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According to what I read, OpenBSD seems to have its own system on encoding and decoding using Base64. However,I cant find any literature whatsoever that can describe it mathematically. my question is, what is the difference between the usual EncodeBase64 DecodeBase64 with the OpenBSD version? how do we calculate it on math?

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I think you're jumping to (unwarranted) conclusions here: the link you provide is a Java implementation of bcrypt and indeed, bcrypt uses a modified base64 definition. See more about the differences on the security stack exchange.

What you should remember is that indeed bcrypt uses a nonstandard base64 dialect (forcing it to provide implementations for base64 within its implementation) but that bcrypt implementations can be supposed to be interoperable as they all need to implement the proprietary encoding scheme.

The link to the Hacker Noon article mentioned on SSE is interesting because it digs deeper into the peculiarities of bcrypt, including its use of a proprietary encoding scheme. Definitely worth a visit.

I am not sure that I answer your full question here but at least the provided pointers should allow you to dig deeper into the subject.

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I have the c++ version of the app too and it has the same phrase but I forgot the link sadly

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