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bls12-381.cuh
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#ifndef BLS12_381_CUH
#define BLS12_381_CUH
// Defines to make the code work with both, CUDA and OpenCL
#ifdef __NVCC__
#define DEVICE __device__
#define GLOBAL
#define KERNEL extern "C" __global__
#define LOCAL __shared__
#define CONSTANT __constant__
#define GET_GLOBAL_ID() blockIdx.x * blockDim.x + threadIdx.x
#define GET_GROUP_ID() blockIdx.x
#define GET_LOCAL_ID() threadIdx.x
#define GET_LOCAL_SIZE() blockDim.x
#define BARRIER_LOCAL() __syncthreads()
typedef unsigned char uchar;
#define BLS12_381_CUH_CUDA
#else // OpenCL
#define DEVICE
#define GLOBAL __global
#define KERNEL __kernel
#define LOCAL __local
#define CONSTANT __constant
#define GET_GLOBAL_ID() get_global_id(0)
#define GET_GROUP_ID() get_group_id(0)
#define GET_LOCAL_ID() get_local_id(0)
#define GET_LOCAL_SIZE() get_local_size(0)
#define BARRIER_LOCAL() barrier(CLK_LOCAL_MEM_FENCE)
#endif
#ifdef __NV_CL_C_VERSION
#define BLS12_381_CUH_OPENCL_NVIDIA
#endif
#if defined(__WinterPark__) || defined(__BeaverCreek__) || defined(__Turks__) || \
defined(__Caicos__) || defined(__Tahiti__) || defined(__Pitcairn__) || \
defined(__Capeverde__) || defined(__Cayman__) || defined(__Barts__) || \
defined(__Cypress__) || defined(__Juniper__) || defined(__Redwood__) || \
defined(__Cedar__) || defined(__ATI_RV770__) || defined(__ATI_RV730__) || \
defined(__ATI_RV710__) || defined(__Loveland__) || defined(__GPU__) || \
defined(__Hawaii__)
#define AMD
#endif
// Returns a * b + c + d, puts the carry in d
DEVICE ulong mac_with_carry_64(ulong a, ulong b, ulong c, ulong *d) {
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
ulong lo, hi;
asm("mad.lo.cc.u64 %0, %2, %3, %4;\r\n"
"madc.hi.u64 %1, %2, %3, 0;\r\n"
"add.cc.u64 %0, %0, %5;\r\n"
"addc.u64 %1, %1, 0;\r\n"
: "=l"(lo), "=l"(hi) : "l"(a), "l"(b), "l"(c), "l"(*d));
*d = hi;
return lo;
#else
ulong lo = a * b + c;
ulong hi = mad_hi(a, b, (ulong)(lo < c));
a = lo;
lo += *d;
hi += (lo < a);
*d = hi;
return lo;
#endif
}
// Returns a + b, puts the carry in d
DEVICE ulong add_with_carry_64(ulong a, ulong *b) {
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
ulong lo, hi;
asm("add.cc.u64 %0, %2, %3;\r\n"
"addc.u64 %1, 0, 0;\r\n"
: "=l"(lo), "=l"(hi) : "l"(a), "l"(*b));
*b = hi;
return lo;
#else
ulong lo = a + *b;
*b = lo < a;
return lo;
#endif
}
// Returns a * b + c + d, puts the carry in d
DEVICE uint mac_with_carry_32(uint a, uint b, uint c, uint *d) {
ulong res = (ulong)a * b + c + *d;
*d = res >> 32;
return res;
}
// Returns a + b, puts the carry in b
DEVICE uint add_with_carry_32(uint a, uint *b) {
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
uint lo, hi;
asm("add.cc.u32 %0, %2, %3;\r\n"
"addc.u32 %1, 0, 0;\r\n"
: "=r"(lo), "=r"(hi) : "r"(a), "r"(*b));
*b = hi;
return lo;
#else
uint lo = a + *b;
*b = lo < a;
return lo;
#endif
}
// Reverse the given bits. It's used by the FFT kernel.
DEVICE uint bitreverse(uint n, uint bits) {
uint r = 0;
for(int i = 0; i < bits; i++) {
r = (r << 1) | (n & 1);
n >>= 1;
}
return r;
}
#ifdef BLS12_381_CUH_CUDA
// CUDA doesn't support local buffers ("dynamic shared memory" in CUDA lingo) as function
// arguments, but only a single globally defined extern value. Use `uchar` so that it is always
// allocated by the number of bytes.
