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fix: fixed reduction parameter error (#31)
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* fixed reduction parameter error

redc_param previously was only large enough to cover barrett reductions whose input was < 2^{modulus_bits * 2 + 2}

this was insufficient for elliptic curve arithmetic in bignum

redc_param is now large enough to cover barrett reduction inputs of at least 16 * modulus^2

additonally, library upgraded to not trigger compiler warnings for nargo 0.35.0

* format

* Update src/runtime_bignum_test.nr

Co-authored-by: Maxim Vezenov <[email protected]>

* Update src/runtime_bignum_test.nr

Co-authored-by: Maxim Vezenov <[email protected]>

* Update src/runtime_bignum_test.nr

Co-authored-by: Maxim Vezenov <[email protected]>

---------

Co-authored-by: Tom French <[email protected]>
Co-authored-by: Maxim Vezenov <[email protected]>
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@zac-williamson @TomAFrench @vezenovm
3 people authored Oct 10, 2024
1 parent 862ac51 commit c312ef7
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199 changes: 106 additions & 93 deletions src/bignum_test.nr
View file
Original file line number Diff line number Diff line change
Expand Up @@ -16,87 +16,46 @@ struct Test2048Params{}
fn silence_warning() {
let _ = Test2048Params {};
}

impl RuntimeBigNumParamsTrait<18> for Test2048Params {
fn modulus_bits() -> u32 {
2048
}
}
impl BigNumParamsTrait<18> for Test2048Params {
fn get_instance() -> BigNumInstance<18, Self> {
let modulus: [Field; 18] = [
0x0000000000000000000000000000000000c0a197a5ae0fcdceb052c9732614fe,
0x0000000000000000000000000000000000656ae034423283422243918ab83be3,
0x00000000000000000000000000000000006bf590da48a7c1070b7d5aabaac678,
0x00000000000000000000000000000000000cce39f530238b606f24b296e2bda9,
0x000000000000000000000000000000000001e1fef9bb9c1c3ead98f226f1bfa0,
0x0000000000000000000000000000000000ad8c1c816e12e0ed1379055e373abf,
0x0000000000000000000000000000000000cebe80e474f753aa9d1461c435123d,
0x0000000000000000000000000000000000aee5a18ceedef88d115a8b93c167ad,
0x0000000000000000000000000000000000268ba83c4a65c4307427fc495d9e44,
0x0000000000000000000000000000000000dd2777926848667b7df79f342639d4,
0x0000000000000000000000000000000000f455074c96855ca0068668efe7da3d,
0x00000000000000000000000000000000005ddba6b30bbc168bfb3a1225f27d65,
0x0000000000000000000000000000000000591fec484f36707524133bcd6f4258,
0x000000000000000000000000000000000059641b756766aeebe66781dd01d062,
0x000000000000000000000000000000000058bc5eaff4b165e142bf9e2480eebb,
0x0000000000000000000000000000000000667a3964f08e06df772ce64b229a72,
0x00000000000000000000000000000000009c1fdb18907711bfe3e3c1cf918395,
