Pairing signatures without distortion maps

Definitions and mathematical statement

In a usual pairing-signature setting, $G_1,G_2,G_T$ are groups of prime order $r$ with an efficiently computable nondegenerate bilinear map \[ e:G_1\times G_2\to G_T. \] The source asks to give particular previously cited pairing-based signature constructions and security proofs that do not rely on distortion maps. The cited constructions themselves are not identified in the supplied statement.

For an elliptic-curve pairing, a standard distortion-map role is to move a point or subgroup within the ambient torsion so that a pairing which would be degenerate on the original cyclic subgroup becomes nontrivial. For example one may seek an efficiently computable $\psi$ with $e(P,\psi(P))\ne1$. This is not the same general notion as an arbitrary efficient homomorphism between two designated source groups. The source does not state the precise distortion-map convention to be removed.

A complete task would specify each scheme's key-generation, signing and verification algorithms, its group/pairing assumptions, and its security model and theorem. Typical security formulations concern existential unforgeability under chosen-message attacks, but the source does not select that notion, its exact variant, or standard-versus-random-oracle assumptions. The requested change is to eliminate reliance on distortion maps while retaining the relevant functionality and proof; no unspecified generic signature scheme is silently substituted for the omitted cited constructions.

Short English statement

Reformulate the referenced pairing signatures and prove their required security without using distortion maps. The original scheme references and security hypotheses are missing.

The supplied summary labels the task solved, naming Hess (2003), Boneh--Lynn--Shacham, Type-3/asymmetric variants, Verheul, and Chatterjee--Sarkar, and saying modern implementations avoid distortion maps. It gives no exact scheme/proof match and informally equates absence of distortion maps with absence of inter-source-group maps. These claims are retained as reported context, not a proof of every omitted construction. Current primary-literature audit remains required.

Sources

• Local UnsolvedMath registry, permanent ID 8800060, source code catalogue entry, "Pairing signatures without distortion maps," proposed by Tanja Lange. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• IPAM, Workshop on Number Theory and Cryptography --- Open Problems (2006), Attachment 6239, PDF page 48, printed slide 46, schedule locator https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; pdf-and-legacy-ppt visual-audit extraction, accessed 2026-07-29.
• Supplied summary attributions: Hess (2003), Boneh--Lynn--Shacham, Verheul, and Chatterjee--Sarkar. Full references, exact Type-3 adaptations, and matching security theorems are absent.
• Source provenance reports inspection of 17 PDFs and two legacy PPT decks among 19 surviving attachments, excluding repetitions and solved definitions; this is supplied provenance, not a new inspection claimed here.
• Current primary-literature status audit and redistribution terms NEEDS_REVIEW for the publicly accessible author/publisher copy, separately from the dataset licence.