extern LOCAL uchar cuda_shared[];
typedef uint uint32_t;
typedef int int32_t;
typedef uint limb;
DEVICE inline uint32_t add_cc(uint32_t a, uint32_t b) {
uint32_t r;
asm volatile ("add.cc.u32 %0, %1, %2;" : "=r"(r) : "r"(a), "r"(b));
return r;
}
DEVICE inline uint32_t addc_cc(uint32_t a, uint32_t b) {
uint32_t r;
asm volatile ("addc.cc.u32 %0, %1, %2;" : "=r"(r) : "r"(a), "r"(b));
return r;
}
DEVICE inline uint32_t addc(uint32_t a, uint32_t b) {
uint32_t r;
asm volatile ("addc.u32 %0, %1, %2;" : "=r"(r) : "r"(a), "r"(b));
return r;
}
DEVICE inline uint32_t madlo(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("mad.lo.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madlo_cc(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("mad.lo.cc.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madloc_cc(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("madc.lo.cc.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madloc(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("madc.lo.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madhi(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("mad.hi.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madhi_cc(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("mad.hi.cc.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madhic_cc(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("madc.hi.cc.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
DEVICE inline uint32_t madhic(uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
asm volatile ("madc.hi.u32 %0, %1, %2, %3;" : "=r"(r) : "r"(a), "r"(b), "r"(c));
return r;
}
typedef struct {
int32_t _position;
} chain_t;
DEVICE inline
void chain_init(chain_t *c) {
c->_position = 0;
}
DEVICE inline
uint32_t chain_add(chain_t *ch, uint32_t a, uint32_t b) {
uint32_t r;
ch->_position++;
if(ch->_position==1)
r=add_cc(a, b);
else
r=addc_cc(a, b);
return r;
}
DEVICE inline
uint32_t chain_madlo(chain_t *ch, uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
ch->_position++;
if(ch->_position==1)
r=madlo_cc(a, b, c);
else
r=madloc_cc(a, b, c);
return r;
}
DEVICE inline
uint32_t chain_madhi(chain_t *ch, uint32_t a, uint32_t b, uint32_t c) {
uint32_t r;
ch->_position++;
if(ch->_position==1)
r=madhi_cc(a, b, c);
else
r=madhic_cc(a, b, c);
return r;
}
#endif
#define blstrs__scalar__Scalar_limb uint
#define blstrs__scalar__Scalar_LIMBS 8
#define blstrs__scalar__Scalar_LIMB_BITS 32
#define blstrs__scalar__Scalar_INV 4294967295
typedef struct { blstrs__scalar__Scalar_limb val[blstrs__scalar__Scalar_LIMBS]; } blstrs__scalar__Scalar;
CONSTANT blstrs__scalar__Scalar blstrs__scalar__Scalar_ONE = { { 4294967294, 1, 215042, 1485092858, 3971764213, 2576109551, 2898593135, 405057881 } };
CONSTANT blstrs__scalar__Scalar blstrs__scalar__Scalar_P = { { 1, 4294967295, 4294859774, 1404937218, 161601541, 859428872, 698187080, 1944954707 } };
CONSTANT blstrs__scalar__Scalar blstrs__scalar__Scalar_R2 = { { 4092763245, 3382307216, 2274516003, 728559051, 1918122383, 97719446, 2673475345, 122214873 } };
CONSTANT blstrs__scalar__Scalar blstrs__scalar__Scalar_ZERO = { { 0, 0, 0, 0, 0, 0, 0, 0 } };
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_sub_nvidia(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
asm("sub.cc.u32 %0, %0, %8;\r\n"
"subc.cc.u32 %1, %1, %9;\r\n"
"subc.cc.u32 %2, %2, %10;\r\n"
"subc.cc.u32 %3, %3, %11;\r\n"
"subc.cc.u32 %4, %4, %12;\r\n"
"subc.cc.u32 %5, %5, %13;\r\n"
"subc.cc.u32 %6, %6, %14;\r\n"
"subc.u32 %7, %7, %15;\r\n"
:"+r"(a.val[0]), "+r"(a.val[1]), "+r"(a.val[2]), "+r"(a.val[3]), "+r"(a.val[4]), "+r"(a.val[5]), "+r"(a.val[6]), "+r"(a.val[7])
:"r"(b.val[0]), "r"(b.val[1]), "r"(b.val[2]), "r"(b.val[3]), "r"(b.val[4]), "r"(b.val[5]), "r"(b.val[6]), "r"(b.val[7]));
return a;
}
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_add_nvidia(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
asm("add.cc.u32 %0, %0, %8;\r\n"
"addc.cc.u32 %1, %1, %9;\r\n"
"addc.cc.u32 %2, %2, %10;\r\n"
"addc.cc.u32 %3, %3, %11;\r\n"
"addc.cc.u32 %4, %4, %12;\r\n"
"addc.cc.u32 %5, %5, %13;\r\n"
"addc.cc.u32 %6, %6, %14;\r\n"
"addc.u32 %7, %7, %15;\r\n"
:"+r"(a.val[0]), "+r"(a.val[1]), "+r"(a.val[2]), "+r"(a.val[3]), "+r"(a.val[4]), "+r"(a.val[5]), "+r"(a.val[6]), "+r"(a.val[7])
:"r"(b.val[0]), "r"(b.val[1]), "r"(b.val[2]), "r"(b.val[3]), "r"(b.val[4]), "r"(b.val[5]), "r"(b.val[6]), "r"(b.val[7]));
return a;
}
#endif
// FinalityLabs - 2019
// Arbitrary size prime-field arithmetic library (add, sub, mul, pow)
#define blstrs__scalar__Scalar_BITS (blstrs__scalar__Scalar_LIMBS * blstrs__scalar__Scalar_LIMB_BITS)
#if blstrs__scalar__Scalar_LIMB_BITS == 32
#define blstrs__scalar__Scalar_mac_with_carry mac_with_carry_32
#define blstrs__scalar__Scalar_add_with_carry add_with_carry_32
#elif blstrs__scalar__Scalar_LIMB_BITS == 64
