			if (insn->imm == BPF_FUNC_tail_call) {
				ret = visit_abnormal_return_insn(env, t);
				if (ret)
					return ret;
			}
		} else if (insn->src_reg == BPF_PSEUDO_KFUNC_CALL) {
			struct bpf_call_arg_meta meta;

			ret = bpf_fetch_kfunc_arg_meta(env, insn->imm, insn->off, &meta);
			if (ret == 0 && bpf_is_iter_next_kfunc(&meta)) {
				mark_prune_point(env, t);
				/* Checking and saving state checkpoints at iter_next() call
				 * is crucial for fast convergence of open-coded iterator loop
				 * logic, so we need to force it. If we don't do that,
				 * is_state_visited() might skip saving a checkpoint, causing
				 * unnecessarily long sequence of not checkpointed
				 * instructions and jumps, leading to exhaustion of jump
				 * history buffer, and potentially other undesired outcomes.
				 * It is expected that with correct open-coded iterators
				 * convergence will happen quickly, so we don't run a risk of
				 * exhausting memory.
				 */
				mark_force_checkpoint(env, t);
			}
			/* Same as helpers, if called in a non-sleepable context
			 * program will be rejected anyway, so we should end up
			 * with precise sleepable marks on subprogs, except for
			 * dead code elimination.
			 */
			if (ret == 0 && bpf_is_kfunc_sleepable(&meta))
				mark_subprog_might_sleep(env, t);
			if (ret == 0 && bpf_is_kfunc_pkt_changing(&meta))
				mark_subprog_changes_pkt_data(env, t);
			if (ret == 0 && bpf_is_throw_kfunc(insn))
				mark_subprog_might_throw(env, t);
		}
		return visit_func_call_insn(t, insns, env, insn->src_reg == BPF_PSEUDO_CALL);

	case BPF_JA:
		if (BPF_SRC(insn->code) == BPF_X)
			return visit_gotox_insn(t, env);

		if (BPF_CLASS(insn->code) == BPF_JMP)
			off = insn->off;
		else
			off = insn->imm;

		/* unconditional jump with single edge */
		ret = push_insn(t, t + off + 1, FALLTHROUGH, env);
		if (ret)
			return ret;

		mark_prune_point(env, t + off + 1);
		mark_jmp_point(env, t + off + 1);

		return ret;

	default:
		/* conditional jump with two edges */
		mark_prune_point(env, t);
		if (bpf_is_may_goto_insn(insn))
			mark_force_checkpoint(env, t);

		ret = push_insn(t, t + 1, FALLTHROUGH, env);
		if (ret)
			return ret;

		return push_insn(t, t + insn->off + 1, BRANCH, env);
	}
}

/* non-recursive depth-first-search to detect loops in BPF program
 * loop == back-edge in directed graph
 */
int bpf_check_cfg(struct bpf_verifier_env *env)
{
	int insn_cnt = env->prog->len;
	int *insn_stack, *insn_state;
	int ex_insn_beg, i, ret = 0;

	insn_state = env->cfg.insn_state = kvzalloc_objs(int, insn_cnt,
							 GFP_KERNEL_ACCOUNT);
	if (!insn_state)
		return -ENOMEM;

	insn_stack = env->cfg.insn_stack = kvzalloc_objs(int, insn_cnt,
							 GFP_KERNEL_ACCOUNT);
	if (!insn_stack) {
		kvfree(insn_state);
		return -ENOMEM;
	}

	ex_insn_beg = env->exception_callback_subprog
		      ? env->subprog_info[env->exception_callback_subprog].start
		      : 0;

	insn_state[0] = DISCOVERED; /* mark 1st insn as discovered */
	insn_stack[0] = 0; /* 0 is the first instruction */
	env->cfg.cur_stack = 1;

walk_cfg:
	while (env->cfg.cur_stack > 0) {
		int t = insn_stack[env->cfg.cur_stack - 1];

		ret = visit_insn(t, env);
		switch (ret) {
		case DONE_EXPLORING:
			insn_state[t] = EXPLORED;
			env->cfg.cur_stack--;
			break;
		case KEEP_EXPLORING:
			break;
		default:
			if (ret > 0) {
				verifier_bug(env, "visit_insn internal bug");
				ret = -EFAULT;
			}
			goto err_free;
		}
	}

	if (env->cfg.cur_stack < 0) {
		verifier_bug(env, "pop stack internal bug");
		ret = -EFAULT;
		goto err_free;
	}

	if (ex_insn_beg && insn_state[ex_insn_beg] != EXPLORED) {
		insn_state[ex_insn_beg] = DISCOVERED;
		insn_stack[0] = ex_insn_beg;
		env->cfg.cur_stack = 1;
		goto walk_cfg;
	}

	for (i = 0; i < insn_cnt; i++) {
		struct bpf_insn *insn = &env->prog->insnsi[i];

