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Minimum sample

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.


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
p = 0x2523648240000001ba344d80000000086121000000000013a700000000000013
BLS12-381 4 r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
BN381 2 r = 0x240026400f3d82b2e42de125b00158405b710818ac000007e0042f008e3e00000000001080046200000000000000000d
p = 0x240026400f3d82b2e42de125b00158405b710818ac00000840046200950400000000001380052e000000000000000013



This is a struct of Fp. The value is stored as Montgomery representation.


This is a struct of Fr. The value is stored as Montgomery representation.


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.


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.


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.


This is a struct of GT which has a member mclBnFp d[12].


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.


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

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 1.
  • The cost of verification is not small, so set doVerify = 0 carefully if necessary.
  • This is not thread safe.

Setter / Getter


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.



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
  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 (experimental)

void mclBn_setETHserialization(int ETHserialization);


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 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))


void mclBn_pairing(mclBnGT *z, const mclBnG1 *x, const mclBnG2 *y);


void mclBn_millerLoop(mclBnGT *z, const mclBnG1 *x, const mclBnG2 *y);


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
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


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
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