#define blstrs__scalar__Scalar_mac_with_carry mac_with_carry_64
#define blstrs__scalar__Scalar_add_with_carry add_with_carry_64
#endif
// Greater than or equal
DEVICE bool blstrs__scalar__Scalar_gte(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
for(char i = blstrs__scalar__Scalar_LIMBS - 1; i >= 0; i--){
if(a.val[i] > b.val[i])
return true;
if(a.val[i] < b.val[i])
return false;
}
return true;
}
// Equals
DEVICE bool blstrs__scalar__Scalar_eq(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
for(uchar i = 0; i < blstrs__scalar__Scalar_LIMBS; i++)
if(a.val[i] != b.val[i])
return false;
return true;
}
// Normal addition
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
#define blstrs__scalar__Scalar_add_ blstrs__scalar__Scalar_add_nvidia
#define blstrs__scalar__Scalar_sub_ blstrs__scalar__Scalar_sub_nvidia
#else
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_add_(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
bool carry = 0;
for(uchar i = 0; i < blstrs__scalar__Scalar_LIMBS; i++) {
blstrs__scalar__Scalar_limb old = a.val[i];
a.val[i] += b.val[i] + carry;
carry = carry ? old >= a.val[i] : old > a.val[i];
}
return a;
}
blstrs__scalar__Scalar blstrs__scalar__Scalar_sub_(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
bool borrow = 0;
for(uchar i = 0; i < blstrs__scalar__Scalar_LIMBS; i++) {
blstrs__scalar__Scalar_limb old = a.val[i];
a.val[i] -= b.val[i] + borrow;
borrow = borrow ? old <= a.val[i] : old < a.val[i];
}
return a;
}
#endif
// Modular subtraction
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_sub(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
blstrs__scalar__Scalar res = blstrs__scalar__Scalar_sub_(a, b);
if(!blstrs__scalar__Scalar_gte(a, b)) res = blstrs__scalar__Scalar_add_(res, blstrs__scalar__Scalar_P);
return res;
}
// Modular addition
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_add(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
blstrs__scalar__Scalar res = blstrs__scalar__Scalar_add_(a, b);
if(blstrs__scalar__Scalar_gte(res, blstrs__scalar__Scalar_P)) res = blstrs__scalar__Scalar_sub_(res, blstrs__scalar__Scalar_P);
return res;
}
#ifdef BLS12_381_CUH_CUDA
// Code based on the work from Supranational, with special thanks to Niall Emmart:
//
// We would like to acknowledge Niall Emmart at Nvidia for his significant
// contribution of concepts and code for generating efficient SASS on
// Nvidia GPUs. The following papers may be of interest:
// Optimizing Modular Multiplication for NVIDIA's Maxwell GPUs
// https://ieeexplore.ieee.org/document/7563271
//
// Faster modular exponentiation using double precision floating point
// arithmetic on the GPU
// https://ieeexplore.ieee.org/document/8464792
DEVICE void blstrs__scalar__Scalar_reduce(uint32_t accLow[blstrs__scalar__Scalar_LIMBS], uint32_t np0, uint32_t fq[blstrs__scalar__Scalar_LIMBS]) {
// accLow is an IN and OUT vector
// , i must be even
const uint32_t count = blstrs__scalar__Scalar_LIMBS;
uint32_t accHigh[blstrs__scalar__Scalar_LIMBS];
uint32_t bucket=0, lowCarry=0, highCarry=0, q;
int32_t i, j;
#pragma unroll
for(i=0;i<count;i++)
accHigh[i]=0;
// bucket is used so we don't have to push a carry all the way down the line
#pragma unroll
for(j=0;j<count;j++) { // main iteration
if(j%2==0) {
add_cc(bucket, 0xFFFFFFFF);
accLow[0]=addc_cc(accLow[0], accHigh[1]);
bucket=addc(0, 0);
q=accLow[0]*np0;
chain_t chain1;
chain_init(&chain1);
#pragma unroll
for(i=0;i<count;i+=2) {
accLow[i]=chain_madlo(&chain1, q, fq[i], accLow[i]);
accLow[i+1]=chain_madhi(&chain1, q, fq[i], accLow[i+1]);
}
lowCarry=chain_add(&chain1, 0, 0);
chain_t chain2;
chain_init(&chain2);
for(i=0;i<count-2;i+=2) {
accHigh[i]=chain_madlo(&chain2, q, fq[i+1], accHigh[i+2]); // note the shift down
accHigh[i+1]=chain_madhi(&chain2, q, fq[i+1], accHigh[i+3]);
}
accHigh[i]=chain_madlo(&chain2, q, fq[i+1], highCarry);
accHigh[i+1]=chain_madhi(&chain2, q, fq[i+1], 0);
}
else {
add_cc(bucket, 0xFFFFFFFF);
accHigh[0]=addc_cc(accHigh[0], accLow[1]);
bucket=addc(0, 0);
q=accHigh[0]*np0;
chain_t chain3;
chain_init(&chain3);
#pragma unroll
for(i=0;i<count;i+=2) {
accHigh[i]=chain_madlo(&chain3, q, fq[i], accHigh[i]);
accHigh[i+1]=chain_madhi(&chain3, q, fq[i], accHigh[i+1]);
}
highCarry=chain_add(&chain3, 0, 0);
chain_t chain4;
chain_init(&chain4);
for(i=0;i<count-2;i+=2) {
accLow[i]=chain_madlo(&chain4, q, fq[i+1], accLow[i+2]); // note the shift down
accLow[i+1]=chain_madhi(&chain4, q, fq[i+1], accLow[i+3]);
}
accLow[i]=chain_madlo(&chain4, q, fq[i+1], lowCarry);
accLow[i+1]=chain_madhi(&chain4, q, fq[i+1], 0);
}
}
// at this point, accHigh needs to be shifted back a word and added to accLow
// we'll use one other trick. Bucket is either 0 or 1 at this point, so we
// can just push it into the carry chain.