		if (insn_state[i] != EXPLORED) {
			verbose(env, "unreachable insn %d\n", i);
			bpf_diag_program_structure(
				env, i, "unreachable instruction",
				"Remove the unreachable instruction or add valid control flow that reaches it.",
				"Instruction %d is not reachable from the program entry point.", i);
			ret = -EINVAL;
			goto err_free;
		}
		if (bpf_is_ldimm64(insn)) {
			if (insn_state[i + 1] != 0) {
				verbose(env, "jump into the middle of ldimm64 insn %d\n", i);
				bpf_diag_program_structure(
					env, i, "jump into ldimm64 immediate",
					"Target the first instruction of the ldimm64 pair, or restructure the jump target.",
					"Control flow reaches the second half of the ldimm64 instruction pair that starts at instruction %d.",
					i);
				ret = -EINVAL;
				goto err_free;
			}
			i++; /* skip second half of ldimm64 */
		}
	}
	ret = 0; /* cfg looks good */
	env->prog->aux->changes_pkt_data = env->subprog_info[0].changes_pkt_data;
	env->prog->aux->might_sleep = env->subprog_info[0].might_sleep;

err_free:
	kvfree(insn_state);
	kvfree(insn_stack);
	env->cfg.insn_state = env->cfg.insn_stack = NULL;
	return ret;
}

/*
 * For each subprogram 'i' fill array env->cfg.insn_subprogram sub-range
 * [env->subprog_info[i].postorder_start, env->subprog_info[i+1].postorder_start)
 * with indices of 'i' instructions in postorder.
 */
int bpf_compute_postorder(struct bpf_verifier_env *env)
{
	u32 cur_postorder, i, top, stack_sz, s;
	int *stack = NULL, *postorder = NULL, *state = NULL;
	struct bpf_iarray *succ;

	postorder = kvzalloc_objs(int, env->prog->len, GFP_KERNEL_ACCOUNT);
	state = kvzalloc_objs(int, env->prog->len, GFP_KERNEL_ACCOUNT);
	stack = kvzalloc_objs(int, env->prog->len, GFP_KERNEL_ACCOUNT);
	if (!postorder || !state || !stack) {
		kvfree(postorder);
		kvfree(state);
		kvfree(stack);
		return -ENOMEM;
	}
	cur_postorder = 0;
	for (i = 0; i < env->subprog_cnt; i++) {
		env->subprog_info[i].postorder_start = cur_postorder;
		stack[0] = env->subprog_info[i].start;
		stack_sz = 1;
		do {
			top = stack[stack_sz - 1];
			state[top] |= DISCOVERED;
			if (state[top] & EXPLORED) {
				postorder[cur_postorder++] = top;
				stack_sz--;
				continue;
			}
			succ = bpf_insn_successors(env, top);
			for (s = 0; s < succ->cnt; ++s) {
				if (!state[succ->items[s]]) {
					stack[stack_sz++] = succ->items[s];
					state[succ->items[s]] |= DISCOVERED;
				}
			}
			state[top] |= EXPLORED;
		} while (stack_sz);
	}
	env->subprog_info[i].postorder_start = cur_postorder;
	env->cfg.insn_postorder = postorder;
	env->cfg.cur_postorder = cur_postorder;
	kvfree(stack);
	kvfree(state);
	return 0;
}

/*
 * Compute strongly connected components (SCCs) on the CFG.
 * Assign an SCC number to each instruction, recorded in env->insn_aux[*].scc.
 * If instruction is a sole member of its SCC and there are no self edges,
 * assign it SCC number of zero.
 * Uses a non-recursive adaptation of Tarjan's algorithm for SCC computation.
 */
int bpf_compute_scc(struct bpf_verifier_env *env)
{
	const u32 NOT_ON_STACK = U32_MAX;

	struct bpf_insn_aux_data *aux = env->insn_aux_data;
	const u32 insn_cnt = env->prog->len;
	int stack_sz, dfs_sz, err = 0;
	u32 *stack, *pre, *low, *dfs;
	u32 i, j, t, w;
	u32 next_preorder_num;
	u32 next_scc_id;
	bool assign_scc;
	struct bpf_iarray *succ;

