**The n-Diffie-Hellman Problem and its Applications**

*Liqun Chen and Yu Chen*

**Abstract: **The main contributions of this paper are twofold. On the one hand, the twin Diffie-Hellman (twin DH) problem proposed by Cash, Kiltz and Shoup is extended to the $n$-Diffie-Hellman ($n$-DH) problem for an arbitrary integer $n$, and this new problem is shown to be at least as hard as the ordinary DH problem. Like the twin DH problem, the $n$-DH problem remains hard even in the presence of a decision oracle that recognizes solution to the problem. On the other hand, observe that the double-size key in the Cash et al. twin DH based encryption scheme can be replaced by two separated keys each for one entity, that results in a 2-party encryption scheme which holds the same security feature as the original scheme but removes the key redundancy. This idea is further extended to an $n$-party case, which is also known as $n$-out-of-$n$ encryption. As examples, a variant of ElGamal encryption and a variant of Boneh-Franklin IBE have been presented; both of them have proved to be CCA secure under the computational DH assumption and the computational bilinear Diffie-Hellman (BDH) assumption respectively, in the random oracle model. The two schemes are efficient, due partially to the size of their ciphertext, which is independent to the value $n$.

**Category / Keywords: **public-key cryptography / the (strong) $n$-DH assumption, the (strong) $n$-BDH assumption, multiple public key encryption, multiple identity-based encryption

**Publication Info: **An extended abstract of this paper appears in the Proceedings of the 14th Information Security Conference (ISC 2011).

**Date: **received 25 Jul 2011, last revised 8 Oct 2011

**Contact author: **liqun chen at hp com

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Note: **Revise several lapses

**Version: **20111009:020634 (All versions of this report)

**Short URL: **ia.cr/2011/397

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