Paper 2009/075

Security of Practical Cryptosystems Using Merkle-Damgard Hash Function in the Ideal Cipher Model

Yusuke Naito, Kazuki Yoneyama, Lei Wang, and Kazuo Ohta

Abstract

Since the Merkle-Damgård (MD) type hash functions are differentiable from ROs even when compression functions are modeled by ideal primitives, there is no guarantee as to the security of cryptosystems when ROs are instantiated with structural hash functions. In this paper, we study the security of the instantiated cryptosystems whereas the hash functions have the well known structure of Merkle-Damgård construction with Stam's type-II compression function (denoted MD-TypeII) in the Ideal Cipher Model (ICM). Note that since the Type-II scheme includes the Davies-Meyer compression function, SHA-256 and SHA-1 have the MD-TypeII structure. We show that OAEP, RSA-KEM, PSEC-KEM, ECIES-KEM and many other encryption schemes are secure when using the MD-TypeII hash function. In order to show this, we customize the indifferentiability framework of Maurer, Renner and Holenstein. We call the customized framework ``indifferentiability with condition''. In this framework, for some condition $\alpha$ that cryptosystem $C$ satisfies, if hash function $H$ is indifferentiable from RO under condition $\alpha$, $C$ is secure when RO is instantiated with $H$. We note the condition of ``prefix-free'' that the above schemes satisfy. We show that the MD-TypeII hash function is indifferentiable from RO under this condition. When the output length of RO is incompatible with that of the hash function, the output size is expanded by Key Derivation Functions (KDFs). Since a KDF is specified as MGF1 in RSA's PKCS $\#$1 V2.1, its security discussion is important in practice. We show that, KDFs using the MD-TypeII hash function (KDF-MD-TypeII) are indifferentiable from ROs under this condition of ``prefix-free''. Therefore, we can conclude that the above practical encryption schemes are secure even when ROs are instantiated with (KDF-)MD-TypeII hash functions. Dodis, Ristenpart and Shrimpton showed that FDH, PSS, Fiat-Shamir, and so on are secure when RO is instantiated with the MD-TypeII hash function in the ICM, their analyses use the different approach from our approach called indifferentiability from public-use RO (pub-RO). They showed that the above cryptosystems are secure in the pub-RO model and the MD-TypeII hash function is indifferentiable from pub-RO. Since their analyses did not consider the structure of KDFs, there might exist some attack using a KDF's structure. We show that KDFs using pub-RO (KDF-pub-RO) is differentiable from pub-RO. Thus, we cannot trivially extend the result of Dodis et al to the indifferentiability for KDF-MD-TypeII hash functions. We propose a new oracle called private interface leak RO (privleak-RO). We show that KDF-pub-ROs are indifferentiable from privleak-ROs and the above cryptosystems are secure in the privleak-RO model. Therefore, by combining the result of Dodis et al. with our result, we can conclude that the above cryptosystems are secure when ROs are instantiated with KDF-MD-TypeII hash functions. Since OAEP, RSA-KEM, PSEC-KEM, ECIES-KEM and many other encryption schemes are insecure in the pub-RO (privleak-RO) model, we cannot confirm the security of these encryption schemes from the approach of Dodis et al. Therefore, the result of Dodis et al can be supplemented with our result. Consequently, from the two results we can confirm the security of almost practical cryptosystems when ROs are instantiated with (KDF-)MD-TypeII hash functions.