0x00000000000000000000000000000000000000000000000000000000000000b8
];
// fn double_modulus() -> [Field; 18] {[
// 0x000000000000000000000000000000000181432f4b5c1f9b9d60a592e64c29fc,
// 0x0000000000000000000000000000000001cad5c06884650684448723157077c6,
// 0x0000000000000000000000000000000001d7eb21b4914f820e16fab557558cef,
// 0x0000000000000000000000000000000001199c73ea604716c0de49652dc57b51,
// 0x000000000000000000000000000000000103c3fdf37738387d5b31e44de37f3f,
// 0x00000000000000000000000000000000015b183902dc25c1da26f20abc6e757d,
// 0x00000000000000000000000000000000019d7d01c8e9eea7553a28c3886a247a,
// 0x00000000000000000000000000000000015dcb4319ddbdf11a22b5172782cf5a,
// 0x00000000000000000000000000000000014d17507894cb8860e84ff892bb3c88,
// 0x0000000000000000000000000000000001ba4eef24d090ccf6fbef3e684c73a7,
// 0x0000000000000000000000000000000001e8aa0e992d0ab9400d0cd1dfcfb47a,
// 0x0000000000000000000000000000000001bbb74d6617782d17f674244be4faca,
// 0x0000000000000000000000000000000001b23fd8909e6ce0ea4826779ade84af,
// 0x0000000000000000000000000000000001b2c836eacecd5dd7cccf03ba03a0c3,
// 0x0000000000000000000000000000000001b178bd5fe962cbc2857f3c4901dd75,
// 0x0000000000000000000000000000000001ccf472c9e11c0dbeee59cc964534e3,
// 0x0000000000000000000000000000000001383fb63120ee237fc7c7839f230729,
// 0x0000000000000000000000000000000000000000000000000000000000000170
// ]}
let redc_param: [Field; 18] = [
0x000000000000000000000000000000000091697def7100cd5cf8d890b4ef2ec3,
0x00000000000000000000000000000000006765ba8304214dac764d3f4adc3185,
0x000000000000000000000000000000000048404bd14d927ea230e60d4bebf940,
0x00000000000000000000000000000000007c4d53a23bacc251ecbfc4b7ba5a0b,
0x000000000000000000000000000000000093eaf3499474a6f5b2fff83f1259c8,
0x00000000000000000000000000000000005bff4c737b97281f1a5f2384a8c16d,
0x000000000000000000000000000000000061b4cf2f55358476b5323782999055,
0x00000000000000000000000000000000001e7a804e8eacfe3a2a5673bc3885b8,
0x0000000000000000000000000000000000eabadeae4282906c817adf70eab4ae,
0x0000000000000000000000000000000000166f7df257fe2bf27f0809aceed9b0,
0x00000000000000000000000000000000007d90fb7428901b8bed11f6b81e36bf,
0x0000000000000000000000000000000000f36e6ba885c60b7024c563605df7e0,
0x000000000000000000000000000000000052b7c58d2fb5d2c8478963ae6d4a44,
0x000000000000000000000000000000000036ee761de26635f114ccc3f7d74f85,
0x0000000000000000000000000000000000e3fb726a10cf2220897513f05243de,
0x0000000000000000000000000000000000f43a26bbd732496eb4d828591b8056,
0x0000000000000000000000000000000000ff4e42304e60fb3a54fca735499f2c,
0x0000000000000000000000000000000000000000000000000000000000000162
];
BigNumInstance::new(modulus, redc_param)
test2048_Instance
}
fn modulus_bits() -> u32 {
2048
}
}