chain_t chain5;
chain_init(&chain5);
chain_add(&chain5, bucket, 0xFFFFFFFF); // push the carry into the chain
#pragma unroll
for(i=0;i<count-1;i++)
accLow[i]=chain_add(&chain5, accLow[i], accHigh[i+1]);
accLow[i]=chain_add(&chain5, accLow[i], highCarry);
}
// Requirement: yLimbs >= xLimbs
DEVICE inline
void blstrs__scalar__Scalar_mult_v1(uint32_t *x, uint32_t *y, uint32_t *xy) {
const uint32_t xLimbs = blstrs__scalar__Scalar_LIMBS;
const uint32_t yLimbs = blstrs__scalar__Scalar_LIMBS;
const uint32_t xyLimbs = blstrs__scalar__Scalar_LIMBS * 2;
uint32_t temp[blstrs__scalar__Scalar_LIMBS * 2];
uint32_t carry = 0;
#pragma unroll
for (int32_t i = 0; i < xyLimbs; i++) {
temp[i] = 0;
}
#pragma unroll
for (int32_t i = 0; i < xLimbs; i++) {
chain_t chain1;
chain_init(&chain1);
#pragma unroll
for (int32_t j = 0; j < yLimbs; j++) {
if ((i + j) % 2 == 1) {
temp[i + j - 1] = chain_madlo(&chain1, x[i], y[j], temp[i + j - 1]);
temp[i + j] = chain_madhi(&chain1, x[i], y[j], temp[i + j]);
}
}
if (i % 2 == 1) {
temp[i + yLimbs - 1] = chain_add(&chain1, 0, 0);
}
}
#pragma unroll
for (int32_t i = xyLimbs - 1; i > 0; i--) {
temp[i] = temp[i - 1];
}
temp[0] = 0;
#pragma unroll
for (int32_t i = 0; i < xLimbs; i++) {
chain_t chain2;
chain_init(&chain2);
#pragma unroll
for (int32_t j = 0; j < yLimbs; j++) {
if ((i + j) % 2 == 0) {
temp[i + j] = chain_madlo(&chain2, x[i], y[j], temp[i + j]);
temp[i + j + 1] = chain_madhi(&chain2, x[i], y[j], temp[i + j + 1]);
}
}
if ((i + yLimbs) % 2 == 0 && i != yLimbs - 1) {
temp[i + yLimbs] = chain_add(&chain2, temp[i + yLimbs], carry);
temp[i + yLimbs + 1] = chain_add(&chain2, temp[i + yLimbs + 1], 0);
carry = chain_add(&chain2, 0, 0);
}
if ((i + yLimbs) % 2 == 1 && i != yLimbs - 1) {
carry = chain_add(&chain2, carry, 0);
}
}
#pragma unroll
for(int32_t i = 0; i < xyLimbs; i++) {
xy[i] = temp[i];
}
}
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_mul_nvidia(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
// Perform full multiply
limb ab[2 * blstrs__scalar__Scalar_LIMBS];
blstrs__scalar__Scalar_mult_v1(a.val, b.val, ab);
uint32_t io[blstrs__scalar__Scalar_LIMBS];
#pragma unroll
for(int i=0;i<blstrs__scalar__Scalar_LIMBS;i++) {
io[i]=ab[i];
}
blstrs__scalar__Scalar_reduce(io, blstrs__scalar__Scalar_INV, blstrs__scalar__Scalar_P.val);
// Add io to the upper words of ab
ab[blstrs__scalar__Scalar_LIMBS] = add_cc(ab[blstrs__scalar__Scalar_LIMBS], io[0]);
int j;
#pragma unroll
for (j = 1; j < blstrs__scalar__Scalar_LIMBS - 1; j++) {
ab[j + blstrs__scalar__Scalar_LIMBS] = addc_cc(ab[j + blstrs__scalar__Scalar_LIMBS], io[j]);
}
ab[2 * blstrs__scalar__Scalar_LIMBS - 1] = addc(ab[2 * blstrs__scalar__Scalar_LIMBS - 1], io[blstrs__scalar__Scalar_LIMBS - 1]);
blstrs__scalar__Scalar r;
#pragma unroll
for (int i = 0; i < blstrs__scalar__Scalar_LIMBS; i++) {
r.val[i] = ab[i + blstrs__scalar__Scalar_LIMBS];
}
if (blstrs__scalar__Scalar_gte(r, blstrs__scalar__Scalar_P)) {
r = blstrs__scalar__Scalar_sub_(r, blstrs__scalar__Scalar_P);
}
return r;
}
#endif
// Modular multiplication
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_mul_default(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
/* CIOS Montgomery multiplication, inspired from Tolga Acar's thesis:
* https://www.microsoft.com/en-us/research/wp-content/uploads/1998/06/97Acar.pdf
* Learn more:
* https://en.wikipedia.org/wiki/Montgomery_modular_multiplication
* https://alicebob.cryptoland.net/understanding-the-montgomery-reduction-algorithm/
*/
blstrs__scalar__Scalar_limb t[blstrs__scalar__Scalar_LIMBS + 2] = {0};
for(uchar i = 0; i < blstrs__scalar__Scalar_LIMBS; i++) {
blstrs__scalar__Scalar_limb carry = 0;
for(uchar j = 0; j < blstrs__scalar__Scalar_LIMBS; j++)
t[j] = blstrs__scalar__Scalar_mac_with_carry(a.val[j], b.val[i], t[j], &carry);
t[blstrs__scalar__Scalar_LIMBS] = blstrs__scalar__Scalar_add_with_carry(t[blstrs__scalar__Scalar_LIMBS], &carry);
t[blstrs__scalar__Scalar_LIMBS + 1] = carry;
carry = 0;
blstrs__scalar__Scalar_limb m = blstrs__scalar__Scalar_INV * t[0];
blstrs__scalar__Scalar_mac_with_carry(m, blstrs__scalar__Scalar_P.val[0], t[0], &carry);
for(uchar j = 1; j < blstrs__scalar__Scalar_LIMBS; j++)
t[j - 1] = blstrs__scalar__Scalar_mac_with_carry(m, blstrs__scalar__Scalar_P.val[j], t[j], &carry);
t[blstrs__scalar__Scalar_LIMBS - 1] = blstrs__scalar__Scalar_add_with_carry(t[blstrs__scalar__Scalar_LIMBS], &carry);
t[blstrs__scalar__Scalar_LIMBS] = t[blstrs__scalar__Scalar_LIMBS + 1] + carry;
}
blstrs__scalar__Scalar result;
for(uchar i = 0; i < blstrs__scalar__Scalar_LIMBS; i++) result.val[i] = t[i];
if(blstrs__scalar__Scalar_gte(result, blstrs__scalar__Scalar_P)) result = blstrs__scalar__Scalar_sub_(result, blstrs__scalar__Scalar_P);
return result;
}
#ifdef BLS12_381_CUH_CUDA
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_mul(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
return blstrs__scalar__Scalar_mul_nvidia(a, b);
}
#else
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_mul(blstrs__scalar__Scalar a, blstrs__scalar__Scalar b) {
return blstrs__scalar__Scalar_mul_default(a, b);
}
#endif
// Squaring is a special case of multiplication which can be done ~1.5x faster.
