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Replacing SHA-2 with SHA-3 Enhances Generic Security of \(\mathtt {HMAC}\)

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Topics in Cryptology - CT-RSA 2016 (CT-RSA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9610))

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Abstract

In this paper, we study the MAC- and the PRF-security of \(\mathtt {HMAC}\) in the sense of generic security when replacing SHA-2 with SHA-3. We first consider the generic security of the SHA-3-based \(\mathtt {HMAC}\) construction: \(\mathtt {Sponge}\)-based \(\mathtt {HMAC}\). We provide (nearly) tight upper-bounds on the MAC- and the PRF-security of \(\mathtt {Sponge}\)-based \(\mathtt {HMAC}\), which are \(O(\frac{nq}{2^n})\) and \(O(\frac{q^2}{2^{n}})\), respectively. Here, q is the number of queries to \(\mathtt {HMAC}\) and \(n\) is the output length of the hash function.

We then compare the MAC- and the PRF-security of \(\mathtt {Sponge}\)-based \(\mathtt {HMAC}\) with those of the SHA-2-based \(\mathtt {HMAC}\) constructions: \(\mathtt {MD}\)- (Merkle-Damgård) or \(\mathtt {ChopMD}\)-based \(\mathtt {HMAC}\). It was proven that the upper-bounds on the MAC- and the PRF-security of \(\mathtt {MD}\)-based \(\mathtt {HMAC}\) are both \(O(\frac{\ell q^2}{2^n})\), and those for \(\mathtt {ChopMD}\)-based \(\mathtt {HMAC}\) are both \(O(\frac{q^2}{2^{n}} + \frac{\ell q^2}{2^{n+t}})\). Here, q is the number of queries to \(\mathtt {HMAC}\), \(\ell \) is the maximum query length, \(n\) is the output length of the hash function, and t is the number of truncated bits in \(\mathtt {ChopMD}\). Hence, replacing SHA-2 with SHA-3 enhances the MAC-security of \(\mathtt {HMAC}\). Replacing SHA-2 having the \(\mathtt {MD}\) construction with SHA-3 enhances the PRF-security of \(\mathtt {HMAC}\), and if \(\ell > 2^t\) then replacing SHA-2 having the \(\mathtt {ChopMD}\) construction with SHA-3 enhances the PRF-security of \(\mathtt {HMAC}\).

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Notes

  1. 1.

    Although the proofs in [2, 3, 11] deal with only \(\mathtt {HMAC}\_\mathtt {MD}\), these can be applied to \(\mathtt {HMAC}\_\mathtt {ChopMD}\). Their proofs only depend on \(\ell \) and the output length of a compression function. The upper-bounds for \(\mathtt {HMAC}\_\mathtt {ChopMD}\) are obtained by adding the probability corresponding with the truncated output length.

  2. 2.

    Note that for the sake of simplicity, we omit the discussion for the probability of recovering a secret key that is \(O(\frac{Q}{2^k})\).

  3. 3.

    Note that \(|K'| = r\) holds. Hence one can recover \(K'\) with \(Q \le 2^{r}\). In the case of SHA3-512 (\(c=1024\), \(r=574\) and \(n = 512\)), \(2^r \le 2^{b/2}\) holds, thereby, in order to obtain the bounds, we require the assumption that \(Q \le 2^{574}\).

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Acknowledgements

Lei Wang is supported by Major State Basic Research Development Program (973 Plan) (2013CB338004), National Natural Science Foundation of China (61472250), Innovation Plan of Science and Technology of Shanghai (14511100300), Doctoral Fund of Ministry of Education of China (20120073110094), and National Natural Science Foundation of China (NO. 61402288).

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  1. Mitsubishi Electric Corporation, Kanagawa, Japan

    Yusuke Naito

  2. Shanghai Jiao Tong University, Shanghai, China

    Lei Wang

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  1. Yusuke Naito
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Correspondence to Yusuke Naito .

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  1. NEC Cloud Systems Research Laboratories, Kawasaki, Japan

    Kazue Sako

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Naito, Y., Wang, L. (2016). Replacing SHA-2 with SHA-3 Enhances Generic Security of \(\mathtt {HMAC}\) . In: Sako, K. (eds) Topics in Cryptology - CT-RSA 2016. CT-RSA 2016. Lecture Notes in Computer Science(), vol 9610. Springer, Cham. https://doi.org/10.1007/978-3-319-29485-8_23

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  • DOI: https://doi.org/10.1007/978-3-319-29485-8_23

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