bpf: improve the general precision of tnum_mul

This commit addresses a challenge explained in an open question ("How
can we incorporate correlation in unknown bits across partial
products?") left by Harishankar et al. in their paper:
https://arxiv.org/abs/2105.05398

When LSB(a) is uncertain, we know for sure that it is either 0 or 1,
from which we could find two possible partial products and take a
union. Experiment shows that applying this technique in long
multiplication improves the precision in a significant number of cases
(at the cost of losing precision in a relatively lower number of
cases).

This commit also removes the value-mask decomposition technique
employed by Harishankar et al., as its direct incorporation did not
result in any improvements for the new algorithm.

Signed-off-by: Nandakumar Edamana <[email protected]>

Author: nandedamana

include/linux/tnum.h

@@ -57,6 +57,9 @@ bool tnum_overlap(struct tnum a, struct tnum b);
 /* Return a tnum representing numbers satisfying both @a and @b */
 struct tnum tnum_intersect(struct tnum a, struct tnum b);
 
+/* Returns a tnum representing numbers satisfying either @a or @b */
+struct tnum tnum_union(struct tnum t1, struct tnum t2);
+
 /* Return @a with all but the lowest @size bytes cleared */
 struct tnum tnum_cast(struct tnum a, u8 size);
 

kernel/bpf/tnum.c

@@ -116,31 +116,39 @@ struct tnum tnum_xor(struct tnum a, struct tnum b)
 	return TNUM(v & ~mu, mu);
 }
 
-/* Generate partial products by multiplying each bit in the multiplier (tnum a)
- * with the multiplicand (tnum b), and add the partial products after
- * appropriately bit-shifting them. Instead of directly performing tnum addition
- * on the generated partial products, equivalenty, decompose each partial
- * product into two tnums, consisting of the value-sum (acc_v) and the
- * mask-sum (acc_m) and then perform tnum addition on them. The following paper
- * explains the algorithm in more detail: https://arxiv.org/abs/2105.05398.
+/* Perform long multiplication, iterating through the trits in a.
+ * Inside `else if (a.mask & 1)`, instead of simply multiplying b with LSB(a)'s
+ * uncertainty and accumulating directly, we find two possible partial products
+ * (one for LSB(a) = 0 and another for LSB(a) = 1), and add their union to the
+ * accumulator. This addresses an issue pointed out in an open question ("How
+ * can we incorporate correlation in unknown bits across partial products?")
+ * left by Harishankar et al. (https://arxiv.org/abs/2105.05398), improving
+ * the general precision significantly.
  */
 struct tnum tnum_mul(struct tnum a, struct tnum b)
 {
-	u64 acc_v = a.value * b.value;
-	struct tnum acc_m = TNUM(0, 0);
+	struct tnum acc = TNUM(0, 0);
 
 	while (a.value || a.mask) {
 		/* LSB of tnum a is a certain 1 */
 		if (a.value & 1)
-			acc_m = tnum_add(acc_m, TNUM(0, b.mask));
+			acc = tnum_add(acc, b);
 		/* LSB of tnum a is uncertain */
-		else if (a.mask & 1)
-			acc_m = tnum_add(acc_m, TNUM(0, b.value | b.mask));
+		else if (a.mask & 1) {
+			/* acc = tnum_union(acc_0, acc_1), where acc_0 and
+			 * acc_1 are partial accumulators for cases
+			 * LSB(a) = certain 0 and LSB(a) = certain 1.
+			 * acc_0 = acc + 0 * b = acc.
+			 * acc_1 = acc + 1 * b = tnum_add(acc, b).
+			 */
+
+			acc = tnum_union(acc, tnum_add(acc, b));
+		}
 		/* Note: no case for LSB is certain 0 */
 		a = tnum_rshift(a, 1);
 		b = tnum_lshift(b, 1);
 	}
-	return tnum_add(TNUM(acc_v, 0), acc_m);
+	return acc;
 }
 
 bool tnum_overlap(struct tnum a, struct tnum b)
@@ -163,6 +171,19 @@ struct tnum tnum_intersect(struct tnum a, struct tnum b)
 	return TNUM(v & ~mu, mu);
 }
 