// https://stackoverflow.com/a/16388571/1348497
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_sqr(blstrs__scalar__Scalar a) {
return blstrs__scalar__Scalar_mul(a, a);
}
// Left-shift the limbs by one bit and subtract by modulus in case of overflow.
// Faster version of blstrs__scalar__Scalar_add(a, a)
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_double(blstrs__scalar__Scalar a) {
for(uchar i = blstrs__scalar__Scalar_LIMBS - 1; i >= 1; i--)
a.val[i] = (a.val[i] << 1) | (a.val[i - 1] >> (blstrs__scalar__Scalar_LIMB_BITS - 1));
a.val[0] <<= 1;
if(blstrs__scalar__Scalar_gte(a, blstrs__scalar__Scalar_P)) a = blstrs__scalar__Scalar_sub_(a, blstrs__scalar__Scalar_P);
return a;
}
// Modular exponentiation (Exponentiation by Squaring)
// https://en.wikipedia.org/wiki/Exponentiation_by_squaring
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_pow(blstrs__scalar__Scalar base, uint exponent) {
blstrs__scalar__Scalar res = blstrs__scalar__Scalar_ONE;
while(exponent > 0) {
if (exponent & 1)
res = blstrs__scalar__Scalar_mul(res, base);
exponent = exponent >> 1;
base = blstrs__scalar__Scalar_sqr(base);
}
return res;
}
// Store squares of the base in a lookup table for faster evaluation.
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_pow_lookup(GLOBAL blstrs__scalar__Scalar *bases, uint exponent) {
blstrs__scalar__Scalar res = blstrs__scalar__Scalar_ONE;
uint i = 0;
while(exponent > 0) {
if (exponent & 1)
res = blstrs__scalar__Scalar_mul(res, bases[i]);
exponent = exponent >> 1;
i++;
}
return res;
}
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_mont(blstrs__scalar__Scalar a) {
return blstrs__scalar__Scalar_mul(a, blstrs__scalar__Scalar_R2);
}
DEVICE blstrs__scalar__Scalar blstrs__scalar__Scalar_unmont(blstrs__scalar__Scalar a) {
blstrs__scalar__Scalar one = blstrs__scalar__Scalar_ZERO;
one.val[0] = 1;
return blstrs__scalar__Scalar_mul(a, one);
}
// Get `i`th bit (From most significant digit) of the field.
DEVICE bool blstrs__scalar__Scalar_get_bit(blstrs__scalar__Scalar l, uint i) {
return (l.val[blstrs__scalar__Scalar_LIMBS - 1 - i / blstrs__scalar__Scalar_LIMB_BITS] >> (blstrs__scalar__Scalar_LIMB_BITS - 1 - (i % blstrs__scalar__Scalar_LIMB_BITS))) & 1;
}
// Get `window` consecutive bits, (Starting from `skip`th bit) from the field.