global test2048_Instance: BigNumInstance<18, Test2048Params> = BigNumInstance {
modulus: [
0xc0a197a5ae0fcdceb052c9732614fe, 0x656ae034423283422243918ab83be3, 0x6bf590da48a7c1070b7d5aabaac678, 0x0cce39f530238b606f24b296e2bda9, 0x01e1fef9bb9c1c3ead98f226f1bfa0, 0xad8c1c816e12e0ed1379055e373abf, 0xcebe80e474f753aa9d1461c435123d, 0xaee5a18ceedef88d115a8b93c167ad, 0x268ba83c4a65c4307427fc495d9e44, 0xdd2777926848667b7df79f342639d4, 0xf455074c96855ca0068668efe7da3d, 0x5ddba6b30bbc168bfb3a1225f27d65, 0x591fec484f36707524133bcd6f4258, 0x59641b756766aeebe66781dd01d062, 0x58bc5eaff4b165e142bf9e2480eebb, 0x667a3964f08e06df772ce64b229a72, 0x9c1fdb18907711bfe3e3c1cf918395, 0xb8
],
double_modulus: [
0x0181432f4b5c1f9b9d60a592e64c29fc, 0x01cad5c06884650684448723157077c6, 0x01d7eb21b4914f820e16fab557558cef, 0x01199c73ea604716c0de49652dc57b51, 0x0103c3fdf37738387d5b31e44de37f3f, 0x015b183902dc25c1da26f20abc6e757d, 0x019d7d01c8e9eea7553a28c3886a247a, 0x015dcb4319ddbdf11a22b5172782cf5a, 0x014d17507894cb8860e84ff892bb3c88, 0x01ba4eef24d090ccf6fbef3e684c73a7, 0x01e8aa0e992d0ab9400d0cd1dfcfb47a, 0x01bbb74d6617782d17f674244be4faca, 0x01b23fd8909e6ce0ea4826779ade84af, 0x01b2c836eacecd5dd7cccf03ba03a0c3, 0x01b178bd5fe962cbc2857f3c4901dd75, 0x01ccf472c9e11c0dbeee59cc964534e3, 0x01383fb63120ee237fc7c7839f230729, 0x0170
],
modulus_u60: U60Repr { limbs: [
0x0eb052c9732614fe, 0x0c0a197a5ae0fcdc, 0x022243918ab83be3, 0x0656ae0344232834, 0x070b7d5aabaac678, 0x06bf590da48a7c10, 0x6f24b296e2bda9, 0xcce39f530238b6, 0x0ead98f226f1bfa0, 0x1e1fef9bb9c1c3, 0x0d1379055e373abf, 0x0ad8c1c816e12e0e, 0x0a9d1461c435123d, 0x0cebe80e474f753a, 0x0d115a8b93c167ad, 0x0aee5a18ceedef88, 0x7427fc495d9e44, 0x0268ba83c4a65c43, 0x0b7df79f342639d4, 0x0dd2777926848667, 0x068668efe7da3d, 0x0f455074c96855ca, 0x0bfb3a1225f27d65, 0x05ddba6b30bbc168, 0x0524133bcd6f4258, 0x0591fec484f36707, 0x0be66781dd01d062, 0x059641b756766aee, 0x0142bf9e2480eebb, 0x058bc5eaff4b165e, 0x0f772ce64b229a72, 0x0667a3964f08e06d, 0x0fe3e3c1cf918395, 0x09c1fdb18907711b, 0xb8, 0x00]},
modulus_u60_x4: U60Repr { limbs: [
0x0eb052c9732614fe, 0x0c0a197a5ae0fcdc, 0x022243918ab83be3, 0x0656ae0344232834, 0x070b7d5aabaac678, 0x06bf590da48a7c10, 0x6f24b296e2bda9, 0xcce39f530238b6, 0x0ead98f226f1bfa0, 0x1e1fef9bb9c1c3, 0x0d1379055e373abf, 0x0ad8c1c816e12e0e, 0x0a9d1461c435123d, 0x0cebe80e474f753a, 0x0d115a8b93c167ad, 0x0aee5a18ceedef88, 0x7427fc495d9e44, 0x0268ba83c4a65c43, 0x0b7df79f342639d4, 0x0dd2777926848667, 0x068668efe7da3d, 0x0f455074c96855ca, 0x0bfb3a1225f27d65, 0x05ddba6b30bbc168, 0x0524133bcd6f4258, 0x0591fec484f36707, 0x0be66781dd01d062, 0x059641b756766aee, 0x0142bf9e2480eebb, 0x058bc5eaff4b165e, 0x0f772ce64b229a72, 0x0667a3964f08e06d, 0x0fe3e3c1cf918395, 0x09c1fdb18907711b, 0xb8, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00] },
redc_param: [
0x1697def7100cd5cf8d890b4ef2ec3f, 0x765ba8304214dac764d3f4adc31859, 0x8404bd14d927ea230e60d4bebf9406, 0xc4d53a23bacc251ecbfc4b7ba5a0b4, 0x3eaf3499474a6f5b2fff83f1259c87, 0xbff4c737b97281f1a5f2384a8c16d9, 0x1b4cf2f55358476b53237829990555, 0xe7a804e8eacfe3a2a5673bc3885b86, 0xabadeae4282906c817adf70eab4ae1, 0x66f7df257fe2bf27f0809aceed9b0e, 0xd90fb7428901b8bed11f6b81e36bf1, 0x36e6ba885c60b7024c563605df7e07, 0x2b7c58d2fb5d2c8478963ae6d4a44f, 0x6ee761de26635f114ccc3f7d74f855, 0x3fb726a10cf2220897513f05243de3, 0x43a26bbd732496eb4d828591b8056e, 0xf4e42304e60fb3a54fca735499f2cf, 0x162f
]
};
type Fq = BigNum<3, BNParams>;

