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Instantiability of RSA-OAEP Under Chosen-Plaintext Attack

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Abstract

We show that the widely deployed RSA-OAEP encryption scheme of Bellare and Rogaway (Eurocrypt 1994), which combines RSA with two rounds of an underlying Feistel network whose hash ( i.e., round) functions are modeled as random oracles, meets indistinguishability under chosen-plaintext attack (IND-CPA) in the standard model based on simple, non-interactive, and non-interdependent assumptions on RSA and the hash functions. To prove this, we first give a result on a more general notion called “padding-based” encryption, saying that such a scheme is IND-CPA if (1) its underlying padding transform satisfies a “fooling" condition against small-range distinguishers on a class of high-entropy input distributions, and (2) its trapdoor permutation is sufficiently lossy as defined by Peikert and Waters (STOC 2008). We then show that the first round of OAEP satisfies condition (1) if its hash function is t-wise independent for t roughly proportional to the allowed message length. We clarify that this result requires the hash function to be keyed, and for its key to be included in the public key of RSA-OAEP. We also show that RSA satisfies condition (2) under the \(\Phi \)-Hiding Assumption of Cachin et al. (Eurocrypt 1999). This is the first positive result about the instantiability of RSA-OAEP. In particular, it increases confidence that chosen-plaintext attacks are unlikely to be found against the scheme. In contrast, RSA-OAEP’s predecessor in PKCS #1 v1.5 was shown to be vulnerable to such attacks by Coron et al. (Eurocrypt 2000).

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Notes

  1. 1.

    We often use the same terminology for ‘f-OAEP,’ which refers to OAEP using an abstract TDP f, with the meaning hopefully clear from context.

  2. 2.

    Such schemes were called “simple embedding schemes” by Bellare and Rogaway [5], who discussed them only on an intuitive level.

  3. 3.

    In the formal definition, we actually consider an “external” distinguisher who gets the extractor seed; see Sect. 3 for details.

  4. 4.

    In particular, this result requires that G is a keyed hash function whose key is included in the public key for OAEP. On the other hand, cryptographic hash functions are typically unkeyed. But see “Using unkeyed hash functions” below.

  5. 5.

    We remark that the recent attacks on \(\Phi \)A [56] are for moduli of a special form that does not include RSA.

  6. 6.

    Note, however, that their result does not rule out such a proof based on other properties of the TDP, non-black-box assumptions on the hash functions, or in the case of a specific TDP like RSA.

  7. 7.

    In particular, their security notion does not imply IND-CPA since they consider random messages. We also point out that it remains an open question whether NM-PRGs can be constructed.

  8. 8.

    We note that [49] actually defines lossy trapdoor functions, but the extension to permutations is straightforward.

  9. 9.

    This is done by choosing a uniform \((1/2-c)k\)-bit number x until \(p = x e + 1\) is a prime.

  10. 10.

    Additionally, in practice the encryption exponent e is usually fixed. This can be addressed by parameterizing E\(\Phi \)A by a fixed e instead of choosing it at random. Note that for \(e = 3\) one should make both \(e~|~p-1\) and \(e~|~q-1\) in the lossy case (otherwise the assumption is false [16]).

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Acknowledgments

We thank Mihir Bellare, Alexandra Boldyreva, Dan Brown, Yevgeniy Dodis, Mathias Herrmann, Jason Hinek, Arjen Lenstra, Alex May, Phil Rogaway, and the anonymous reviewers of Crypto 2010 and the Journal of Cryptology for helpful comments. In particular, we thank Dan for reminding us of [16, Remark2,p. 6], Alex and Mathias for pointing out the improved attacks in Sect. 5.3, Phil for encouraging us to consider the case of small e more closely and for telling us that KI security as defined in Appendix 8 was previously considered by [44], and Yevgeniy for suggesting the statement of Lemma 4.5 (our original lemma was specific to OAEP).

