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a portable and fast pairing-based cryptography library
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# C API
## New features
```
void mclBn_setOriginalG2cofactor(int enable);
```
Use faster multiplication of `G2` with cofactor if `enable = 1`.
This is enabled if `mclBn_setMapToMode(MCL_MAP_TO_MODE_ETH2)`.
if `enable = 0`, then [the fast algorithm (mulByCofactorBLS12)](https://github.com/herumi/mcl/blob/master/include/mcl/bn.hpp#L463) is used.
## Minimum sample
[sample/pairing_c.c](sample/pairing_c.c) is a sample of how to use BLS12-381 pairing.
```
cd mcl
make -j4
make bin/pairing_c.exe && bin/pairing_c.exe
```
## Header and libraries
To use BLS12-381, include `mcl/bn_c384_256.h` and link
- libmclbn384_256.{a,so}
- libmcl.{a,so} ; core library
`384_256` means the max bit size of `Fp` is 384 and that size of `Fr` is 256.
## Notation
The elliptic equation of a curve E is `E: y^2 = x^3 + b`.
- `Fp` ; a finite field of a prime order `p`, where curves is defined over.
- `Fr` ; a finite field of a prime order `r`.
- `Fp2` ; the field extension over Fp with degree 2. Fp[i] / (i^2 + 1).
- `Fp6` ; the field extension over Fp2 with degree 3. Fp2[v] / (v^3 - Xi) where Xi = i + 1.
- `Fp12` ; the field extension over Fp6 with degree 2. Fp6[w] / (w^2 - v).
- `G1` ; the cyclic subgroup of E(Fp).
- `G2` ; the cyclic subgroup of the inverse image of E'(Fp^2) under a twisting isomorphism from E' to E.
- `GT` ; the cyclie subgroup of Fp12.
- `G1`, `G2` and `GT` have the order `r`.
The pairing e: G1 x G2 -> GT is the optimal ate pairing.
mcl treats `G1` and `G2` as an additive group and `GT` as a multiplicative group.
- `mclSize` ; `unsigned int` if WebAssembly else `size_t`
### Curve Parameter
r = |G1| = |G2| = |GT|
curveType | b| r and p |
------------|--|------------------|
BN254 | 2|r = 0x2523648240000001ba344d8000000007ff9f800000000010a10000000000000d <br> p = 0x2523648240000001ba344d80000000086121000000000013a700000000000013 |
BLS12-381 | 4|r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 <br> p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab |
BN381 | 2|r = 0x240026400f3d82b2e42de125b00158405b710818ac000007e0042f008e3e00000000001080046200000000000000000d <br> p = 0x240026400f3d82b2e42de125b00158405b710818ac00000840046200950400000000001380052e000000000000000013 |
## Structures
### `mclBnFp`
This is a struct of `Fp`. The value is stored as Montgomery representation.
### `mclBnFr`
This is a struct of `Fr`. The value is stored as Montgomery representation.
### `mclBnFp2`
This is a struct of `Fp2` which has a member `mclBnFp d[2]`.
An element `x` of `Fp2` is represented as `x = d[0] + d[1] i` where `i^2 = -1`.
### `mclBnG1`
This is a struct of `G1` which has three members `x`, `y`, `z` of type `mclBnFp`.
An element `P` of `G1` is represented as `P = [x:y:z]` of a Jacobi coordinate.
### `mclBnG2`
This is a struct of `G2` which has three members `x`, `y`, `z` of type `mclBnFp2`.
An element `Q` of `G2` is represented as `Q = [x:y:z]` of a Jacobi coordinate.
### `mclBnGT`
This is a struct of `GT` which has a member `mclBnFp d[12]`.
### sizeof
library |MCLBN_FR_UNIT_SIZE|MCLBN_FP_UNIT_SIZE|sizeof Fr|sizeof Fp|
------------------|------------------|------------------|---------|---------|
libmclbn256.a | 4 | 4 | 32 | 32 |
libmclbn384_256.a | 4 | 6 | 32 | 48 |
libmclbn384.a | 6 | 6 | 48 | 48 |
## Thread safety
All functions except for initialization and changing global setting are thread-safe.
## Initialization
Initialize mcl library. Call this function at first before calling the other functions.
