**DDH-like Assumptions Based on Extension Rings**

*Ronald Cramer and Ivan Damgaard and Eike Kiltz and Sarah Zakarias and Angela Zottarel*

**Abstract: **We introduce and study a new type of DDH-like assumptions based on
groups of prime order q. Whereas standard DDH is based on encoding
elements of F_{q} ``in the exponent'' of elements in the group, we
ask what happens if instead we put in the exponent elements of the
extension ring R_f= \F_{q}[X]/(f) where f can be any degree-d
polynomial. We show that solving the decision problem that follows
naturally reduces to the case where f is irreducible. This variant
is called the d-DDH problem, where 1-DDH is standard
DDH. Essentially any known cryptographic construction based on DDH can
be immediately generalized to use instead d-DDH, and we show in the
generic group model that d-DDH is harder than DDH. This means that
virtually any application of DDH can now be realized with the same
(amortized) efficiency, but under a potentially weaker assumption. On
the negative side, we also show that d-DDH, just like DDH, is easy
in bilinear groups. This motivates our suggestion of a different type
of assumption, the d-vector DDH problems (VDDH), which are based on
f(X)= X^d, but with a twist to avoid the problems with reducible
polynomials. We show in the generic group model that VDDH is hard in
bilinear groups and that in fact the problems become harder with
increasing d and hence form an infinite hierarchy. We show that
hardness of VDDH implies CCA-secure encryption, efficient
Naor-Reingold style pseudorandom functions, and auxiliary input secure
encryption, a strong form of leakage resilience. This can be seen as
an alternative to the known family of k-linear assumptions.

**Category / Keywords: **public-key cryptography / DDH, Public Key Encryption, PRF, Leakage Resilient Encryption

**Date: **received 30 May 2011, last revised 7 Mar 2012

**Contact author: **angela at cs au dk

**Available format(s): **PDF | BibTeX Citation

**Version: **20120307:124831 (All versions of this report)

**Short URL: **ia.cr/2011/280

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