      Part of this work was done, while E.K. was at CWI, Amsterdam. E.K. is funded by ERC Project ERCC (FP7/615074) and the German Federal Ministry for Education and Research. Part of this work was done while A.O. was at Georgia Institute of Technology, supported in part by NSF award #0545659 and NSF Cyber Trust award #0831184. A.S. was supported in part by NSF awards #0747294, 0729171.

      Eike Kiltz was partially supported by DFG grant KI 795/4-1 and ERC Project ERCC (FP7/615074). Adam Smith was funded by US National Science Foundation award CCF-0747294.

Author information

Correspondence to Eike Kiltz.

Additional information

A preliminary version of this paper appears in Advances in Cryptology—CRYPTO 2010, 30th Annual International Cryptology Conference, T. Rabin ed., LNCS, Springer, 2010. This is the full version.

Communicated by Kenneth Paterson.

Appendices

Appendix 1: Proof of Lemma 4.5

We introduce the following notation for the proof. For a random variable V with range \({\mathcal {V}}\), we define the collision probability of V as \(\mathrm {Col}(V) = \Pr \left[ \, V = V' \,\right] = \sum _{v \in {\mathcal {V}}} P_V(v)^2\) where \(V'\) is an independent copy of V, and for an event \({\mathcal {E}}\) we define the conditional collision probability \(\mathrm {Col}_{{\mathcal {E}}}(V) = {\Pr }\left[ \, V = V'\,\left| \right. \,{\mathcal {E}}\,\right] \). For random variables VW, we define the square of the 2-distance as \(D(V,W) = \sum _v \big (P_V(v) - P_W(v)\big )^2\).

Writing \({\mathbf{E}}_k\) for expectation over the choice of random k from \({\mathcal {K}}\), we have

$$\begin{aligned}&\Delta \bigl ((K,g(X,h(K,X))), (K,g(X,U))\bigr ) \,=\,{\mathbf{E}}_k\bigl [ \Delta \bigl (g(X,h(k,X)), g(X,U)\bigr ) \bigr ] \end{aligned}$$
(9)
$$\begin{aligned}&\,\le \,\frac{1}{2} {\mathbf{E}}_k\biggl [ \sqrt{|S| \, D\bigl (g(X,h(k,X)), g(X,U)\bigr )} \biggr ] \nonumber \\&\quad \,\le \,\frac{1}{2} \sqrt{|S| \, {\mathbf{E}}_k \bigl [ D\bigl (g(X,h(k,X)), g(X,U)\bigr ) \bigl ]} \end{aligned}$$
(10)

where the first inequality is by Cauchy-Swartz and the second is by Jensen’s inequality. We now show

$$\begin{aligned}{\mathbf{E}}_k\bigl [ D\bigl (g(X,h(k,X)), g(X,U)\bigr ) \bigl ] \,\le \,\mathrm {Col}(X) \, \end{aligned}$$

from which the theorem follows. Write \((X,Y_k) = (X,h(k,X))\) for an arbitrary but fixed k. Then

$$\begin{aligned} D\bigl (g(X,Y_k), g(X,U)\bigr )&\,=\,\sum _s \big ( P_{g(X,Y_k)}(s) - P_{g(X,U)}(s) \big )^2 \\&\,=\,\sum _s P_{g(X,Y_k)}(s)^2 - 2\sum _s P_{g(X,Y_k)}(s)P_{g(X,U)}(s) \\&\quad + \sum _s P_{g(X,U)}(s)^2 \, . \end{aligned}$$

Using the Kronecker delta \(\delta _{s,s'}\) which equals 1 if \(s =s'\) and else 0 for all \(s,s' \in S\), we can write \(P_{g(X,Y_k)}(s) = \sum _x P_X(x) \delta _{g(x,h(k,x)),s}\), and thus

$$\begin{aligned} \sum _s P_{g(X,Y_k)}(s)^2&\,=\,\sum _s \bigg (\sum _{x} P_X(x) \delta _{g(x,h(k,x)),s} \bigg ) \bigg ( \sum _{x'} P_X(x') \delta _{g(x',h(k,x')),s} \bigg ) \\&\,=\,\sum _{x,x'} P_X(x) P_X(x') \delta _{g(h(k,x)),g(h(k,x'))}. \end{aligned}$$