```
int mclBn_init(int curve, int compiledTimeVar);
```
- `curve` ; specify the curve type
- MCL_BN254 ; BN254 (a little faster if including `mcl/bn_c256.h` and linking `libmclbn256.{a,so}`)
- MCL_BN_SNARK1 ; the same parameter used in libsnark
- MCL_BLS12_381 ; BLS12-381
- MCL_BN381_1 ; BN381 (include `mcl/bn_c384.h` and link `libmclbn384.{a,so}`)
- `compiledTimeVar` ; set `MCLBN_COMPILED_TIME_VAR`, which macro is used to make sure that
the values are the same when the library is built and used.
- return 0 if success.
- This is not thread safe.
## Global setting
```
int mclBn_setMapToMode(int mode);
```
The map-to-G2 function if `mode = MCL_MAP_TO_MODE_HASH_TO_CURVE`.
### Control to verify that a point of the elliptic curve has the order `r`.
This function affects `setStr()` and `deserialize()` for G1/G2.
```
void mclBn_verifyOrderG1(int doVerify);
void mclBn_verifyOrderG2(int doVerify);
```
- verify if `doVerify` is 1 or does not. The default parameter is 0 because the cost of verification is not small.
- Set `doVerify = 1` if considering subgroup attack is necessary.
- This is not thread safe.
## Setter / Getter
### Clear
Set `x` is zero.
```
void mclBnFr_clear(mclBnFr *x);
void mclBnFp_clear(mclBnFp *x);
void mclBnFp2_clear(mclBnFp2 *x);
void mclBnG1_clear(mclBnG1 *x);
void mclBnG2_clear(mclBnG2 *x);
void mclBnGT_clear(mclBnGT *x);
```
### Set `x` to `y`.
```
void mclBnFp_setInt(mclBnFp *y, mclInt x);
void mclBnFr_setInt(mclBnFr *y, mclInt x);
void mclBnGT_setInt(mclBnGT *y, mclInt x);
```
### Set `buf[0..bufSize-1]` to `x` with masking according to the following way.
```
int mclBnFp_setLittleEndian(mclBnFp *x, const void *buf, mclSize bufSize);
int mclBnFr_setLittleEndian(mclBnFr *x, const void *buf, mclSize bufSize);
```
1. set x = buf[0..bufSize-1] as little endian
2. x &= (1 << bitLen(r)) - 1
3. if (x >= r) x &= (1 << (bitLen(r) - 1)) - 1
- always return 0
### Set (`buf[0..bufSize-1]` mod `p` or `r`) to `x`.
```
int mclBnFp_setLittleEndianMod(mclBnFp *x, const void *buf, mclSize bufSize);
int mclBnFr_setLittleEndianMod(mclBnFr *x, const void *buf, mclSize bufSize);
```
- return 0 if bufSize <= (sizeof(*x) * 8 * 2) else -1
### Get little endian byte sequence `buf` corresponding to `x`
```
mclSize mclBnFr_getLittleEndian(void *buf, mclSize maxBufSize, const mclBnFr *x);
mclSize mclBnFp_getLittleEndian(void *buf, mclSize maxBufSize, const mclBnFp *x);
```
- write `x` to `buf` as little endian
- return the written size if sucess else 0
- NOTE: `buf[0] = 0` and return 1 if `x` is zero.
### Serialization
### Serialize
```
mclSize mclBnFr_serialize(void *buf, mclSize maxBufSize, const mclBnFr *x);
mclSize mclBnG1_serialize(void *buf, mclSize maxBufSize, const mclBnG1 *x);
mclSize mclBnG2_serialize(void *buf, mclSize maxBufSize, const mclBnG2 *x);
mclSize mclBnGT_serialize(void *buf, mclSize maxBufSize, const mclBnGT *x);
mclSize mclBnFp_serialize(void *buf, mclSize maxBufSize, const mclBnFp *x);
mclSize mclBnFp2_serialize(void *buf, mclSize maxBufSize, const mclBnFp2 *x);
```
- serialize `x` into `buf[0..maxBufSize-1]`
- return written byte size if success else 0
### Serialization format
- `Fp`(resp. `Fr`) ; a little endian byte sequence with a fixed size
- the size is the return value of `mclBn_getFpByteSize()` (resp. `mclBn_getFpByteSize()`).
- `G1` ; a compressed fixed size
- the size is equal to `mclBn_getG1ByteSize()` (=`mclBn_getFpByteSize()`).
