We reported a secure scan design approach using shift register equivalents (SR-equivalents, for short) that are functionally equivalent but not structurally equivalent to shift registers [10] and also introduced generalized shift registers (GSRs, for short) to apply them to secure scan design [11]-[ 13]. In this paper, we combine both concepts of SR-equivalents and GSRs and consider the synthesis problem of SR-equivalent GSRs, i.e., how to modify a given GSR to an SR-equivalent GSR. We also consider the enumeration problem of SR-equivalent GFSRs, i.e., the cardinality of the class of SR-equivalent GSRs to clarify the security level of the secure scan architecture.