We use the pairwise independence of h to rewrite this in terms of collision probabilities:

$$\begin{aligned} {\mathbf{E}}_k\Bigl [\sum _s P_{g(X,Y_k)}(s)^2\bigr ]&\,=\,&\sum _{x,x'} P_X(x) P_X(x') {\mathbf{E}}_k[\delta _{g(x,h(k,x)),g(x',h(k,x'))}] \nonumber \\&\,=\,&\mathrm {Col}(X)~+~ \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1-\mathrm {Col}(X))\,, \end{aligned}$$
(11)

where the subscript \({\mathcal {E}}\) denotes (conditioning on) the event that \(X \ne X'\). That is,

$$\begin{aligned} \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) \,=\,{\Pr }\left[ \, g(X,U) = g(X',U')\,\left| \right. \,X \ne X'\,\right] . \end{aligned}$$

Similarly,

$$\begin{aligned} \sum _s P_{g(X,Y_k)}(s) P_{g(X,U)}(s)&\,=\,\sum _s \bigg (\sum _{x} P_X(x) \delta _{g(x,h(k,x)),s} \bigg ) \\&\quad \bigg (\sum _{x',u} P_X(x') P_U(u) \delta _{g(x',u),s} \bigg ) \\&\quad \,=\,\sum _{x} \sum _{x'} \sum _u P_X(x) P_X(x') P_U(u) \delta _{g(x,h(k,x)),g(x',u)} \end{aligned}$$

so that

$$\begin{aligned} {\mathbf{E}}_k\Bigl [\sum _s P_{g(X,Y_k)}(s) P_{g(X,U)}(s)\bigr ]&\,=\,\sum _{x} \sum _{x'} \sum _u P_X(x) P_X(x') P_U(u) {\mathbf{E}}_k [\delta _{g(x,h(k,x)),g(x',u)}] \\&\,=\,\mathrm {Col}(g(X,U)) \,=\,\mathrm {Col}_{\overline{{\mathcal {E}}}}(g(X,U)) \mathrm {Col}(X)\\&\quad + \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1-\mathrm {Col}(X)) \, . \end{aligned}$$

where \({\mathcal {E}}\) is defined as above. Note that the only difference between the expression above and that in (11) is that even when \(X=X'\), a collision is not guaranteed.

Finally,

$$\begin{aligned} \sum _s P_{g(X,U)}(s)^2&\,=\,\mathrm {Col}(g(X,U))\\&\,=\,\mathrm {Col}_{\overline{{\mathcal {E}}}}(g(X,U)) \mathrm {Col}(X) + \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1-\mathrm {Col}(X)) \end{aligned}$$

as well. By combining the above, we have

$$\begin{aligned} {\mathbf{E}}_k\bigl [ D\bigl (g(X,Y_k), f(X,U)\bigr ) \bigr ]= & {} \mathrm {Col}(X) + \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1-\mathrm {Col}(X)) \\&-\,2 (\mathrm {Col}_{\overline{{\mathcal {E}}}}(g(X,U)) \mathrm {Col}(X)\\&+\,\mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1-\mathrm {Col}(X))) \\&+\,\mathrm {Col}_{\overline{{\mathcal {E}}}}(g(X,U)) \mathrm {Col}(X) + \mathrm {Col}_{{\mathcal {E}}}(g(X,U)) (1{-}\mathrm {Col}(X)) \\= & {} (1-\mathrm {Col}_{\overline{{\mathcal {E}}}}(g(X,U)))\mathrm {Col}(X)\\\le & {} \mathrm {Col}(X). \end{aligned}$$

