This is a fully-functional Unity project (Unity 2017.3, using .NET 4.5) that demonstrates sign-up and sign-in for AWS Cognito user accounts. This is not (Jan 2018) supported by the official AWS SDK release. I have rebuilt the SDK using this fork, DLLs are included.
See this tutorial for instructions on how to set up the AWS side, and things you need to do in the code to make it work: http://www.gradientspace.com/tutorials/2018/1/25/using-aws-cognito-user-accounts-with-unity
The Sign-in code, and particularly the math parts, are borrowed from this article by Marcus Lachinger: http://blog.mmlac.com/aws-cognito-srp-login-c-sharp-dot-net/. His github is here: https://github.com/mmlac
Also using the Hkdf class from this gist: https://gist.github.com/CodesInChaos/8710228
Also includes a complied BouncyCastle dll, if you want to rebuild from source, it is here: https://github.com/bcgit/bc-csharp
My code is released under MIT license, but these other things, have their own licenses.