DEVICE uint blstrs__scalar__Scalar_get_bits(blstrs__scalar__Scalar l, uint skip, uint window) {
uint ret = 0;
for(uint i = 0; i < window; i++) {
ret <<= 1;
ret |= blstrs__scalar__Scalar_get_bit(l, skip + i);
}
return ret;
}
#define blstrs__fp__Fp_limb uint
#define blstrs__fp__Fp_LIMBS 12
#define blstrs__fp__Fp_LIMB_BITS 32
#define blstrs__fp__Fp_INV 4294770685
typedef struct { blstrs__fp__Fp_limb val[blstrs__fp__Fp_LIMBS]; } blstrs__fp__Fp;
CONSTANT blstrs__fp__Fp blstrs__fp__Fp_ONE = { { 196605, 1980301312, 3289120770, 3958636555, 1405573306, 1598593111, 1884444485, 2010011731, 2723605613, 1543969431, 4202751123, 368467651 } };
CONSTANT blstrs__fp__Fp blstrs__fp__Fp_P = { { 4294945451, 3120496639, 2975072255, 514588670, 4138792484, 1731252896, 4085584575, 1685539716, 1129032919, 1260103606, 964683418, 436277738 } };
CONSTANT blstrs__fp__Fp blstrs__fp__Fp_R2 = { { 473175878, 4108263220, 164693233, 175564454, 1284880085, 2380613484, 2476573632, 1743489193, 3038352685, 2591637125, 2462770090, 295210981 } };
CONSTANT blstrs__fp__Fp blstrs__fp__Fp_ZERO = { { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 } };
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
DEVICE blstrs__fp__Fp blstrs__fp__Fp_sub_nvidia(blstrs__fp__Fp a, blstrs__fp__Fp b) {
asm("sub.cc.u32 %0, %0, %12;\r\n"
"subc.cc.u32 %1, %1, %13;\r\n"
"subc.cc.u32 %2, %2, %14;\r\n"
"subc.cc.u32 %3, %3, %15;\r\n"
"subc.cc.u32 %4, %4, %16;\r\n"
"subc.cc.u32 %5, %5, %17;\r\n"
"subc.cc.u32 %6, %6, %18;\r\n"
"subc.cc.u32 %7, %7, %19;\r\n"
"subc.cc.u32 %8, %8, %20;\r\n"
"subc.cc.u32 %9, %9, %21;\r\n"
"subc.cc.u32 %10, %10, %22;\r\n"
"subc.u32 %11, %11, %23;\r\n"
:"+r"(a.val[0]), "+r"(a.val[1]), "+r"(a.val[2]), "+r"(a.val[3]), "+r"(a.val[4]), "+r"(a.val[5]), "+r"(a.val[6]), "+r"(a.val[7]), "+r"(a.val[8]), "+r"(a.val[9]), "+r"(a.val[10]), "+r"(a.val[11])
:"r"(b.val[0]), "r"(b.val[1]), "r"(b.val[2]), "r"(b.val[3]), "r"(b.val[4]), "r"(b.val[5]), "r"(b.val[6]), "r"(b.val[7]), "r"(b.val[8]), "r"(b.val[9]), "r"(b.val[10]), "r"(b.val[11]));
return a;
}
DEVICE blstrs__fp__Fp blstrs__fp__Fp_add_nvidia(blstrs__fp__Fp a, blstrs__fp__Fp b) {
asm("add.cc.u32 %0, %0, %12;\r\n"
"addc.cc.u32 %1, %1, %13;\r\n"
"addc.cc.u32 %2, %2, %14;\r\n"
"addc.cc.u32 %3, %3, %15;\r\n"
"addc.cc.u32 %4, %4, %16;\r\n"
"addc.cc.u32 %5, %5, %17;\r\n"
"addc.cc.u32 %6, %6, %18;\r\n"
"addc.cc.u32 %7, %7, %19;\r\n"
"addc.cc.u32 %8, %8, %20;\r\n"
"addc.cc.u32 %9, %9, %21;\r\n"
"addc.cc.u32 %10, %10, %22;\r\n"
"addc.u32 %11, %11, %23;\r\n"
:"+r"(a.val[0]), "+r"(a.val[1]), "+r"(a.val[2]), "+r"(a.val[3]), "+r"(a.val[4]), "+r"(a.val[5]), "+r"(a.val[6]), "+r"(a.val[7]), "+r"(a.val[8]), "+r"(a.val[9]), "+r"(a.val[10]), "+r"(a.val[11])
:"r"(b.val[0]), "r"(b.val[1]), "r"(b.val[2]), "r"(b.val[3]), "r"(b.val[4]), "r"(b.val[5]), "r"(b.val[6]), "r"(b.val[7]), "r"(b.val[8]), "r"(b.val[9]), "r"(b.val[10]), "r"(b.val[11]));
return a;
}
#endif
// FinalityLabs - 2019
// Arbitrary size prime-field arithmetic library (add, sub, mul, pow)
#define blstrs__fp__Fp_BITS (blstrs__fp__Fp_LIMBS * blstrs__fp__Fp_LIMB_BITS)
#if blstrs__fp__Fp_LIMB_BITS == 32
#define blstrs__fp__Fp_mac_with_carry mac_with_carry_32
#define blstrs__fp__Fp_add_with_carry add_with_carry_32
#elif blstrs__fp__Fp_LIMB_BITS == 64
#define blstrs__fp__Fp_mac_with_carry mac_with_carry_64
#define blstrs__fp__Fp_add_with_carry add_with_carry_64
#endif
// Greater than or equal
DEVICE bool blstrs__fp__Fp_gte(blstrs__fp__Fp a, blstrs__fp__Fp b) {
for(char i = blstrs__fp__Fp_LIMBS - 1; i >= 0; i--){
if(a.val[i] > b.val[i])
return true;
if(a.val[i] < b.val[i])
return false;
}
return true;
}
// Equals
DEVICE bool blstrs__fp__Fp_eq(blstrs__fp__Fp a, blstrs__fp__Fp b) {
for(uchar i = 0; i < blstrs__fp__Fp_LIMBS; i++)
if(a.val[i] != b.val[i])
return false;
return true;
}
// Normal addition
