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.

Importantly, we update BPF_MUL to allow signed bound multiplication (i.e. multiplying negative bounds) as well as allow u64 inputs to take on values from [0, U64_MAX]. We perform signed multiplication on two bounds [a,b] and [c,d] by multiplying every combination of the bounds (i.e. a*c, a*d, b*c, and b*d) and checking for overflow of each product. If there is an overflow, we mark the signed bounds unbounded [S64_MIN, S64_MAX]. In the case of no overflow, we take the minimum of these products to be the resulting smin, and the maximum to be the resulting smax.

The key idea here is that if there’s no possibility of overflow, either when multiplying signed bounds or unsigned bounds, we can safely multiply the respective bounds; otherwise, we set the bounds that exhibit overflow (during multiplication) to 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_umin = 0; *dst_umax = U64_MAX; } ...

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 8-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 and more precise for some 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 harishankar.vishwanathan@gmail.com Signed-off-by: Harishankar Vishwanathan harishankar.vishwanathan@gmail.com Co-developed-by: Srinivas Narayana srinivas.narayana@rutgers.edu Signed-off-by: Srinivas Narayana srinivas.narayana@rutgers.edu Co-developed-by: Santosh Nagarakatte santosh.nagarakatte@rutgers.edu Signed-off-by: Santosh Nagarakatte santosh.nagarakatte@rutgers.edu Signed-off-by: Matan Shachnai m.shachnai@gmail.com --- kernel/bpf/verifier.c | 80 +++++++++++++++++++------------------------ 1 file changed, 36 insertions(+), 44 deletions(-)

diff --git a/kernel/bpf/verifier.c b/kernel/bpf/verifier.c index f27274e933e5..419f75c2cabb 100644 --- a/kernel/bpf/verifier.c +++ b/kernel/bpf/verifier.c @@ -14084,64 +14084,56 @@ static void scalar_min_max_sub(struct bpf_reg_state *dst_reg, static void scalar32_min_max_mul(struct bpf_reg_state *dst_reg, struct bpf_reg_state *src_reg) { - s32 smin_val = src_reg->s32_min_value; - u32 umin_val = src_reg->u32_min_value; - u32 umax_val = src_reg->u32_max_value; + s32 *dst_smin = &dst_reg->s32_min_value; + s32 *dst_smax = &dst_reg->s32_max_value; + u32 *dst_umin = &dst_reg->u32_min_value; + u32 *dst_umax = &dst_reg->u32_max_value; + s32 tmp_prod[4];

- if (smin_val < 0 || dst_reg->s32_min_value < 0) { - /* Ain't nobody got time to multiply that sign */ - __mark_reg32_unbounded(dst_reg); - return; - } - /* Both values are positive, so we can work with unsigned and - * copy the result to signed (unless it exceeds S32_MAX). - */ - if (umax_val > U16_MAX || dst_reg->u32_max_value > U16_MAX) { - /* Potential overflow, we know nothing */ - __mark_reg32_unbounded(dst_reg); - return; + if (check_mul_overflow(*dst_umax, src_reg->u32_max_value, dst_umax) || + check_mul_overflow(*dst_umin, src_reg->u32_min_value, dst_umin)) { + /* Overflow possible, we know nothing */ + *dst_umin = 0; + *dst_umax = U32_MAX; } - dst_reg->u32_min_value *= umin_val; - dst_reg->u32_max_value *= umax_val; - if (dst_reg->u32_max_value > S32_MAX) { + if (check_mul_overflow(*dst_smin, src_reg->s32_min_value, &tmp_prod[0]) || + check_mul_overflow(*dst_smin, src_reg->s32_max_value, &tmp_prod[1]) || + check_mul_overflow(*dst_smax, src_reg->s32_min_value, &tmp_prod[2]) || + check_mul_overflow(*dst_smax, src_reg->s32_max_value, &tmp_prod[3])) { /* Overflow possible, we know nothing */ - dst_reg->s32_min_value = S32_MIN; - dst_reg->s32_max_value = S32_MAX; + *dst_smin = S32_MIN; + *dst_smax = S32_MAX; } else { - dst_reg->s32_min_value = dst_reg->u32_min_value; - dst_reg->s32_max_value = dst_reg->u32_max_value; + *dst_smin = min_array(tmp_prod, 4); + *dst_smax = max_array(tmp_prod, 4); } }

static void scalar_min_max_mul(struct bpf_reg_state *dst_reg, struct bpf_reg_state *src_reg) { - s64 smin_val = src_reg->smin_value; - u64 umin_val = src_reg->umin_value; - u64 umax_val = src_reg->umax_value; + s64 *dst_smin = &dst_reg->smin_value; + s64 *dst_smax = &dst_reg->smax_value; + u64 *dst_umin = &dst_reg->umin_value; + u64 *dst_umax = &dst_reg->umax_value; + s64 tmp_prod[4];

- if (smin_val < 0 || dst_reg->smin_value < 0) { - /* Ain't nobody got time to multiply that sign */ - __mark_reg64_unbounded(dst_reg); - return; - } - /* 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; + 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_umin = 0; + *dst_umax = U64_MAX; } - dst_reg->umin_value *= umin_val; - dst_reg->umax_value *= umax_val; - if (dst_reg->umax_value > S64_MAX) { + if (check_mul_overflow(*dst_smin, src_reg->smin_value, &tmp_prod[0]) || + check_mul_overflow(*dst_smin, src_reg->smax_value, &tmp_prod[1]) || + check_mul_overflow(*dst_smax, src_reg->smin_value, &tmp_prod[2]) || + check_mul_overflow(*dst_smax, src_reg->smax_value, &tmp_prod[3])) { /* Overflow possible, we know nothing */ - dst_reg->smin_value = S64_MIN; - dst_reg->smax_value = S64_MAX; + *dst_smin = S64_MIN; + *dst_smax = S64_MAX; } else { - dst_reg->smin_value = dst_reg->umin_value; - dst_reg->smax_value = dst_reg->umax_value; + *dst_smin = min_array(tmp_prod, 4); + *dst_smax = max_array(tmp_prod, 4); } }