+/* Returns a tnum with the uncertainty from both a and b, and in addition, new
+ * uncertainty at any position that a and b disagree. This represents a
+ * superset of the union of the concrete sets of both a and b. Despite the
+ * overapproximation, it is optimal.
+ */
+struct tnum tnum_union(struct tnum a, struct tnum b)
+{
+	u64 v = a.value & b.value;
+	u64 mu = (a.value ^ b.value) | a.mask | b.mask;
+
+	return TNUM(v & ~mu, mu);
+}
+
 struct tnum tnum_cast(struct tnum a, u8 size)
 {
 	a.value &= (1ULL << (size * 8)) - 1;

tools/testing/selftests/bpf/prog_tests/verifier.c

@@ -59,6 +59,7 @@
 #include "verifier_meta_access.skel.h"
 #include "verifier_movsx.skel.h"
 #include "verifier_mtu.skel.h"
+#include "verifier_mul.skel.h"
 #include "verifier_netfilter_ctx.skel.h"
 #include "verifier_netfilter_retcode.skel.h"
 #include "verifier_bpf_fastcall.skel.h"
@@ -194,6 +195,7 @@ void test_verifier_may_goto_1(void)           { RUN(verifier_may_goto_1); }
 void test_verifier_may_goto_2(void)           { RUN(verifier_may_goto_2); }
 void test_verifier_meta_access(void)          { RUN(verifier_meta_access); }
 void test_verifier_movsx(void)                 { RUN(verifier_movsx); }
+void test_verifier_mul(void)                  { RUN(verifier_mul); }
 void test_verifier_netfilter_ctx(void)        { RUN(verifier_netfilter_ctx); }
 void test_verifier_netfilter_retcode(void)    { RUN(verifier_netfilter_retcode); }
 void test_verifier_bpf_fastcall(void)         { RUN(verifier_bpf_fastcall); }

tools/testing/selftests/bpf/progs/verifier_mul.c

@@ -0,0 +1,74 @@
+// SPDX-License-Identifier: GPL-2.0
+/* Copyright (c) 2025 Nandakumar Edamana */
+#include <linux/bpf.h>
+#include <bpf/bpf_helpers.h>
+#include <bpf/bpf_tracing.h>
+#include "bpf_misc.h"
+
+/* The programs here are meant to test the abstract multiplication
+ * technique(s) used by the verifier. Using assembly to prevent
+ * compiler optimizations.
+ */
+
+SEC("fentry/bpf_fentry_test1")
+void BPF_PROG(mul_0, int x)
+{
+	asm volatile (" \
+	call %[bpf_get_prandom_u32];\
+	r0 *= 0;\
+	if r0 != 0 goto l0_%=;\
+	r0 = 0;\
+	goto l1_%=;\
+l0_%=:\
+	r0 = 1;\
+l1_%=:\
+" :
+	: __imm(bpf_get_prandom_u32)
+	: __clobber_all);
+}
+
+SEC("fentry/bpf_fentry_test1")
+__failure __msg("At program exit the register R0 has smin=1 smax=1 should have been in [0, 0]")
+void BPF_PROG(mul_uncertain, int x)
+{
+	asm volatile (" \
+	call %[bpf_get_prandom_u32];\
+	r0 *= 0x3;\
+	if r0 != 0 goto l0_%=;\
+	r0 = 0;\
+	goto l1_%=;\
+l0_%=:\
+	r0 = 1;\
+l1_%=:\
+" :
+	: __imm(bpf_get_prandom_u32)
+	: __clobber_all);
+}
+
+SEC("fentry/bpf_fentry_test1")
+void BPF_PROG(mul_precise, int x)
+{
+	/* First, force the verifier to be uncertain about the value:
+	 *     unsigned int a = (bpf_get_prandom_u32() & 0x2) | 0x1;
+	 *
+	 * Assuming the verifier is using tnum, a must be tnum{.v=0x1, .m=0x2}.
+	 * Then a * 0x3 would be m0m1 (m for uncertain). Added imprecision would
+	 * cause the following to fail, because the required return value is 0.:
+	 *     return (a * 0x3) & 0x4);
+	 */
+
+	asm volatile ("call %[bpf_get_prandom_u32];\
+	r0 &= 0x2;\
+	r0 |= 0x1;\
+	r0 *= 0x3;\
+	r0 &= 0x4;\
+	if r0 != 0 goto l0_%=;\
+	r0 = 0;\
+	goto l1_%=;\
+l0_%=:\
+	r0 = 1;\
+l1_%=:\
+" :
+	: __imm(bpf_get_prandom_u32)
+	: __clobber_all);
+}