#if defined(BLS12_381_CUH_OPENCL_NVIDIA) || defined(BLS12_381_CUH_CUDA)
#define blstrs__fp__Fp_add_ blstrs__fp__Fp_add_nvidia
#define blstrs__fp__Fp_sub_ blstrs__fp__Fp_sub_nvidia
#else
DEVICE blstrs__fp__Fp blstrs__fp__Fp_add_(blstrs__fp__Fp a, blstrs__fp__Fp b) {
bool carry = 0;
for(uchar i = 0; i < blstrs__fp__Fp_LIMBS; i++) {
blstrs__fp__Fp_limb old = a.val[i];
a.val[i] += b.val[i] + carry;
carry = carry ? old >= a.val[i] : old > a.val[i];
}
return a;
}
blstrs__fp__Fp blstrs__fp__Fp_sub_(blstrs__fp__Fp a, blstrs__fp__Fp b) {
bool borrow = 0;
for(uchar i = 0; i < blstrs__fp__Fp_LIMBS; i++) {
blstrs__fp__Fp_limb old = a.val[i];
a.val[i] -= b.val[i] + borrow;
borrow = borrow ? old <= a.val[i] : old < a.val[i];
}
return a;
}
#endif
// Modular subtraction
DEVICE blstrs__fp__Fp blstrs__fp__Fp_sub(blstrs__fp__Fp a, blstrs__fp__Fp b) {
blstrs__fp__Fp res = blstrs__fp__Fp_sub_(a, b);
if(!blstrs__fp__Fp_gte(a, b)) res = blstrs__fp__Fp_add_(res, blstrs__fp__Fp_P);
return res;
}
// Modular addition
DEVICE blstrs__fp__Fp blstrs__fp__Fp_add(blstrs__fp__Fp a, blstrs__fp__Fp b) {
blstrs__fp__Fp res = blstrs__fp__Fp_add_(a, b);
if(blstrs__fp__Fp_gte(res, blstrs__fp__Fp_P)) res = blstrs__fp__Fp_sub_(res, blstrs__fp__Fp_P);
return res;
}
#ifdef BLS12_381_CUH_CUDA
// Code based on the work from Supranational, with special thanks to Niall Emmart:
//
// We would like to acknowledge Niall Emmart at Nvidia for his significant
// contribution of concepts and code for generating efficient SASS on
// Nvidia GPUs. The following papers may be of interest:
// Optimizing Modular Multiplication for NVIDIA's Maxwell GPUs
// https://ieeexplore.ieee.org/document/7563271
//
// Faster modular exponentiation using double precision floating point
// arithmetic on the GPU
// https://ieeexplore.ieee.org/document/8464792
DEVICE void blstrs__fp__Fp_reduce(uint32_t accLow[blstrs__fp__Fp_LIMBS], uint32_t np0, uint32_t fq[blstrs__fp__Fp_LIMBS]) {
// accLow is an IN and OUT vector
// count must be even
const uint32_t count = blstrs__fp__Fp_LIMBS;
uint32_t accHigh[blstrs__fp__Fp_LIMBS];
uint32_t bucket=0, lowCarry=0, highCarry=0, q;
int32_t i, j;
#pragma unroll
for(i=0;i<count;i++)
accHigh[i]=0;
// bucket is used so we don't have to push a carry all the way down the line
#pragma unroll
for(j=0;j<count;j++) { // main iteration
if(j%2==0) {
add_cc(bucket, 0xFFFFFFFF);
accLow[0]=addc_cc(accLow[0], accHigh[1]);
bucket=addc(0, 0);
q=accLow[0]*np0;
chain_t chain1;
chain_init(&chain1);
#pragma unroll
for(i=0;i<count;i+=2) {
accLow[i]=chain_madlo(&chain1, q, fq[i], accLow[i]);
accLow[i+1]=chain_madhi(&chain1, q, fq[i], accLow[i+1]);
}
lowCarry=chain_add(&chain1, 0, 0);
chain_t chain2;
chain_init(&chain2);
for(i=0;i<count-2;i+=2) {
accHigh[i]=chain_madlo(&chain2, q, fq[i+1], accHigh[i+2]); // note the shift down
accHigh[i+1]=chain_madhi(&chain2, q, fq[i+1], accHigh[i+3]);
}
accHigh[i]=chain_madlo(&chain2, q, fq[i+1], highCarry);
accHigh[i+1]=chain_madhi(&chain2, q, fq[i+1], 0);
}
else {
add_cc(bucket, 0xFFFFFFFF);
accHigh[0]=addc_cc(accHigh[0], accLow[1]);
bucket=addc(0, 0);
q=accHigh[0]*np0;
chain_t chain3;
chain_init(&chain3);
#pragma unroll
for(i=0;i<count;i+=2) {
accHigh[i]=chain_madlo(&chain3, q, fq[i], accHigh[i]);
accHigh[i+1]=chain_madhi(&chain3, q, fq[i], accHigh[i+1]);
}
highCarry=chain_add(&chain3, 0, 0);
chain_t chain4;
chain_init(&chain4);
for(i=0;i<count-2;i+=2) {
accLow[i]=chain_madlo(&chain4, q, fq[i+1], accLow[i+2]); // note the shift down
accLow[i+1]=chain_madhi(&chain4, q, fq[i+1], accLow[i+3]);
}
accLow[i]=chain_madlo(&chain4, q, fq[i+1], lowCarry);
accLow[i+1]=chain_madhi(&chain4, q, fq[i+1], 0);
}
}
// at this point, accHigh needs to be shifted back a word and added to accLow
// we'll use one other trick. Bucket is either 0 or 1 at this point, so we
// can just push it into the carry chain.
