This script provides a starting point for implementing elliptic curve point multiplication in CUDA. To achieve high performance, additional optimizations and use of specialized libraries, if available, will be required.

Python script:

import pycuda.driver as cuda
import pycuda.autoinit
from pycuda.compiler import SourceModule
import numpy as np

# Определение параметров кривой secp256k1
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
a = 0x0000000000000000000000000000000000000000000000000000000000000000
b = 0x0000000000000000000000000000000000000000000000000000000000000007
Gx = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
Gy = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8

# CUDA Kernel для сложения точек
mod = SourceModule("""
    __device__
    int add_mod(int a, int b, int p) {
        int sum = a + b;
        if (sum >= p) {
            sum -= p;
        }
        return sum;
    }

    __device__
    int sub_mod(int a, int b, int p) {
        int diff = a - b;
        if (diff < 0) {
            diff += p;
        }
        return diff;
    }

    __device__
    int mul_mod(int a, int b, int p) {
        return (int)(((long long)a * b) % p);
    }

    __device__
    int pow_mod(int base, int exp, int p) {
        int res = 1;
        base %= p;
        while (exp > 0) {
            if (exp % 2 == 1) res = mul_mod(res, base, p);
            base = mul_mod(base, base, p);
            exp /= 2;
        }
        return res;
    }

    __device__
    int inv_mod(int a, int p) {
        return pow_mod(a, p - 2, p);
    }

    __device__
    void point_add(int x1, int y1, int x2, int y2, int p, int a,
                   int *x3, int *y3) {
        if (x1 == x2 && y1 == y2) {
            // Удвоение точки
            int slope = mul_mod(sub_mod(mul_mod(3, pow_mod(x1, 2, p), p), a, p),
                               inv_mod(mul_mod(2, y1, p), p), p);
            *x3 = sub_mod(sub_mod(mul_mod(slope, slope, p), mul_mod(2, x1, p), p),a,p);
            *y3 = sub_mod(mul_mod(slope, sub_mod(x1, *x3, p), p), y1, p);
        } else {
            // Сложение точек
            int slope = mul_mod(sub_mod(y2, y1, p), inv_mod(sub_mod(x2, x1, p), p), p);
            *x3 = sub_mod(sub_mod(mul_mod(slope, slope, p), x1, p), x2, p);
            *y3 = sub_mod(mul_mod(slope, sub_mod(x1, *x3, p), p), y1, p);
        }
    }

    __global__
    void scalar_multiply(int scalar, int Gx, int Gy, int p, int a,
                         int *resultX, int *resultY) {
        int idx = blockIdx.x * blockDim.x + threadIdx.x;
        int x = Gx;
        int y = Gy;
        int x_temp, y_temp;

        for (int i = 1; i < scalar; i++) {
            point_add(x, y, Gx, Gy, p, a, &x_temp, &y_temp);
            x = x_temp;
            y = y_temp;
        }

        resultX[idx] = x;
        resultY[idx] = y;
    }
""")

# Получаем функцию из скомпилированного модуля
scalar_multiply = mod.get_function("scalar_multiply")

def multiply_point(scalar):
    """
    Умножает базовую точку G на скаляр с использованием CUDA.
    """
    # Выделяем память на GPU
    result_x = np.zeros((1), dtype=np.int32)
    result_y = np.zeros((1), dtype=np.int32)
    result_x_gpu = cuda.mem_alloc(result_x.nbytes)
    result_y_gpu = cuda.mem_alloc(result_y.nbytes)

    # Передаем данные на GPU
    cuda.memcpy_htod(result_x_gpu, result_x)
    cuda.memcpy_htod(result_y_gpu, result_y)

    # Запускаем CUDA kernel
    block_size = 1
    grid_size = 1
    scalar_multiply(np.int32(scalar), np.int32(Gx), np.int32(Gy), np.int32(p), np.int32(a),
                    result_x_gpu, result_y_gpu,
                    block=(block_size, 1, 1), grid=(grid_size, 1, 1))

    # Получаем результат с GPU
    cuda.memcpy_dtoh(result_x, result_x_gpu)
    cuda.memcpy_dtoh(result_y, result_y_gpu)

    return result_x[0], result_y[0]

# Пример использования
scalar = 2  # Пример скаляра
result_x, result_y = multiply_point(scalar)
print(f"Результат умножения точки на скаляр {scalar}:")
print(f"x: {result_x}")
print(f"y: {result_y}")

Code explanation:

1. Import libraries:
   • pycuda.driver: For low-level control of CUDA.
   • pycuda.autoinit: To automatically initialize CUDA.
   • pycuda.compiler: To compile CUDA code from strings.
   • numpy: For working with data arrays.
2. Definition of secp256k1 curve parameters:
   • p: A prime number that determines the order of the field.
   • a, b: Coefficients of the equation of an elliptic curve.
   • Gx, Gy: Coordinates of the base point G.
3. CUDA Kernel:
   • Defined as a string using SourceModule.
   • Functions add_mod, sub_mod, mul_mod, pow_mod, inv_mod: implement the modular arithmetic needed for operations on an elliptic curve.
   • Function point_add: implements addition of points on an elliptic curve.
   • Function scalar_multiply: A CUDA kernel that performs point-scalar multiplication. It adds a point G to itself sequentially scalartimes.
4. Function multiply_point:
   • Allocates memory on the GPU to store the results (x and y coordinates).
   • Transfers data to the GPU.
   • Starts the CUDA kernel using scalar_multiply.
   • Gets results from the GPU and returns them.
5. Example of use:
   • Sets the scalar scalar.
   • Calls a function multiply_pointto multiply the base point by a scalar.
   • Displays results.

Notes:

• This code provides a basic implementation of secp256k1 elliptic curve point multiplication in CUDA.
• For real applications, optimizations such as using more efficient algorithms (e.g. double-and-add), more parallelism, and memory access optimizations need to be considered.
• The code uses sequential addition of points, which is not optimal for large scalars. It is recommended to use the double-and-add algorithm.
• Optimization of CUDA code can include the use of shared memory, reducing thread divergence, and other methods.
• This implementation assumes that you have the CUDA toolkit and PyCUDA installed and configured correctly.