- `G2` ; a compressed fixed size
- the size is equal to `mclBn_getG1ByteSize() * 2`.
pseudo-code to serialize of `P` of `G1` (resp. `G2`)
```
size = mclBn_getG1ByteSize() # resp. mclBn_getG1ByteSize() * 2
if P is zero:
return [0] * size
else:
P = P.normalize()
s = P.x.serialize()
# x in Fp2 is odd <=> x.a is odd
if P.y is odd: # resp. P.y.d[0] is odd
s[byte-length(s) - 1] |= 0x80
return s
```
### Ethereum serialization mode for BLS12-381
```
void mclBn_setETHserialization(int ETHserialization);
```
- serialize according to [serialization of BLS12-381](https://github.com/zkcrypto/pairing/blob/master/src/bls12_381/README.md#serialization) if BLS12-381 is used and `ETHserialization = 1` (default 0).
### Deserialize
```
mclSize mclBnFr_deserialize(mclBnFr *x, const void *buf, mclSize bufSize);
mclSize mclBnG1_deserialize(mclBnG1 *x, const void *buf, mclSize bufSize);
mclSize mclBnG2_deserialize(mclBnG2 *x, const void *buf, mclSize bufSize);
mclSize mclBnGT_deserialize(mclBnGT *x, const void *buf, mclSize bufSize);
mclSize mclBnFp_deserialize(mclBnFp *x, const void *buf, mclSize bufSize);
mclSize mclBnFp2_deserialize(mclBnFp2 *x, const void *buf, mclSize bufSize);
```
- deserialize `x` from `buf[0..bufSize-1]`
- return read size if success else 0
## String conversion
### Get string
```
mclSize mclBnFr_getStr(char *buf, mclSize maxBufSize, const mclBnFr *x, int ioMode);
mclSize mclBnG1_getStr(char *buf, mclSize maxBufSize, const mclBnG1 *x, int ioMode);
mclSize mclBnG2_getStr(char *buf, mclSize maxBufSize, const mclBnG2 *x, int ioMode);
mclSize mclBnGT_getStr(char *buf, mclSize maxBufSize, const mclBnGT *x, int ioMode);
mclSize mclBnFp_getStr(char *buf, mclSize maxBufSize, const mclBnFp *x, int ioMode);
```
- write `x` to `buf` according to `ioMode`
- `ioMode`
- 10 ; decimal number
- 16 ; hexadecimal number
- `MCLBN_IO_EC_PROJ` ; output as Jacobi coordinate
- return `strlen(buf)` if success else 0.
The meaning of the output of `G1`.
- `0` ; infinity
- `1 <x> <y>` ; affine coordinate
- `4 <x> <y> <z>` ; Jacobi coordinate
- the element `<x>` of `G2` outputs `d[0] d[1]`.
### Set string
```
int mclBnFr_setStr(mclBnFr *x, const char *buf, mclSize bufSize, int ioMode);
int mclBnG1_setStr(mclBnG1 *x, const char *buf, mclSize bufSize, int ioMode);
int mclBnG2_setStr(mclBnG2 *x, const char *buf, mclSize bufSize, int ioMode);
int mclBnGT_setStr(mclBnGT *x, const char *buf, mclSize bufSize, int ioMode);
int mclBnFp_setStr(mclBnFp *x, const char *buf, mclSize bufSize, int ioMode);
```
- set `buf[0..bufSize-1]` to `x` accoring to `ioMode`
- return 0 if success else -1
If you want to use the same generators of BLS12-381 with [zkcrypto](https://github.com/zkcrypto/pairing/tree/master/src/bls12_381#g2) then,
```
mclBnG1 P;
mclBnG1_setStr(&P, "1 3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569", 10);
mclBnG2 Q;
mclBnG2_setStr(&Q, "1 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582");
```
## Set random value
Set `x` by cryptographically secure pseudo random number generator.
```
int mclBnFr_setByCSPRNG(mclBnFr *x);
int mclBnFp_setByCSPRNG(mclBnFp *x);
```
### Change random generator function
```
void mclBn_setRandFunc(
void *self,
unsigned int (*readFunc)(void *self, void *buf, unsigned int bufSize)
);
```
- `self` ; user-defined pointer
- `readFunc` ; user-defined function, which writes random `bufSize` bytes to `buf` and returns `bufSize` if success else returns 0.
- `readFunc` must be thread-safe.
- Set the default random function if `self == 0` and `readFunc == 0`.
