**On the Regularity of Lossy RSA: Improved Bounds and Applications to Padding-Based Encryption**

*Adam Smith and Ye Zhang*

**Abstract: **We provide new bounds on how close to regular the map x |--> x^e is on arithmetic progressions in Z_N, assuming e | Phi(N) and N is composite. We use these bounds to analyze the security of natural cryptographic problems related to RSA, based on the well-studied Phi-Hiding assumption. For example, under this assumption, we show that RSA PKCS #1 v1.5 is secure against chosen-plaintext attacks for messages of length roughly (log N)/4 bits, whereas the previous analysis, due to Lewko et al (2013), applies only to messages of length less than (log N)/32.

In addition to providing new bounds, we also show that a key lemma of Lewko et al. is incorrect. We prove a weaker version of the claim which is nonetheless sufficient for most, though not all, of their applications.

Our technical results can be viewed as showing that exponentiation in Z_N is a deterministic extractor for every source that is uniform on an arithmetic progression. Previous work showed this type of statement only on average over a large class of sources, or for much longer progressions (that is, sources with much more entropy).

**Category / Keywords: **public-key cryptography / PKCS, Phi-Hiding, Regularity

**Original Publication**** (in the same form): **IACR-TCC-2015

**Date: **received 12 Jan 2015

**Contact author: **yxz169 at cse psu edu

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

**Version: **20150114:164955 (All versions of this report)

**Short URL: **ia.cr/2015/027

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