the name used on the EAD card is the same that is on both the I-821D & I-765 application application. seeing how you sent in your BC or passport with both names on them, then both names would be used on your SSN as well.
i too have 2 last names, even though i dont use my mother's birth last name. but seeing how both last names are on my passport and my USCIS applications, then it will be on my EAD card too.
SSA goes off the forms of ID that you give them. so, if your EAD and passport or BC has both last names, then that is what they are going to use.
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Date Sent 09/18/2012 Date Received 09/21/12
Date of Biometrics: October 24, 2012
2nd RFE update: 03/18/13 Received: 03/21/13
(asking for BC, 07 residency & HS transcript)
Date of Approval: 07/09/13 Date EAD Received: 07/13/13