Utilizing the Double-Precision Floating-Point Computing Power of GPUs for RSA Acceleration

Asymmetric cryptographic algorithm (e.g., RSA and Elliptic Curve Cryptography) implementations on Graphics Processing Units (GPUs) have been researched for over a decade. The basic idea of most previous contributions is exploiting the highly parallel GPU architecture and porting the integer-based algorithms from general-purpose CPUs to GPUs, to offer high performance. However, the great potential cryptographic computing power of GPUs, especially by the more powerful floating-point instructions, has not been comprehensively investigated in fact. In this paper, we fully exploit the floating-point computing power of GPUs, by various designs, including the floating-point-based Montgomery multiplication/exponentiation algorithm and Chinese Remainder Theorem (CRT) implementation in GPU. And for practical usage of the proposed algorithm, a new method is performed to convert the input/output between octet strings and floating-point numbers, fully utilizing GPUs and further promoting the overall performance by about 5%. The performance of RSA-2048/3072/4096 decryption on NVIDIA GeForce GTX TITAN reaches 42,211/12,151/5,790 operations per second, respectively, which achieves 13 times the performance of the previous fastest floating-point-based implementation (published in Eurocrypt 2009). The RSA-4096 decryption precedes the existing fastest integer-based result by 23%

Analysis of Parallel Montgomery Multiplication in CUDA

For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared

Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs

Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure correction has to be solved at every time step in the dynamical core of many numerical weather prediction models, and equations of a very similar structure arise in global ocean models, subsurface flow simulations and gas and oil reservoir modelling. The elliptic solve is often the bottleneck of the forecast, and an algorithmically optimal method has to be used and implemented efficiently. Graphics Processing Units have been shown to be highly efficient for a wide range of applications in scientific computing, and recently iterative solvers have been parallelised on these architectures. We describe the GPU implementation and optimisation of a Preconditioned Conjugate Gradient (PCG) algorithm for the solution of a three dimensional anisotropic elliptic PDE for the pressure correction in NWP. Our implementation exploits the strong vertical anisotropy of the elliptic operator in the construction of a suitable preconditioner. As the algorithm is memory bound, performance can be improved significantly by reducing the amount of global memory access. We achieve this by using a matrix-free implementation which does not require explicit storage of the matrix and instead recalculates the local stencil. Global memory access can also be reduced by rewriting the algorithm using loop fusion and we show that this further reduces the runtime on the GPU. We demonstrate the performance of our matrix-free GPU code by comparing it to a sequential CPU implementation and to a matrix-explicit GPU code which uses existing libraries. The absolute performance of the algorithm for different problem sizes is quantified in terms of floating point throughput and global memory bandwidth.Comment: 18 pages, 7 figure