- This is not thread safe.
## Arithmetic operations
### neg / inv / sqr / add / sub / mul / div of `Fr`, `Fp`, `Fp2`, `GT`.
```
void mclBnFr_neg(mclBnFr *y, const mclBnFr *x);
void mclBnFr_inv(mclBnFr *y, const mclBnFr *x);
void mclBnFr_sqr(mclBnFr *y, const mclBnFr *x);
void mclBnFr_add(mclBnFr *z, const mclBnFr *x, const mclBnFr *y);
void mclBnFr_sub(mclBnFr *z, const mclBnFr *x, const mclBnFr *y);
void mclBnFr_mul(mclBnFr *z, const mclBnFr *x, const mclBnFr *y);
void mclBnFr_div(mclBnFr *z, const mclBnFr *x, const mclBnFr *y);
void mclBnFp_neg(mclBnFp *y, const mclBnFp *x);
void mclBnFp_inv(mclBnFp *y, const mclBnFp *x);
void mclBnFp_sqr(mclBnFp *y, const mclBnFp *x);
void mclBnFp_add(mclBnFp *z, const mclBnFp *x, const mclBnFp *y);
void mclBnFp_sub(mclBnFp *z, const mclBnFp *x, const mclBnFp *y);
void mclBnFp_mul(mclBnFp *z, const mclBnFp *x, const mclBnFp *y);
void mclBnFp_div(mclBnFp *z, const mclBnFp *x, const mclBnFp *y);
void mclBnFp2_neg(mclBnFp2 *y, const mclBnFp2 *x);
void mclBnFp2_inv(mclBnFp2 *y, const mclBnFp2 *x);
void mclBnFp2_sqr(mclBnFp2 *y, const mclBnFp2 *x);
void mclBnFp2_add(mclBnFp2 *z, const mclBnFp2 *x, const mclBnFp2 *y);
void mclBnFp2_sub(mclBnFp2 *z, const mclBnFp2 *x, const mclBnFp2 *y);
void mclBnFp2_mul(mclBnFp2 *z, const mclBnFp2 *x, const mclBnFp2 *y);
void mclBnFp2_div(mclBnFp2 *z, const mclBnFp2 *x, const mclBnFp2 *y);
void mclBnGT_inv(mclBnGT *y, const mclBnGT *x); // y = a - bw for x = a + bw where Fp12 = Fp6[w]
void mclBnGT_sqr(mclBnGT *y, const mclBnGT *x);
void mclBnGT_mul(mclBnGT *z, const mclBnGT *x, const mclBnGT *y);
void mclBnGT_div(mclBnGT *z, const mclBnGT *x, const mclBnGT *y);
```
- use `mclBnGT_invGeneric` for an element in Fp12 - GT.
- NOTE: The following functions do NOT return a GT element because GT is multiplicative group.
```
void mclBnGT_neg(mclBnGT *y, const mclBnGT *x);
void mclBnGT_add(mclBnGT *z, const mclBnGT *x, const mclBnGT *y);
void mclBnGT_sub(mclBnGT *z, const mclBnGT *x, const mclBnGT *y);
```
### Square root of `x`.
```
int mclBnFr_squareRoot(mclBnFr *y, const mclBnFr *x);
int mclBnFp_squareRoot(mclBnFp *y, const mclBnFp *x);
int mclBnFp2_squareRoot(mclBnFp2 *y, const mclBnFp2 *x);
```
- `y` is one of square root of `x` if `y` exists.
- return 0 if success else -1
### add / sub / dbl / neg for `G1` and `G2`.
```
void mclBnG1_neg(mclBnG1 *y, const mclBnG1 *x);
void mclBnG1_dbl(mclBnG1 *y, const mclBnG1 *x);
void mclBnG1_add(mclBnG1 *z, const mclBnG1 *x, const mclBnG1 *y);
void mclBnG1_sub(mclBnG1 *z, const mclBnG1 *x, const mclBnG1 *y);
void mclBnG2_neg(mclBnG2 *y, const mclBnG2 *x);
void mclBnG2_dbl(mclBnG2 *y, const mclBnG2 *x);
void mclBnG2_add(mclBnG2 *z, const mclBnG2 *x, const mclBnG2 *y);
void mclBnG2_sub(mclBnG2 *z, const mclBnG2 *x, const mclBnG2 *y);
