F1 -> H1B is resume required? Do I provide it or the company hiring me?

[rinko_par]

I am currently an F-1 visa holder on OPT. I have previously worked (illegally) while I was in school and that is the only working experience I have.
It would be helpful if I can put that down on my resume, but I also have concern that eventually if my employer is willing to give me H1B sponsorship, the same resume is going to be summitted to the Immigration. And they will find out that I have illegally worked during my F1 period.
So my question would be, do I provide the resume? or will the comany that hires me provide the resume directly to the immigration?

Thanks in advance for any contribution to my question!!