Implementation of Elliptic Curve Point Multiplication on CUDA and Using Special Libraries

BY KEYHUNTER 17.03.2025
Implementation of Elliptic Curve Point Multiplication on CUDA and Using Special Libraries

Elliptic curve cryptography (ECC) is one of the most promising areas in cryptography due to its high cryptographic strength with relatively small key sizes. One of the key components of ECC is the multiplication of points on an elliptic curve, which requires significant computing resources. In this article, we will consider how to implement ECC point multiplication on the CUDA platform and use special libraries to optimize computations.

Theoretical Foundations of ECC

Elliptic curve cryptography is based on the mathematical properties of elliptic curves. Each point on the curve can be represented as a pair of coordinates (x,y)(x, y)(x,y). Multiplying points on an elliptic curve is the process of finding a new point that is the result of multiplying a given point by a scalar. This process involves successive operations of adding and doubling points.

Implementation of Point Multiplication in CUDA

To implement point multiplication on CUDA, it is necessary to use the massive parallelism model provided by this platform. The main steps of implementation include:

  1. Data Initialization : Passing elliptic curve parameters and points as arrays to the device.
  2. Parallelization of computation : Breaking down the point multiplication process into smaller tasks that can be executed in parallel on GPU cores.
  3. Computation : Using CUDA cores to perform point addition and doubling operations.

An example of code for a CUDA kernel performing point multiplication might look like this:

where__global__ void eccPointMultiplicationKernel(int* resultX, int* resultY, int* scalar, int* basePointX, int* basePointY, int* curveParams) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    
    // Вычисление умножения точки на скаляр
    if (idx < scalar[0]) {
        int tempX, tempY;
        // Инициализация начальной точки
        tempX = basePointX[0];
        tempY = basePointY[0];
        
        // Умножение точки на скаляр
        for (int i = 1; i <= scalar[0]; i++) {
            // Удвоение точки
            if (i % 2 == 0) {
                tempX = doublePoint(tempX, tempY, curveParams);
                tempY = doublePoint(tempY, tempX, curveParams);
            }
            // Сложение точек
            else {
                tempX = addPoints(tempX, tempY, basePointX[0], basePointY[0], curveParams);
                tempY = addPoints(tempY, tempX, basePointY[0], basePointX[0], curveParams);
            }
        }
        
        // Запись результата
        resultX[idx] = tempX;
        resultY[idx] = tempY;
    }
}

// Функции для удвоения и сложения точек
int doublePoint(int x, int y, int* curveParams) {
    // Реализация алгоритма удвоения точки
}

int addPoints(int x1, int y1, int x2, int y2, int* curveParams) {
    // Реализация алгоритма сложения точек
}

Using Special Libraries

To optimize ECC computations on CUDA, you can use special libraries such as cuECC or the implementation from the HareInWeed/gec repository. These libraries provide ready-made functions for point multiplication and other ECC operations optimized for running on the GPU.

For example, the cuECC library is designed to work with the secp256k1 curve and allows to significantly increase the performance of ECC calculations due to parallelization on the GPU.

Future Works

In the future, we can consider integrating dot multiplication algorithms with other cryptographic protocols to create high-performance and secure systems. In addition, optimizing existing implementations to run on different GPU models can further improve computational efficiency.

There are currently no dedicated CUDA-compatible libraries that directly implement elliptic curve cryptography (ECC) algorithms on GPUs. However, developers can use libraries optimized for parallel computing on GPUs, such as cuBLAS or cuSPARSE, to implement their own ECC algorithms.

Implementation of Elliptic Curve Point Multiplication on CUDA and Using Special Libraries

To implement ECC on CUDA, the following approaches can be used:

  1. Creating your own implementations : Develop your own functions for elliptic curve operations using CUDA to parallelize the computations. This requires a deep understanding of the mathematical foundations of ECC and familiarity with CUDA.
  2. Using libraries for parallel computing : Libraries like cuBLAS or cuSPARSE can be useful for optimizing computations involving matrix operations that may be part of an ECC implementation.
  3. Adapting existing libraries : If there are ECC libraries written in C or C++, they can be adapted to work with CUDA by moving the computational parts into CUDA cores.

Some libraries that might be useful for working with cryptography on the GPU, although they are not specifically focused on ECC:

  • OpenSSL : Although not optimized for CUDA, it can be used in conjunction with CUDA for some cryptographic tasks.
  • cuECC : Not an official library, but can be implemented as a userland library for ECC on CUDA.

For remote access to GPUs, the rCUDA framework can be used , which allows using the CUDA API on remote machines without modifying the code 1 . However, this is not a library for ECC, but rather a tool for remote access to GPU resources.