fn test_eq<BigNum, let N: u32>(_: BigNum, __: [Field; N]) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([1, 2, 3, 4]);
let c = BigNum::__derive_from_seed([2, 2, 3, 4]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let c = unsafe {
BigNum::__derive_from_seed([2, 2, 3, 4])
};

let modulus = BigNum::modulus();
let t0: U60Repr<N, 2> = (U60Repr::from(modulus.get().as_array()));
Expand Down Expand Up @@ -128,17 +87,25 @@ fn test_eq<BigNum, let N: u32>(_: BigNum, __: [Field; N]) where BigNum: BigNumTr
// // // 99689
// // // 929 gates for a 2048 bit mul
fn test_mul<BigNum>(_: BigNum) where BigNum: BigNumTrait + std::ops::Mul + std::ops::Add {
let a: BigNum = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b: BigNum = BigNum::__derive_from_seed([4, 5, 6, 7]);
let a: BigNum = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b: BigNum = unsafe {
BigNum::__derive_from_seed([4, 5, 6, 7])
};

let c = (a + b) * (a + b);
let d = (a * a) + (b * b) + (a * b) + (a * b);
assert(c.eq(d));
}

fn test_add<BigNum>(_: BigNum) where BigNum: BigNumTrait + std::ops::Add + std::ops::Mul + std::cmp::Eq {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b: BigNum = BigNum::__derive_from_seed([4, 5, 6, 7]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b: BigNum = unsafe {
BigNum::__derive_from_seed([4, 5, 6, 7])
};
let one = BigNum::one();
a.validate_in_range();
a.validate_in_field();
Expand All @@ -162,40 +129,62 @@ fn test_add<BigNum>(_: BigNum) where BigNum: BigNumTrait + std::ops::Add + std::
}

fn test_div<BigNum>(_: BigNum) where BigNum: BigNumTrait + std::ops::Div + std::ops::Mul + std::ops::Add + std::cmp::Eq {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([4, 5, 6, 7]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([4, 5, 6, 7])
};

let c = a / b;
assert((b * c) == (a));
}

fn test_invmod<BigNum>(_: BigNum) where BigNum: BigNumTrait + std::cmp::Eq {
let u = BigNum::__derive_from_seed([1, 2, 3, 4]);
let u = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
for _ in 0..1 {
let v = u.__invmod();
let result = u.__mul(v);
let v = unsafe {
u.__invmod()
};
let result = unsafe {
u.__mul(v)
};
let expected = BigNum::one();
assert(result == expected);
}
}

fn assert_is_not_equal<BigNum>(_: BigNum) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([4, 5, 6, 7]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([4, 5, 6, 7])
};

a.assert_is_not_equal(b);
}

fn assert_is_not_equal_fail<BigNum>(_: BigNum) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([1, 2, 3, 4]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};

a.assert_is_not_equal(b);
}

fn assert_is_not_equal_overloaded_lhs_fail<BigNum, let N: u32>(_: BigNum, __: [Field; N]) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([1, 2, 3, 4]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};

let modulus = BigNum::modulus();

Expand All @@ -206,8 +195,12 @@ fn assert_is_not_equal_overloaded_lhs_fail<BigNum, let N: u32>(_: BigNum, __: [F
}

fn assert_is_not_equal_overloaded_rhs_fail<BigNum, let N: u32>(_: BigNum, __: [Field; N]) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([1, 2, 3, 4]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};

let modulus = BigNum::modulus();

Expand All @@ -218,8 +211,12 @@ fn assert_is_not_equal_overloaded_rhs_fail<BigNum, let N: u32>(_: BigNum, __: [F
}

fn assert_is_not_equal_overloaded_fail<BigNum, let N: u32>(_: BigNum, __: [Field; N]) where BigNum: BigNumTrait {
let a = BigNum::__derive_from_seed([1, 2, 3, 4]);
let b = BigNum::__derive_from_seed([1, 2, 3, 4]);
let a = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};

let modulus = BigNum::modulus();

Expand Down Expand Up @@ -466,13 +463,17 @@ fn test_assert_is_not_equal_overloaded_fail_U256() {
type U256 = BigNum<3, U256Params>;
#[test]
fn test_udiv_mod_U256() {
let a: U256 = BigNum::__derive_from_seed([1, 2, 3, 4]);
let a: U256 = unsafe {
BigNum::__derive_from_seed([1, 2, 3, 4])
};
let b: U256 = BigNum::from_array([12, 0, 0]);

let (q, r) = a.udiv_mod(b);

// let qb = q.__mul(b);
let product = q.__mul(b).__add(r);
let product = unsafe {
q.__mul(b).__add(r)
};
assert(product.eq(a));
}