To complete the proof, we can plug the bound above into (10):

$$\begin{aligned}&\Delta \bigl ((K,g(X,h(K,X))), (K,g(X,U))\bigr ) \,\le \,\frac{1}{2} \sqrt{|S| \, {\mathbf{E}}_k \bigl [ D\bigl (g(X,h(k,X)), g(X,U)\bigr ) \bigl ]} \\&\quad \,\le \,\frac{1}{2} \sqrt{|S|\mathrm {Col}(X)}. \end{aligned}$$

By the assumption on the min-entropy of X, the collision probability \(\mathrm {Col}(X)\) is at most \(4 {\hat{\varepsilon }}^2 / |S|\). So the statistical distance \(\Delta \bigl ((K,g(X,h(K,X))), (K,g(X,U))\bigr )\) is at most \({\hat{\varepsilon }}\), as desired.\(\square \)

Appendix 2: Security of OAEP Under Key-Independent Chosen-Plaintext Attack

The commonly-accepted notions of security for encryption ask for privacy with respect to messages that may depend on the public key. We define here a notion of privacy for messages not depending on the public key. We mention that such a definition appears for example in the work of Micali et al. [44] (under the name “three-pass," versus “one-pass," cryptosystem), in the text of Goldreich [30], and in the context of the recent work on deterministic encryption [2].

The definition. To an encryption scheme \(\Pi = ({\mathcal {K}}, {\mathcal {E}},{\mathcal {D}})\) and an adversary \(B = (B_1, B_2)\) we associate

figurem

We require \(|m_0| = |m_1|\) above. Define the indki-cpa advantage of B against \(\Pi \) as

$$\begin{aligned} \mathbf {Adv}^{\mathrm {indki\text{- }cpa}}_{\Pi ,B}(k) = 2 \cdot \Pr \left[ \, \mathbf {Exp}^{\mathrm {indki\text{- }cpa}}_{\Pi ,B}(k) \,{\Rightarrow }\,1 \,\right] - 1. \end{aligned}$$

Remarks. While non-standard, KI security seems adequate for some applications. For example, in [30] Goldreich points out that high-level applications that use encryption as a tool do so in a key-oblivious manner, and Bellare et al. [2] argue that in real life public keys are abstractions hidden in our software, so messages are unlikely to depend on them. KI security also suffices for hybrid encryption.

The result. We can show a standard model instantiation under KI security directly from Lemma 4.5, where G is any pairwise-independent function. This is captured by the theorem below.

Theorem 8.1

Let \(\mathsf {LTDP}= ({\mathcal {F}}, {\mathcal {F}}')\) be an LTDP with residual leakage \(\ell \), and let \(\mathsf {OAEP}\) be the encryption scheme associated to \({\mathcal {F}}\), hash functions GH, and a parameter \(k_0 < k\). Suppose G is pairwise-independent. Let \(\varepsilon > 0\). Then for any \(k_0 \ge \ell + 2 \log (1/\varepsilon ) - 2\) and any INDKI-CPA adversary B against \(\mathsf {OAEP}\), there is a distinguisher D against \(\mathsf {LTDP}\) such that

$$\begin{aligned} \mathbf {Adv}^{\mathrm {indki\text{- }cpa}}_{\mathsf {OAEP},B}(k) \,\le \,\mathbf {Adv}^{\mathrm {ltdp}}_{\mathsf {LTDP},D}(k) + \varepsilon . \end{aligned}$$

Furthermore, the running-time of D is the time to run B.

As we mentioned, the proof is a simple hybrid argument concluding by Lemma 4.5.

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Kiltz, E., O’Neill, A. & Smith, A. Instantiability of RSA-OAEP Under Chosen-Plaintext Attack. J Cryptol 30, 889–919 (2017). https://doi.org/10.1007/s00145-016-9238-4

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Keywords

  • RSA
  • OAEP
  • Padding-based encryption
  • Lossy trapdoor functions
  • Leftover hash lemma
  • Standard model