International Association
for Cryptologic Research

Recently, in post-quantum cryptography migration, it has been shown that an IND-1-CCA-secure key encapsulation mechanism (KEM) is required for replacing an ephemeral Diffie-Hellman (DH) in widely-used protocols, e.g., TLS, Signal, and Noise. IND-1-CCA security is a notion similar to the traditional IND-CCA security except that the adversary is restricted to one single decapsulation query. At EUROCRYPT 2022, based on CPA-secure public-key encryption (PKE), Huguenin-Dumittan and Vaudenay presented two IND-1-CCA KEM constructions called $T_{CH}$ and $T_H$, which are much more efficient than the widely-used IND-CCA-secure Fujisaki-Okamoto (FO) KEMs. The security of $T_{CH}$ was proved in both random oracle model (ROM) and quantum random oracle model (QROM). However, the QROM proof of $T_{CH}$ relies on an additional ciphertext expansion. %requires that the ciphertext size of the resulting KEM is twice as large as the one of the underlying PKE. While, the security of $T_H$ was only proved in the ROM, and the QROM proof is left open. In this paper, we prove the security of $T_H$ and $T_{RH}$ (an implicit variant of $T_H$) in both ROM and QROM with much tighter reductions than Huguenin-Dumittan and Vaudenay's work. In particular, our QROM proof will not lead to ciphertext expansion. Moreover, for $T_{RH}$, $T_H$ and $T_{CH}$, we also show that a $O(1/q)$ ($O(1/q^2)$, resp.) reduction loss is unavoidable in the ROM (QROM, resp.), and thus claim that our ROM proof is optimal in tightness. Finally, we make a comprehensive comparison among the relative strengths of IND-1-CCA and IND-CCA in the ROM and QROM.

Key encapsulation mechanism (KEM) variants of the Fujisaki-Okamoto (FO) transformation (TCC 2017) that turn a weakly-secure public-key encryption (PKE) into an IND-CCA-secure KEM, were widely used among the KEM submissions to the NIST Post-Quantum Cryptography Standardization Project. Under the standard CPA security assumptions, i.e., OW-CPA and IND-CPA, the security of these variants in the quantum random oracle model (QROM) has been proved by black-box reductions, e.g., Jiang et al. (CRYPTO 2018), and by non-black-box reductions (EUROCRYPT 2020). The non-black-box reductions (EUROCRYPT 2020) have a liner security loss, but can only apply to specific \emph{reversible} adversaries with strict \emph{reversible} implementation. On the contrary, the existing black-box reductions in the literature can apply to an arbitrary adversary with an arbitrary implementation, but suffer a quadratic security loss. In this paper, for KEM variants of the FO transformation, we first show the tightness limits of the black-box reductions, and prove that a \emph{measurement-based} reduction in the QROM from breaking the standard OW-CPA (or IND-CPA) security of the underlying PKE to breaking the IND-CCA security of the resulting KEM, will \emph{inevitably} incur a quadratic loss of the security, where ``measurement-based" means the reduction measures a hash query from the adversary and uses the measurement outcome to break the underlying security of PKE. In particular, most black-box reductions for these FO-like KEM variants are of this type, and our results suggest an explanation for the lack of progress in improving this reduction tightness in terms of the degree of security loss. Then, we further show that the quadratic loss is also unavoidable when one turns a search problem into a decision problem using the one-way to hiding technique in a black-box manner, which has been recognized as an essential technique to prove the security of cryptosystems involving quantum random oracles.

The recent post-quantum cryptography standardization project launched by NIST increased the interest in generic key encapsulation mechanism (KEM) constructions in the quantum random oracle (QROM). Based on a OW-CPA-secure public-key encryption (PKE), Hofheinz, Hövelmanns and Kiltz (TCC 2017) first presented two generic constructions of an IND-CCA-secure KEM with quartic security loss in the QROM, one with implicit rejection (a pseudorandom key is return for an invalid ciphertext) and the other with explicit rejection (an abort symbol is returned for an invalid ciphertext). Both are widely used in the NIST Round-1 KEM submissions and the ones with explicit rejection account for 40%. Recently, the security reductions have been improved to quadratic loss under a standard assumption, and be tight under a nonstandard assumption by Jiang et al. (Crypto 2018) and Saito, Xagawa and Yamakawa (Eurocrypt 2018). However, these improvements only apply to the KEM submissions with implicit rejection and the techniques do not seem to carry over to KEMs with explicit rejection.In this paper, we provide three generic constructions of an IND-CCA-secure KEM with explicit rejection, under the same assumptions and with the same tightness in the security reductions as the aforementioned KEM constructions with implicit rejection (Crypto 2018, Eurocrypt 2018). Specifically, we develop a novel approach to verify the validity of a ciphertext in the QROM and use it to extend the proof techniques for KEM constructions with implicit rejection (Crypto 2018, Eurocrypt 2018) to our KEM constructions with explicit rejection. Moreover, using an improved version of one-way to hiding lemma by Ambainis, Hamburg and Unruh (ePrint 2018/904), for two of our constructions, we present tighter reductions to the standard IND-CPA assumption. Our results directly apply to 9 KEM submissions with explicit rejection, and provide tighter reductions than previously known (TCC 2017).

With the gradual progress of NIST’s post-quantum cryptography standardization, the Round-1 KEM proposals have been posted for public to discuss and evaluate. Among the IND-CCA-secure KEM constructions, mostly, an IND-CPA-secure (or OW-CPA-secure) public-key encryption (PKE) scheme is first introduced, then some generic transformations are applied to it. All these generic transformations are constructed in the random oracle model (ROM). To fully assess the post-quantum security, security analysis in the quantum random oracle model (QROM) is preferred. However, current works either lacked a QROM security proof or just followed Targhi and Unruh’s proof technique (TCC-B 2016) and modified the original transformations by adding an additional hash to the ciphertext to achieve the QROM security.In this paper, by using a novel proof technique, we present QROM security reductions for two widely used generic transformations without suffering any ciphertext overhead. Meanwhile, the security bounds are much tighter than the ones derived by utilizing Targhi and Unruh’s proof technique. Thus, our QROM security proofs not only provide a solid post-quantum security guarantee for NIST Round-1 KEM schemes, but also simplify the constructions and reduce the ciphertext sizes. We also provide QROM security reductions for Hofheinz-Hövelmanns-Kiltz modular transformations (TCC 2017), which can help to obtain a variety of combined transformations with different requirements and properties.