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feat(avm): range check gadget (#7967)
This doesnt replace the existing range check - this just sets up the initial work for a range check gadget
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| Original file line number | Diff line number | Diff line change |
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| include "../main.pil"; | ||
| include "../fixed/powers.pil"; | ||
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| namespace range_check(256); | ||
| pol commit clk; | ||
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| // Range check selector | ||
| pol commit sel_rng_chk; | ||
| sel_rng_chk * (1 - sel_rng_chk) = 0; | ||
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| // Witnesses | ||
| // Value to range check | ||
| pol commit value; | ||
| // Number of bits to check against | ||
| pol commit rng_chk_bits; | ||
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| // Bit Size Columns | ||
| // It is enforced (further down) that the selected column is the lowest multiple of 16 that is greater than rng_chk_bits | ||
| // e.g., rng_chk_bits = 10 ===> is_lte_u16, rng_chk_bits = 100 ==> is_lte_u112 | ||
| // If rng_chk_bits is a multiple of 16, a prover is able to choose either is_lte_xx or is_lte_xx(+16), since the dynamic register will prove 0 | ||
| // This isn't a concern and only costs the prover additional compute. | ||
| // TODO: Maybe we can get rid of is_lte_u128 since it's implicit if we have sel_rng_chk and no other is_lte_x | ||
| pol commit is_lte_u16; | ||
| pol commit is_lte_u32; | ||
| pol commit is_lte_u48; | ||
| pol commit is_lte_u64; | ||
| pol commit is_lte_u80; | ||
| pol commit is_lte_u96; | ||
| pol commit is_lte_u112; | ||
| pol commit is_lte_u128; | ||
| is_lte_u16 * (1 - is_lte_u16) = 0; | ||
| is_lte_u32 * (1 - is_lte_u32) = 0; | ||
| is_lte_u48 * (1 - is_lte_u48) = 0; | ||
| is_lte_u64 * (1 - is_lte_u64) = 0; | ||
| is_lte_u80 * (1 - is_lte_u80) = 0; | ||
| is_lte_u96 * (1 - is_lte_u96) = 0; | ||
| is_lte_u112 * (1 - is_lte_u112) = 0; | ||
| is_lte_u128 * (1 - is_lte_u128) = 0; | ||
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| // Mutual Exclusivity condition | ||
| is_lte_u16 + is_lte_u32 + is_lte_u48 + is_lte_u64 + is_lte_u80 + is_lte_u96 + is_lte_u112 + is_lte_u128 = sel_rng_chk; | ||
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| // Eight 16-bit slice registers | ||
| pol commit u16_r0; | ||
| pol commit u16_r1; | ||
| pol commit u16_r2; | ||
| pol commit u16_r3; | ||
| pol commit u16_r4; | ||
| pol commit u16_r5; | ||
| pol commit u16_r6; | ||
| // This register has a (more expensive) set of constraint that enables dynamic range check of bit values between 0 and 16 bits | ||
| pol commit u16_r7; | ||
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| // In each of these relations, the u16_r7 register contains the most significant 16 bits of value. | ||
| pol X_0 = is_lte_u16 * u16_r7; | ||
| pol X_1 = is_lte_u32 * (u16_r0 + u16_r7 * 2**16); | ||
| pol X_2 = is_lte_u48 * (u16_r0 + u16_r1 * 2**16 + u16_r7 * 2**32); | ||
| pol X_3 = is_lte_u64 * (u16_r0 + u16_r1 * 2**16 + u16_r2 * 2**32 + u16_r7 * 2**48); | ||
| pol X_4 = is_lte_u80 * (u16_r0 + u16_r1 * 2**16 + u16_r2 * 2**32 + u16_r3 * 2**48 + u16_r7 * 2**64); | ||
| pol X_5 = is_lte_u96 * (u16_r0 + u16_r1 * 2**16 + u16_r2 * 2**32 + u16_r3 * 2**48 + u16_r4 * 2**64 + u16_r7 * 2**80); | ||
| pol X_6 = is_lte_u112 * (u16_r0 + u16_r1 * 2**16 + u16_r2 * 2**32 + u16_r3 * 2**48 + u16_r4 * 2**64 + u16_r5 * 2**80 + u16_r7 * 2**96); | ||
| pol X_7 = is_lte_u128 * (u16_r0 + u16_r1 * 2**16 + u16_r2 * 2**32 + u16_r3 * 2**48 + u16_r4 * 2**64 + u16_r5 * 2**80 + u16_r6 * 2**96 + u16_r7 * 2**112); | ||
