Large Volumes of Asymmetric Encryption First of all, I apologize in advance if I use any incorrect terminology. I am just getting my feet wet when it comes to cryptography and there is a lot to learn.

For the past 12 months, we have been developing an application infrastructure for a Cloud and Mobile Application (iOS/Android). As part of the infrastructure, we require the ability to encrypt $\text{SHA-}256$ hashes using asymmetric encryption and in some cases, encrypt multiple hashes at one time. A non-decryptable digital signature is not sufficient for our requirements. The simple explanation for this is, the digital signatures we have been using thus far, allows anyone with a corresponding public key, to review the contents to ensure they align with corresponding data in the database row. Ownership of the row of data cannot be transferred or duplicate rows created, without the correct signature. Rows have a signature and an owner and two digital signatures validating the product serial number is valid and that the declared owner, does, in fact, own the serial number.  

We have built a working prototype that was built on top of our existing framework, which uses PostgreSQL and PHP and preliminary testing using $\text{SHA-}256$ and RSA 2048 has yielded the results of being able to encrypt and insert/update 10,000 unique rows of data every 40 seconds. When inserting 10,000 rows of data, there are 10,000 unique digital signatures that have to be created. The bulk of the work is in creating these digital signatures.

We are wondering what the best way to speed this up is? Ultimately, we need to be able to drastically reduce this time and it would be ideal if we could get 10,000 rows signed and inserted in under 5 seconds.

Is there a faster solution for signing digital signatures with a private key? We have discussed writing this component in C#. We are also wondering if Bitcoin mining hardware (ASIC) could be used for asymmetric encryption? If not mining hardware, would GPU based processing be better? Acquiring specialized hardware, should improve the processing time, which is not a problem. 

We eventually need to generate in excess of 1 billion digital signatures per month. Any guidance or help would be greatly appreciated.

EDIT: I forgot to mention we are using OpenSSL with our current prototype.
I would explore the GPU option before spending money on specialized hardware.

I have done something similar using CUDA where I was doing many ECC operations in parallel. The GPU was about 30x - 50x faster than the CPU.

You could probably hash the data on the CPU and then use the GPU for the RSA exponentiation.