	next_preorder_num = 1;
	next_scc_id = 1;
	/*
	 * - 'stack' accumulates vertices in DFS order, see invariant comment below;
	 * - 'pre[t] == p' => preorder number of vertex 't' is 'p';
	 * - 'low[t] == n' => smallest preorder number of the vertex reachable from 't' is 'n';
	 * - 'dfs' DFS traversal stack, used to emulate explicit recursion.
	 */
	stack = kvcalloc(insn_cnt, sizeof(int), GFP_KERNEL_ACCOUNT);
	pre = kvcalloc(insn_cnt, sizeof(int), GFP_KERNEL_ACCOUNT);
	low = kvcalloc(insn_cnt, sizeof(int), GFP_KERNEL_ACCOUNT);
	dfs = kvcalloc(insn_cnt, sizeof(*dfs), GFP_KERNEL_ACCOUNT);
	if (!stack || !pre || !low || !dfs) {
		err = -ENOMEM;
		goto exit;
	}
	/*
	 * References:
	 * [1] R. Tarjan "Depth-First Search and Linear Graph Algorithms"
	 * [2] D. J. Pearce "A Space-Efficient Algorithm for Finding Strongly Connected Components"
	 *
	 * The algorithm maintains the following invariant:
	 * - suppose there is a path 'u' ~> 'v', such that 'pre[v] < pre[u]';
	 * - then, vertex 'u' remains on stack while vertex 'v' is on stack.
	 *
	 * Consequently:
	 * - If 'low[v] < pre[v]', there is a path from 'v' to some vertex 'u',
	 *   such that 'pre[u] == low[v]'; vertex 'u' is currently on the stack,
	 *   and thus there is an SCC (loop) containing both 'u' and 'v'.
	 * - If 'low[v] == pre[v]', loops containing 'v' have been explored,
	 *   and 'v' can be considered the root of some SCC.
	 *
	 * Here is a pseudo-code for an explicitly recursive version of the algorithm:
	 *
	 *    NOT_ON_STACK = insn_cnt + 1
	 *    pre = [0] * insn_cnt
	 *    low = [0] * insn_cnt
	 *    scc = [0] * insn_cnt
	 *    stack = []
	 *
	 *    next_preorder_num = 1
	 *    next_scc_id = 1
	 *
	 *    def recur(w):
	 *        nonlocal next_preorder_num
	 *        nonlocal next_scc_id
	 *
	 *        pre[w] = next_preorder_num
	 *        low[w] = next_preorder_num
	 *        next_preorder_num += 1
	 *        stack.append(w)
	 *        for s in successors(w):
	 *            # Note: for classic algorithm the block below should look as:
	 *            #
	 *            # if pre[s] == 0:
	 *            #     recur(s)
	 *            #     low[w] = min(low[w], low[s])
	 *            # elif low[s] != NOT_ON_STACK:
	 *            #     low[w] = min(low[w], pre[s])
	 *            #
	 *            # But replacing both 'min' instructions with 'low[w] = min(low[w], low[s])'
	 *            # does not break the invariant and makes iterative version of the algorithm
	 *            # simpler. See 'Algorithm #3' from [2].
	 *
	 *            # 's' not yet visited
	 *            if pre[s] == 0:
	 *                recur(s)
	 *            # if 's' is on stack, pick lowest reachable preorder number from it;
	 *            # if 's' is not on stack 'low[s] == NOT_ON_STACK > low[w]',
	 *            # so 'min' would be a noop.
	 *            low[w] = min(low[w], low[s])
	 *
	 *        if low[w] == pre[w]:
	 *            # 'w' is the root of an SCC, pop all vertices
	 *            # below 'w' on stack and assign same SCC to them.
	 *            while True:
	 *                t = stack.pop()
	 *                low[t] = NOT_ON_STACK
	 *                scc[t] = next_scc_id
	 *                if t == w:
	 *                    break
	 *            next_scc_id += 1
	 *
	 *    for i in range(0, insn_cnt):
	 *        if pre[i] == 0:
	 *            recur(i)
	 *
	 * Below implementation replaces explicit recursion with array 'dfs'.
	 */
	for (i = 0; i < insn_cnt; i++) {
		if (pre[i])
			continue;
		stack_sz = 0;
		dfs_sz = 1;
		dfs[0] = i;
dfs_continue:
		while (dfs_sz) {
			w = dfs[dfs_sz - 1];
			if (pre[w] == 0) {
				low[w] = next_preorder_num;
				pre[w] = next_preorder_num;
				next_preorder_num++;
				stack[stack_sz++] = w;
			}
			/* Visit 'w' successors */
			succ = bpf_insn_successors(env, w);
			for (j = 0; j < succ->cnt; ++j) {
				if (pre[succ->items[j]]) {
					low[w] = min(low[w], low[succ->items[j]]);
				} else {
					dfs[dfs_sz++] = succ->items[j];
					goto dfs_continue;
				}
			}
			/*
			 * Preserve the invariant: if some vertex above in the stack
			 * is reachable from 'w', keep 'w' on the stack.
			 */
			if (low[w] < pre[w]) {
				dfs_sz--;
				goto dfs_continue;
			}
			/*
			 * Assign SCC number only if component has two or more elements,
			 * or if component has a self reference, or if instruction is a
			 * callback calling function (implicit loop).
			 */
			assign_scc = stack[stack_sz - 1] != w;	/* two or more elements? */
			for (j = 0; j < succ->cnt; ++j) {	/* self reference? */
				if (succ->items[j] == w) {
					assign_scc = true;
					break;
				}
			}
			if (bpf_calls_callback(env, w)) /* implicit loop? */
				assign_scc = true;
			/* Pop component elements from stack */
			do {
				t = stack[--stack_sz];
				low[t] = NOT_ON_STACK;
				if (assign_scc)
					aux[t].scc = next_scc_id;
			} while (t != w);
			if (assign_scc)
				next_scc_id++;
			dfs_sz--;
		}
	}
	env->scc_info = kvzalloc_objs(*env->scc_info, next_scc_id,
				      GFP_KERNEL_ACCOUNT);
	if (!env->scc_info) {
		err = -ENOMEM;
		goto exit;
	}
	env->scc_cnt = next_scc_id;
exit:
	kvfree(stack);
	kvfree(pre);
	kvfree(low);
	kvfree(dfs);
	return err;
}