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| // Since the is_lte_x are mutually exclusive, only one of the Xs will be non-zero | ||
| pol RESULT = X_0 + X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7; | ||
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| #[CHECK_RECOMPOSITION] | ||
| sel_rng_chk * (RESULT - value) = 0; | ||
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| // ===== Dynamic Check Constraints ===== | ||
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| // The number of bits that form the dynamic range check is depending on the claimed lte value and the witness rng_chk_bits | ||
| // claimed is_lte_x | dyn_rng_chk_bits | ||
| // -----------------|----------------- | ||
| // is_lte_u16 | rng_chk_bits | ||
| // is_lte_u32 | rng_chk_bits - 16 | ||
| // is_lte_u48 | rng_chk_bits - 32 | ||
| // is_lte_u64 | rng_chk_bits - 48 | ||
| // is_lte_u80 | rng_chk_bits - 64 | ||
| // is_lte_u96 | rng_chk_bits - 80 | ||
| // is_lte_u112 | rng_chk_bits - 96 | ||
| // is_lte_u128 | rng_chk_bits - 112 | ||
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| // [CALCULATION STEPS] | ||
| // 1) Calculate dyn_rng_chk_bits from the table above | ||
| // 2) Calculate dyn_rng_chk_pow_2 = 2^dyn_rng_chk_bits | ||
| // 3) Calculate dyn_diff = dyn_rng_chk_pow_2 - u16_r7 - 1 | ||
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| // [ASSERTIONS] | ||
| // 1) Assert 0 <= dyn_rng_chk_bits <= 16 (i.e. dyn_rng_chk_bits supports up to a 16-bit number) | ||
| // 2) Assert dyn_diff > 0 (i.e. dyn_diff did not underflow) | ||
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| // === Ensuring dyn_rng_chk_bits is in the range [0,16] === | ||
| // 1) We perform an 8-bit lookup to get dyn_rng_chk_pow_2 - note this is an 8-bit lookup so only constrains dyn_rng_chk_bits to be [0, 255] | ||
| // 2) This value is used in dyn_diff = dyn_rng_chk_pow_2 - u16_r7 - 1 | ||
| // (a) A 16-bit lookup is performed on dyn_diff to check it hasn't underflowed - this constrains it to be between [0, 2^16 - 1] | ||
| // (b) u16_r7 is constrained by a 16-bit lookup table [0, 2^16 - 1] | ||
| // 3) If the value of dyn_rng_chk_pow_2 > 2^16, i.e. dyn_rng_chk_bits is > 16, the condition (2a) will not hold | ||
| // (a) [0, 2^16 - 1] = dyn_rng_chk_pow_2 - [0, 2^16 - 1] - 1 | ||
| // (b) from above, dyn_rng_check_pow_2 must be [0, 2^16] | ||
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| // Some counter-examples | ||
| // Assume a range check that the value 3 fits into 100 bits | ||
| // [A Valid Proof] | ||
| // 1) value = 3, rng_chk_bits = 100, is_lte_u112 = 1 | ||
| // 2) u16_r0 = 3, while all other registers including u16_r7 (the dynamic one) are set to zero - passing #[CHECK_RECOMPOSITION] | ||
| // 3) dyn_rng_chk_bits = 100 - 96 = 4, as per the table above - this passes #[LOOKUP_RNG_CHK_POW_2] | ||
| // 4) dyn_rng_chk_pow_2 = 2^4 = 16 | ||
| // 5) dyn_diff = dyn_rng_chk_pow_2 - u16_r7 - 1 = 16 - 0 - 1 = 15 - passing the range check #[LOOKUP_RNG_CHK_DIFF] | ||
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| // [An Invalid Proof where dyn_rng_chk_bits > 16] | ||
| // 1) value = 3, rng_chk_bits = 100, is_lte_u16 = 1 -- a prover tries to claim the value is between 0 and u16 (which it is but isnt what the range check attesting) | ||
| // 2) u16_r7 = 3, this still passes #[CHECK_RECOMPOSITION] | ||
| // 3) dyn_rng_chk_bits = 100, as per the table - this still passes #[LOOKUP_RNG_CHK_POW_2] | ||
| // 4) dyn_rng_check_pow_2 = 2^100 | ||
| // 5) dyn_diff = dyn_rng_chk_pow_2 - u16_r7 - 1 = 2^100 - 3 - 1 = 2^100 - 4 - this would fail the 16-bit range check #[LOOKUP_RNG_CHK_DIFF] | ||
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| // [An Invalid Proof where dyn_rng_chk_bits < 0] | ||
| // 1) value = 3, rng_chk_bits = 100, is_lte_u128 = 1 -- a prover claims it fits within u112 and u128. | ||
