This paper extends the concept of the specific verification function (SVF), introduced by Cui et al. in formally describing the Naor transformation, from hierarchical identity-based encryption (HIBE) to hierarchical identity-based cryptog-raphy (HIBC), including hierarchical identity-based signature (HIBS), where the Naor transformation can convert a secure (H)IBE scheme to a secure (H)IBS one. We formulate this extended concept, checkable key generation (CKG), give its syntax, and classify it into two types: fully and partially ones. We also define security notions of CKG, named unextractability (UE) against several attacks. Next, we show that an$\boldsymbol{\ell}$-level CKG component can be constructed from an$\boldsymbol{\ell}$-level HIBS scheme, and an$\boldsymbol{\ell}$-level HIBS scheme can be constructed from an$(\ell+1)$-level CKG component. Both give security enhancement of HIBS from a weakly secure scheme to a strongly secure one. In addition, we mention that an HIBS scheme with constant-size signatures can be constructed from a partially CKG component.