chain_t chain5;
chain_init(&chain5);
chain_add(&chain5, bucket, 0xFFFFFFFF); // push the carry into the chain
#pragma unroll
for(i=0;i<count-1;i++)
accLow[i]=chain_add(&chain5, accLow[i], accHigh[i+1]);
accLow[i]=chain_add(&chain5, accLow[i], highCarry);
}
// Requirement: yLimbs >= xLimbs
DEVICE inline
void blstrs__fp__Fp_mult_v1(uint32_t *x, uint32_t *y, uint32_t *xy) {
const uint32_t xLimbs = blstrs__fp__Fp_LIMBS;
const uint32_t yLimbs = blstrs__fp__Fp_LIMBS;
const uint32_t xyLimbs = blstrs__fp__Fp_LIMBS * 2;
uint32_t temp[blstrs__fp__Fp_LIMBS * 2];
uint32_t carry = 0;
#pragma unroll
for (int32_t i = 0; i < xyLimbs; i++) {
temp[i] = 0;
}
#pragma unroll
for (int32_t i = 0; i < xLimbs; i++) {
chain_t chain1;
chain_init(&chain1);
#pragma unroll
for (int32_t j = 0; j < yLimbs; j++) {
if ((i + j) % 2 == 1) {
temp[i + j - 1] = chain_madlo(&chain1, x[i], y[j], temp[i + j - 1]);
temp[i + j] = chain_madhi(&chain1, x[i], y[j], temp[i + j]);
}
}
if (i % 2 == 1) {
temp[i + yLimbs - 1] = chain_add(&chain1, 0, 0);
}
}
#pragma unroll
for (int32_t i = xyLimbs - 1; i > 0; i--) {
temp[i] = temp[i - 1];
}
temp[0] = 0;
#pragma unroll
for (int32_t i = 0; i < xLimbs; i++) {
chain_t chain2;
chain_init(&chain2);
#pragma unroll
for (int32_t j = 0; j < yLimbs; j++) {
if ((i + j) % 2 == 0) {
temp[i + j] = chain_madlo(&chain2, x[i], y[j], temp[i + j]);
temp[i + j + 1] = chain_madhi(&chain2, x[i], y[j], temp[i + j + 1]);
}
}
if ((i + yLimbs) % 2 == 0 && i != yLimbs - 1) {
temp[i + yLimbs] = chain_add(&chain2, temp[i + yLimbs], carry);
temp[i + yLimbs + 1] = chain_add(&chain2, temp[i + yLimbs + 1], 0);
carry = chain_add(&chain2, 0, 0);
}
if ((i + yLimbs) % 2 == 1 && i != yLimbs - 1) {
carry = chain_add(&chain2, carry, 0);
}
}
#pragma unroll
for(int32_t i = 0; i < xyLimbs; i++) {
xy[i] = temp[i];
}
}
DEVICE blstrs__fp__Fp blstrs__fp__Fp_mul_nvidia(blstrs__fp__Fp a, blstrs__fp__Fp b) {
// Perform full multiply
limb ab[2 * blstrs__fp__Fp_LIMBS];
blstrs__fp__Fp_mult_v1(a.val, b.val, ab);
uint32_t io[blstrs__fp__Fp_LIMBS];
#pragma unroll
for(int i=0;i<blstrs__fp__Fp_LIMBS;i++) {
io[i]=ab[i];
}
blstrs__fp__Fp_reduce(io, blstrs__fp__Fp_INV, blstrs__fp__Fp_P.val);
// Add io to the upper words of ab
ab[blstrs__fp__Fp_LIMBS] = add_cc(ab[blstrs__fp__Fp_LIMBS], io[0]);
int j;
#pragma unroll
for (j = 1; j < blstrs__fp__Fp_LIMBS - 1; j++) {
ab[j + blstrs__fp__Fp_LIMBS] = addc_cc(ab[j + blstrs__fp__Fp_LIMBS], io[j]);
}
ab[2 * blstrs__fp__Fp_LIMBS - 1] = addc(ab[2 * blstrs__fp__Fp_LIMBS - 1], io[blstrs__fp__Fp_LIMBS - 1]);
blstrs__fp__Fp r;
#pragma unroll
for (int i = 0; i < blstrs__fp__Fp_LIMBS; i++) {
r.val[i] = ab[i + blstrs__fp__Fp_LIMBS];
}
if (blstrs__fp__Fp_gte(r, blstrs__fp__Fp_P)) {
r = blstrs__fp__Fp_sub_(r, blstrs__fp__Fp_P);
}
return r;
}
#endif
// Modular multiplication
DEVICE blstrs__fp__Fp blstrs__fp__Fp_mul_default(blstrs__fp__Fp a, blstrs__fp__Fp b) {
/* CIOS Montgomery multiplication, inspired from Tolga Acar's thesis:
* https://www.microsoft.com/en-us/research/wp-content/uploads/1998/06/97Acar.pdf
* Learn more:
* https://en.wikipedia.org/wiki/Montgomery_modular_multiplication
* https://alicebob.cryptoland.net/understanding-the-montgomery-reduction-algorithm/
*/
blstrs__fp__Fp_limb t[blstrs__fp__Fp_LIMBS + 2] = {0};
for(uchar i = 0; i < blstrs__fp__Fp_LIMBS; i++) {
blstrs__fp__Fp_limb carry = 0;
for(uchar j = 0; j < blstrs__fp__Fp_LIMBS; j++)
t[j] = blstrs__fp__Fp_mac_with_carry(a.val[j], b.val[i], t[j], &carry);
t[blstrs__fp__Fp_LIMBS] = blstrs__fp__Fp_add_with_carry(t[blstrs__fp__Fp_LIMBS], &carry);
t[blstrs__fp__Fp_LIMBS + 1] = carry;
carry = 0;