| // 2) u16_r0 = 3, while all other registers including u16_r7 (the dynamic one) are set to zero - passing #[CHECK_RECOMPOSITION] | ||
| // 3) dyn_rng_chk_bits = 100 - 112 = -12, as per the table above - this fails #[LOOKUP_RNG_CHK_POW_2] | ||
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| // The number of bits that need to be dynamically range checked. | ||
| pol commit dyn_rng_chk_bits; | ||
| // Valid values for dyn_rng_chk_bits are in the range [0, 16] | ||
| dyn_rng_chk_bits - (rng_chk_bits - (is_lte_u32 * 16) - (is_lte_u48 * 32) - (is_lte_u64 * 48) - (is_lte_u80 * 64) - (is_lte_u96 * 80) - (is_lte_u112 * 96) - (is_lte_u128 * 112)) = 0; | ||
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| // To perform the dynamic range check we also need the value of 2^dyn_rng_chk_bits | ||
| pol commit dyn_rng_chk_pow_2; | ||
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| // This lookup does 2 things (1) Indirectly range checks dyn_rng_chk_bits to not have underflowed and (2) Simplified calculation of 2^dyn_rng_chk_bits | ||
| #[LOOKUP_RNG_CHK_POW_2] | ||
| sel_rng_chk {dyn_rng_chk_bits, dyn_rng_chk_pow_2} in main.sel_rng_8 {main.clk, powers.power_of_2}; | ||
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| // Now we need to perform the dynamic range check itself | ||
| // We check that u16_r7 < dyn_rng_chk_pow_2 ==> dyn_rng_chk_pow_2 - u16_r7 - 1 >= 0 | ||
| pol commit dyn_diff; | ||
| sel_rng_chk * (dyn_diff - (dyn_rng_chk_pow_2 - u16_r7 - 1)) = 0; | ||
| // The value of dyn_diff has to be between [0, 2^16) | ||
| // To check we did not underflow we just range check it | ||
| #[LOOKUP_RNG_CHK_DIFF] | ||
| sel_rng_chk { dyn_diff } in main.sel_rng_16 { main.clk }; | ||
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| // Lookup relations. | ||
| // We only need these relations while we do not support pol in the LHS selector | ||
| pol commit sel_lookup_0; | ||
| pol commit sel_lookup_1; | ||
| pol commit sel_lookup_2; | ||
| pol commit sel_lookup_3; | ||
| pol commit sel_lookup_4; | ||
| pol commit sel_lookup_5; | ||
| pol commit sel_lookup_6; | ||
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| // The lookups are cumulative - i.e. every value greater than 16 bits involve sel_lookup_0 | ||
| // Note that the lookup for the u16_r7 is always active (dynamic range check) | ||
| sel_lookup_0 - (is_lte_u32 + is_lte_u48 + is_lte_u64 + is_lte_u80 + is_lte_u96 + is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_1 - (is_lte_u48 + is_lte_u64 + is_lte_u80 + is_lte_u96 + is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_2 - (is_lte_u64 + is_lte_u80 + is_lte_u96 + is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_3 - (is_lte_u80 + is_lte_u96 + is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_4 - (is_lte_u96 + is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_5 - (is_lte_u112 + is_lte_u128) = 0; | ||
| sel_lookup_6 - is_lte_u128 = 0; | ||
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| #[LOOKUP_RNG_CHK_0] | ||
| sel_lookup_0 { u16_r0 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_1] | ||
| sel_lookup_1 { u16_r1 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_2] | ||
| sel_lookup_2 { u16_r2 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_3] | ||
| sel_lookup_3 { u16_r3 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_4] | ||
| sel_lookup_4 { u16_r4 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_5] | ||
| sel_lookup_5 { u16_r5 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_6] | ||
| sel_lookup_6 { u16_r6 } in main.sel_rng_16 { main.clk }; | ||
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| #[LOOKUP_RNG_CHK_7] | ||
| sel_rng_chk { u16_r7 } in main.sel_rng_16 { main.clk }; | ||
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