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bpf, verifier: Improve precision of BPF_MUL
This patch improves (or maintains) the precision of register value tracking
in BPF_MUL across all possible inputs. It also simplifies
scalar32_min_max_mul() and scalar_min_max_mul().
As it stands,BPF_MUL is composed of three functions:
case BPF_MUL:
tnum_mul();
scalar32_min_max_mul();
scalar_min_max_mul();
The current implementation of scalar_min_max_mul() restricts the u64 input
ranges of dst_reg and src_reg to be within [0, U32_MAX]:
/* Both values are positive, so we can work with unsigned and
* copy the result to signed (unless it exceeds S64_MAX).
*/
if (umax_val > U32_MAX || dst_reg->umax_value > U32_MAX) {
/* Potential overflow, we know nothing */
__mark_reg64_unbounded(dst_reg);
return;
}
This restriction is done to avoid unsigned overflow, which could otherwise
wrap the result around 0, and leave an unsound output where umin > umax. We
also observe that limiting these u64 input ranges to [0, U32_MAX] leads to
a loss of precision. Consider the case where the u64 bounds of dst_reg are
[0, 2^34] and the u64 bounds of src_reg are [0, 2^2]. While the
multiplication of these two bounds doesn't overflow and is sound [0, 2^36],
the current scalar_min_max_mul() would set the entire register state to
unbounded.
The key idea of our patch is that if there’s no possibility of overflow, we
can multiply the unsigned bounds; otherwise, we set the 64-bit bounds to
[0, U64_MAX], marking them as unbounded.
if (check_mul_overflow(*dst_umax, src_reg->umax_value, dst_umax) ||
(check_mul_overflow(*dst_umin, src_reg->umin_value, dst_umin))) {
/* Overflow possible, we know nothing */
dst_reg->umin_value = 0;
dst_reg->umax_value = U64_MAX;
}
...
Now, to update the signed bounds based on the unsigned bounds, we need to
ensure that the unsigned bounds don't cross the signed boundary (i.e., if
((s64)reg->umin_value <= (s64)reg->umax_value)). We observe that this is
done anyway by __reg_deduce_bounds later, so we can just set signed bounds
to unbounded [S64_MIN, S64_MAX]. Deferring the assignment of s64 bounds to
reg_bounds_sync removes the current redundancy in scalar_min_max_mul(),
which currently sets the s64 bounds based on the u64 bounds only in the
case where umin <= umax <= 2^(63)-1.
Below, we provide an example BPF program (below) that exhibits the
imprecision in the current BPF_MUL, where the outputs are all unbounded. In
contrast, the updated BPF_MUL produces a bounded register state:
BPF_LD_IMM64(BPF_REG_1, 11),
BPF_LD_IMM64(BPF_REG_2, 4503599627370624),
BPF_ALU64_IMM(BPF_NEG, BPF_REG_2, 0),
BPF_ALU64_IMM(BPF_NEG, BPF_REG_2, 0),
BPF_ALU64_REG(BPF_AND, BPF_REG_1, BPF_REG_2),
BPF_LD_IMM64(BPF_REG_3, 809591906117232263),
BPF_ALU64_REG(BPF_MUL, BPF_REG_3, BPF_REG_1),
BPF_MOV64_IMM(BPF_REG_0, 1),
BPF_EXIT_INSN(),
Verifier log using the old BPF_MUL:
func#0 @0
0: R1=ctx() R10=fp0
0: (18) r1 = 0xb ; R1_w=11
2: (18) r2 = 0x10000000000080 ; R2_w=0x10000000000080
4: (87) r2 = -r2 ; R2_w=scalar()
5: (87) r2 = -r2 ; R2_w=scalar()
6: (5f) r1 &= r2 ; R1_w=scalar(smin=smin32=0,smax=umax=smax32=umax32=11,var_off=(0x0; 0xb)) R2_w=scalar()
7: (18) r3 = 0xb3c3f8c99262687 ; R3_w=0xb3c3f8c99262687
9: (2f) r3 *= r1 ; R1_w=scalar(smin=smin32=0,smax=umax=smax32=umax32=11,var_off=(0x0; 0xb)) R3_w=scalar()
...
Verifier using the new updated BPF_MUL (more precise bounds at label 9)
func#0 @0
0: R1=ctx() R10=fp0
0: (18) r1 = 0xb ; R1_w=11
2: (18) r2 = 0x10000000000080 ; R2_w=0x10000000000080
4: (87) r2 = -r2 ; R2_w=scalar()
5: (87) r2 = -r2 ; R2_w=scalar()
6: (5f) r1 &= r2 ; R1_w=scalar(smin=smin32=0,smax=umax=smax32=umax32=11,var_off=(0x0; 0xb)) R2_w=scalar()
7: (18) r3 = 0xb3c3f8c99262687 ; R3_w=0xb3c3f8c99262687
9: (2f) r3 *= r1 ; R1_w=scalar(smin=smin32=0,smax=umax=smax32=umax32=11,var_off=(0x0; 0xb)) R3_w=scalar(smin=0,smax=umax=0x7b96bb0a94a3a7cd,var_off=(0x0; 0x7fffffffffffffff))
...
Finally, we proved the soundness of the new scalar_min_max_mul() and
scalar32_min_max_mul() functions. Typically, multiplication operations are
expensive to check with bitvector-based solvers. We were able to prove the
soundness of these functions using Non-Linear Integer Arithmetic (NIA)
theory. Additionally, using Agni [2,3], we obtained the encodings for
scalar32_min_max_mul() and scalar_min_max_mul() in bitvector theory, and
were able to prove their soundness using 16-bit bitvectors (instead of
64-bit bitvectors that the functions actually use).
In conclusion, with this patch,
1. We were able to show that we can improve the overall precision of
BPF_MUL. We proved (using an SMT solver) that this new version of
BPF_MUL is at least as precise as the current version for all inputs.
2. We are able to prove the soundness of the new scalar_min_max_mul() and
scalar32_min_max_mul(). By leveraging the existing proof of tnum_mul
[1], we can say that the composition of these three functions within
BPF_MUL is sound.
[1] https://ieeexplore.ieee.org/abstract/document/9741267
[2] https://link.springer.com/chapter/10.1007/978-3-031-37709-9_12
[3] https://people.cs.rutgers.edu/~sn349/papers/sas24-preprint.pdf
Co-developed-by: Harishankar Vishwanathan <[email protected]>
Signed-off-by: Harishankar Vishwanathan <[email protected]>
Co-developed-by: Srinivas Narayana <[email protected]>
Signed-off-by: Srinivas Narayana <[email protected]>
Co-developed-by: Santosh Nagarakatte <[email protected]>
Signed-off-by: Santosh Nagarakatte <[email protected]>
Signed-off-by: Matan Shachnai <[email protected]>- Loading branch information
