From my question about the Pi's slow internet I learned that the main bottleneck is the dismal CPU speed. Is it possible for me to use the GPU to handle the information instead (or perhaps in addition to the CPU)?
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Unfortunately the simple answer is no.
GPU is not designed for general computing so it just can't do most of the CPU's work. From your other question I understand that you want to use VPN so much of the computing power has to go to encryption/decryption. At least in theory that could probably be done by GPU but it would not be easy to do and it would require a lot of code change in the VPN software you are using. And additional overhead for transferring data from CPU do GPU might make it actually slower than using only CPU.