```
### Convert a point from Jacobi coordinate to affine.
```
void mclBnG1_normalize(mclBnG1 *y, const mclBnG1 *x);
void mclBnG2_normalize(mclBnG2 *y, const mclBnG2 *x);
```
- convert `[x:y:z]` to `[x:y:1]` if `z != 0` else `[*:*:0]`
### scalar multiplication
```
void mclBnG1_mul(mclBnG1 *z, const mclBnG1 *x, const mclBnFr *y);
void mclBnG2_mul(mclBnG2 *z, const mclBnG2 *x, const mclBnFr *y);
void mclBnGT_pow(mclBnGT *z, const mclBnGT *x, const mclBnFr *y);
```
- z = x * y for G1 / G2
- z = pow(x, y) for GT
- use `mclBnGT_powGeneric` for an element in Fp12 - GT.
### multi scalar multiplication
```
void mclBnG1_mulVec(mclBnG1 *z, const mclBnG1 *x, const mclBnFr *y, mclSize n);
void mclBnG2_mulVec(mclBnG2 *z, const mclBnG2 *x, const mclBnFr *y, mclSize n);
void mclBnGT_powVec(mclBnGT *z, const mclBnGT *x, const mclBnFr *y, mclSize n);
```
- z = sum_{i=0}^{n-1} mul(x[i], y[i]) for G1 / G2.
- z = prod_{i=0}^{n-1} pow(x[i], y[i]) for GT.
## hash and mapTo functions
### Set hash of `buf[0..bufSize-1]` to `x`
```
int mclBnFr_setHashOf(mclBnFr *x, const void *buf, mclSize bufSize);
int mclBnFp_setHashOf(mclBnFp *x, const void *buf, mclSize bufSize);
```
- always return 0
- use SHA-256 if sizeof(*x) <= 256 else SHA-512
- set accoring to the same way as `setLittleEndian`
- support the other wasy if you want in the future
### map `x` to G1 / G2.
```
int mclBnFp_mapToG1(mclBnG1 *y, const mclBnFp *x);
int mclBnFp2_mapToG2(mclBnG2 *y, const mclBnFp2 *x);
```
- See `struct MapTo` in `mcl/bn.hpp` for the detail of the algorithm.
- return 0 if success else -1
### hash and map to G1 / G2.
```
int mclBnG1_hashAndMapTo(mclBnG1 *x, const void *buf, mclSize bufSize);
int mclBnG2_hashAndMapTo(mclBnG2 *x, const void *buf, mclSize bufSize);
```
- Combine `setHashOf` and `mapTo` functions
## Pairing operations
The pairing function `e(P, Q)` is consist of two parts:
- `MillerLoop(P, Q)`
- `finalExp(x)`
`finalExp` satisfies the following properties:
- `e(P, Q) = finalExp(MillerLoop(P, Q))`
- `e(P1, Q1) e(P2, Q2) = finalExp(MillerLoop(P1, Q1) MillerLoop(P2, Q2))`
### pairing
```
void mclBn_pairing(mclBnGT *z, const mclBnG1 *x, const mclBnG2 *y);
```
### millerLoop
```
void mclBn_millerLoop(mclBnGT *z, const mclBnG1 *x, const mclBnG2 *y);
```
### finalExp
```
void mclBn_finalExp(mclBnGT *y, const mclBnGT *x);
```
## Variants of MillerLoop
### multi pairing
```
void mclBn_millerLoopVec(mclBnGT *z, const mclBnG1 *x, const mclBnG2 *y, mclSize n);
```
- This function is for multi-pairing
- computes prod_{i=0}^{n-1} MillerLoop(x[i], y[i])
- prod_{i=0}^{n-1} e(x[i], y[i]) = finalExp(prod_{i=0}^{n-1} MillerLoop(x[i], y[i]))
### pairing for a fixed point of G2
```
int mclBn_getUint64NumToPrecompute(void);
void mclBn_precomputeG2(uint64_t *Qbuf, const mclBnG2 *Q);
void mclBn_precomputedMillerLoop(mclBnGT *f, const mclBnG1 *P, const uint64_t *Qbuf);
```
These functions is the same computation of `pairing(P, Q);` as the followings:
```
uint64_t *Qbuf = (uint64_t*)malloc(mclBn_getUint64NumToPrecompute() * sizeof(uint64_t));