Expand Down Expand Up @@ -555,7 +556,9 @@ fn test_2048_bit_quadratic_expression() {

let a_bn: BigNum<18, Test2048Params> = BigNum { limbs: a };
let b_bn: BigNum<18, Test2048Params> = BigNum { limbs: b };
let c_bn = a_bn.__mul(b_bn);
let c_bn = unsafe {
a_bn.__mul(b_bn)
};
assert(c_bn.limbs == c_expected);

a_bn.validate_in_range();
Expand Down Expand Up @@ -584,7 +587,9 @@ fn test_expressions() {
0x0
]
};
let yy = y.__add(y);
let yy = unsafe {
y.__add(y)
};

assert(yy.limbs == z.limbs);

Expand Down Expand Up @@ -616,10 +621,18 @@ fn test_expressions() {
0x0000000000000000000000000000000000000000000000000000000000000f93
]
};
let wx = w.__mul(x);
let uv = uu.__mul(vv);
let y = (uv.__add(wx)).__neg();
let z = uv.__add(wx);
let wx = unsafe {
w.__mul(x)
};
let uv = unsafe {
uu.__mul(vv)
};
let y = unsafe {
(uv.__add(wx)).__neg()
};
let z = unsafe {
uv.__add(wx)
};

BigNum::evaluate_quadratic_expression(
[[uu], [w]],
Expand Down
14 changes: 3 additions & 11 deletions src/fields/U1024.nr
View file
Original file line number Diff line number Diff line change
Expand Up @@ -2,15 +2,11 @@ use crate::BigNumParamsTrait;
use crate::runtime_bignum::BigNumInstance;
use crate::runtime_bignum::BigNumParamsTrait as RuntimeBigNumParamsTrait;
use crate::utils::u60_representation::U60Repr;

pub struct U1024Params {}
pub struct U1024Params {}
impl RuntimeBigNumParamsTrait<9> for U1024Params {
fn modulus_bits() -> u32 {
1025
}
fn has_multiplicative_inverse() -> bool {
false
}
}
impl BigNumParamsTrait<9> for U1024Params {
fn get_instance() -> BigNumInstance<9, Self> {
Expand All @@ -19,11 +15,8 @@ impl BigNumParamsTrait<9> for U1024Params {
fn modulus_bits() -> u32 {
1025
}
fn has_multiplicative_inverse() -> bool {
false
}
}
global U1024_Instance: BigNumInstance<9, U1024Params> = BigNumInstance {
pub global U1024_Instance: BigNumInstance<9, U1024Params> = BigNumInstance {
modulus: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x010000000000000000
],
Expand All @@ -35,7 +28,6 @@ global U1024_Instance: BigNumInstance<9, U1024Params> = BigNumInstance {
modulus_u60_x4: U60Repr { limbs: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00] },
redc_param: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x040000000000000000
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x400000000000000000
]
};

14 changes: 3 additions & 11 deletions src/fields/U2048.nr
View file
Original file line number Diff line number Diff line change
Expand Up @@ -2,15 +2,11 @@ use crate::BigNumParamsTrait;
use crate::runtime_bignum::BigNumInstance;
use crate::runtime_bignum::BigNumParamsTrait as RuntimeBigNumParamsTrait;
use crate::utils::u60_representation::U60Repr;

pub struct U2048Params {}
pub struct U2048Params {}
impl RuntimeBigNumParamsTrait<18> for U2048Params {
fn modulus_bits() -> u32 {
2049
}
fn has_multiplicative_inverse() -> bool {
false
}
}
impl BigNumParamsTrait<18> for U2048Params {
fn get_instance() -> BigNumInstance<18, Self> {
Expand All @@ -19,11 +15,8 @@ impl BigNumParamsTrait<18> for U2048Params {
fn modulus_bits() -> u32 {
2049
}
fn has_multiplicative_inverse() -> bool {
false
}
}
global U2048_Instance: BigNumInstance<18, U2048Params> = BigNumInstance {
pub global U2048_Instance: BigNumInstance<18, U2048Params> = BigNumInstance {
modulus: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0100
],
Expand All @@ -35,7 +28,6 @@ global U2048_Instance: BigNumInstance<18, U2048Params> = BigNumInstance {
modulus_u60_x4: U60Repr { limbs: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0100, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00] },
redc_param: [
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0400
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x4000
]
};

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