Using CUDA for elliptic curve cryptography (ECC) computations can provide several benefits:

  1. Performance improvements : CUDA enables parallel computing on the GPU, which can significantly speed up ECC operations such as elliptic curve point multiplication. This is especially important for applications that require high speed and cryptographic strength 1 3 .
  2. Efficient resource utilization : The CUDA architecture enables efficient management of GPU resources, making it possible to perform complex computations with minimal power and memory consumption 6 .
  3. Scalability : CUDA supports multiple GPUs, allowing computation to scale to handle large amounts of data. This is especially useful in applications that require high performance and parallelism 5 .
  4. Ease of implementation : CUDA provides a simple C++-based programming interface, making it easy to develop and optimize code for GPUs 2 3 .
  5. Support for multiple programming languages : CUDA supports multiple programming languages, including C++, Python, and Fortran, making it accessible to developers with different levels of experience 1 3 .
  6. Memory optimization : CUDA enables the use of shared memory and configurable caches, which improves the performance of memory operations and reduces data access time 4 .

Overall, using CUDA for ECC can significantly improve the performance and efficiency of cryptographic computations, making it attractive for applications that require high speed and security.

There are currently no widely known and tailored implementations of Elliptic Curve Cryptography (ECC) for CUDA that are available as ready-made libraries. However, developers can create their own ECC implementations on CUDA by leveraging the massive parallelism model and optimizing the code for GPU execution.

If you are looking for examples or approaches to implementing ECC on CUDA, you can consider the following steps:

  1. Creating your own implementations : Develop your own functions for elliptic curve operations using CUDA to parallelize the computations. This requires a deep understanding of the mathematical foundations of ECC and familiarity with CUDA.
  2. Using libraries for parallel computing : Libraries like cuBLAS or cuSPARSE can be useful for optimizing computations involving matrix operations that may be part of an ECC implementation.
  3. Adapting existing libraries : If there are ECC libraries written in C or C++, they can be adapted to work with CUDA by moving the computational parts into CUDA cores.

Some research and projects may include ECC implementations on CUDA, but they are not widely available as ready-made libraries. There may be custom implementations or projects in GitHub repositories or other platforms, but they are not mentioned in the provided search results.

Adapting existing ECC implementations to CUDA may face a number of challenges due to the specifics of the GPU architecture and the CUDA programming model. The main challenges include:

1. Optimizing memory access

  • Non-locality of memory access : In CUDA, it is important that threads of the same warp read data from adjacent memory addresses (co-local access). Non-local access can significantly reduce performance due to the increased number of read/write transactions 1 .
  • Bank conflicts in shared memory : When multiple threads access the same memory bank at the same time, this causes latency. Such conflicts require careful optimization 1 .

2. Synchronization of threads

  • Synchronization challenges : Threads within a single block can synchronize using __syncthreads(), but synchronization between blocks requires the use of global memory or atomic operations, which reduces performance 3 .
  • Divergent branches : If threads in the same warp execute different branches of code, this results in sequential execution, which reduces the efficiency of parallelism 1 .

3. Transferring data between CPU and GPU

  • High data transfer latency : Copying data between the CPU and GPU is a slow process, especially when small computational tasks are called frequently. It is necessary to minimize the number of such transfers 1 .

4. Adaptation of ECC algorithms

  • Difficulty of parallelization : ECC algorithms such as curve point multiplication involve sequential operations (e.g. point addition and doubling) that are difficult to parallelize efficiently.
  • Mathematical precision requirements : ECC requires precise arithmetic operations on large numbers (e.g. modular arithmetic), which can be difficult to implement on GPUs without specialized libraries.

5. Dynamic parallelism support

  • Modern CUDA devices support dynamic thread spawning, but this adds overhead and requires careful resource management 3 .

6. Difficulties in adapting libraries

  • Existing ECC implementations on CPUs often use optimizations for serial architecture (e.g., using caches) that are inefficient on GPUs. Porting such implementations requires significant reworking of the algorithms to take into account the specifics of CUDA 1 4 .

Recommendations for overcoming difficulties:

  • Optimize memory access through combined read/write operations.
  • Minimize global memory usage and inter-block synchronization.
  • Use atomic operations to manage data state.
  • Redesign ECC algorithms to maximize parallelism.
  • Perform performance profiling using NVIDIA tools (e.g. Nsight).

Thus, adapting ECC to CUDA requires significant optimization and algorithm redesign efforts to effectively take advantage of the parallel architecture of the GPU.

Conclusion

Implementation of elliptic curve point multiplication on CUDA and use of special libraries allow to significantly increase the performance of ECC calculations. This makes it possible to use ECC in applications that require high speed and cryptographic resistance, such as secure communications and cryptocurrencies.