mclBn_precomputeG2(Qbuf, Q); // precomputing of Q
mclBn_precomputedMillerLoop(f, P, Qbuf); // pairing of any P of G1 and the fixed Q
free(p);
```
```
void mclBn_precomputedMillerLoop2(
mclBnGT *f,
const mclBnG1 *P1, const uint64_t *Q1buf,
const mclBnG1 *P2, const uint64_t *Q2buf
);
```
- compute `MillerLoop(P1, Q1buf) * MillerLoop(P2, Q2buf)`
```
void mclBn_precomputedMillerLoop2mixed(
mclBnGT *f,
const mclBnG1 *P1, const mclBnG2 *Q1,
const mclBnG1 *P2, const uint64_t *Q2buf
);
```
- compute `MillerLoop(P1, Q2) * MillerLoop(P2, Q2buf)`
## Check value
### Check validness
```
int mclBnFr_isValid(const mclBnFr *x);
int mclBnFp_isValid(const mclBnFp *x);
int mclBnG1_isValid(const mclBnG1 *x);
int mclBnG2_isValid(const mclBnG2 *x);
```
- return 1 if true else 0
### Check the order of a point
```
int mclBnG1_isValidOrder(const mclBnG1 *x);
int mclBnG2_isValidOrder(const mclBnG2 *x);
```
- Check whether the order of `x` is valid or not
- return 1 if true else 0
- This function always cheks according to `mclBn_verifyOrderG1` and `mclBn_verifyOrderG2`.
### Is equal / zero / one / isOdd
```
int mclBnFr_isEqual(const mclBnFr *x, const mclBnFr *y);
int mclBnFr_isZero(const mclBnFr *x);
int mclBnFr_isOne(const mclBnFr *x);
int mclBnFr_isOdd(const mclBnFr *x);
int mclBnFp_isEqual(const mclBnFp *x, const mclBnFp *y);
int mclBnFp_isZero(const mclBnFp *x);
int mclBnFp_isOne(const mclBnFp *x);
int mclBnFp_isOdd(const mclBnFp *x);
int mclBnFp2_isEqual(const mclBnFp2 *x, const mclBnFp2 *y);
int mclBnFp2_isZero(const mclBnFp2 *x);
int mclBnFp2_isOne(const mclBnFp2 *x);
int mclBnG1_isEqual(const mclBnG1 *x, const mclBnG1 *y);
int mclBnG1_isZero(const mclBnG1 *x);
int mclBnG2_isEqual(const mclBnG2 *x, const mclBnG2 *y);
int mclBnG2_isZero(const mclBnG2 *x);
int mclBnGT_isEqual(const mclBnGT *x, const mclBnGT *y);
int mclBnGT_isZero(const mclBnGT *x);
int mclBnGT_isOne(const mclBnGT *x);
```
- return 1 if true else 0
### isNegative
```
int mclBnFr_isNegative(const mclBnFr *x);
int mclBnFp_isNegative(const mclBnFr *x);
```
return 1 if x >= half where half = (r + 1) / 2 (resp. (p + 1) / 2).
## Lagrange interpolation
```
int mclBn_FrLagrangeInterpolation(mclBnFr *out, const mclBnFr *xVec, const mclBnFr *yVec, mclSize k);
int mclBn_G1LagrangeInterpolation(mclBnG1 *out, const mclBnFr *xVec, const mclBnG1 *yVec, mclSize k);
int mclBn_G2LagrangeInterpolation(mclBnG2 *out, const mclBnFr *xVec, const mclBnG2 *yVec, mclSize k);
```
- Lagrange interpolation
- recover out = y(0) from {(xVec[i], yVec[i])} for {i=0..k-1}
- return 0 if success else -1
- satisfy that xVec[i] != 0, xVec[i] != xVec[j] for i != j
```
int mclBn_FrEvaluatePolynomial(mclBnFr *out, const mclBnFr *cVec, mclSize cSize, const mclBnFr *x);
int mclBn_G1EvaluatePolynomial(mclBnG1 *out, const mclBnG1 *cVec, mclSize cSize, const mclBnFr *x);
int mclBn_G2EvaluatePolynomial(mclBnG2 *out, const mclBnG2 *cVec, mclSize cSize, const mclBnFr *x);
```
- Evaluate polynomial
- out = f(x) = c[0] + c[1] * x + ... + c[cSize - 1] * x^{cSize - 1}
- return 0 if success else -1
- satisfy cSize >= 1