December 1, 2024: updated to make corrections as shown. The proposed alternative definition of special soundness has not been changed, but its wording has been clarified.
I am co-authoring a book on the Foundations of Cryptographic Authentication with Sukhi Chuhan and Veronica Wojnas, where Chapter 3 is concerned with traditional credentials (as opposed to verifiable credentials or the mDL). Section 3.5 has an extensive treatment of zero knowledge, and in section 3.5.6 we use BBS signatures as an example of anonymous credentials. The chapter is still work in progress: the definition of special soundness proposed here is discussed in Section 3.5.6 but has not been retrofitted yet into sections 3.5.1-5.
As we were studying in detail the BBS paper we saw that the Sigma protocol defined in the paper for proving knowledge of a BBS signature is not special sound, contrary to what the paper claims. Then we thought of a modification to the definition of special soundness, proposed here, that would make it special sound. The proposed definition is more widely applicable than the current one, while still implying proof of knowledge.
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