Provable Security of (Tweakable) Block Ciphers Based on Substitution-Permutation Networks

Substitution-Permutation Networks (SPNs) refer to a family of constructions which build a wn-bit block cipher from n-bit public permutations (often called S-boxes), which alternate keyless and “local” substitution steps utilizing such S-boxes, with keyed and “global” permu- tation steps which are non-cryptographic. Many widely deployed block ciphers are constructed based on the SPNs, but there are essentially no provable-security results about SPNs. In this work, we initiate a comprehensive study of the provable security of SPNs as (possibly tweakable) wn-bit block ciphers, when the underlying n-bit permutation is modeled as a public random permutation. When the permutation step is linear (which is the case for most existing designs), we show that 3 SPN rounds are necessary and sufficient for security. On the other hand, even 1-round SPNs can be secure when non-linearity is allowed. Moreover, 2-round non-linear SPNs can achieve “beyond- birthday” (up to 2 2n/3 adversarial queries) security, and, as the number of non-linear rounds increases, our bounds are meaningful for the number of queries approaching 2 n . Finally, our non-linear SPNs can be made tweakable by incorporating the tweak into the permutation layer, and provide good multi-user security. As an application, our construction can turn two public n-bit permuta- tions (or fixed-key block ciphers) into a tweakable block cipher working on wn-bit inputs, 6n-bit key and an n-bit tweak (for any w ≥ 2); the tweakable block cipher provides security up to 2 2n/3 adversarial queries in the random permutation model, while only requiring w calls to each permutation, and 3w field multiplications for each wn-bit input

Small-Box Cryptography

One of the ultimate goals of symmetric-key cryptography is to find a rigorous theoretical framework for building block ciphers from small components, such as cryptographic S-boxes, and then argue why iterating such small components for sufficiently many rounds would yield a secure construction. Unfortunately, a fundamental obstacle towards reaching this goal comes from the fact that traditional security proofs cannot get security beyond 2^{-n}, where n is the size of the corresponding component. As a result, prior provably secure approaches - which we call "big-box cryptography" - always made n larger than the security parameter, which led to several problems: (a) the design was too coarse to really explain practical constructions, as (arguably) the most interesting design choices happening when instantiating such "big-boxes" were completely abstracted out; (b) the theoretically predicted number of rounds for the security of this approach was always dramatically smaller than in reality, where the "big-box" building block could not be made as ideal as required by the proof. For example, Even-Mansour (and, more generally, key-alternating) ciphers completely ignored the substitution-permutation network (SPN) paradigm which is at the heart of most real-world implementations of such ciphers. In this work, we introduce a novel paradigm for justifying the security of existing block ciphers, which we call small-box cryptography. Unlike the "big-box" paradigm, it allows one to go much deeper inside the existing block cipher constructions, by only idealizing a small (and, hence, realistic!) building block of very small size n, such as an 8-to-32-bit S-box. It then introduces a clean and rigorous mixture of proofs and hardness conjectures which allow one to lift traditional, and seemingly meaningless, "at most 2^{-n}" security proofs for reduced-round idealized variants of the existing block ciphers, into meaningful, full-round security justifications of the actual ciphers used in the real world. We then apply our framework to the analysis of SPN ciphers (e.g, generalizations of AES), getting quite reasonable and plausible concrete hardness estimates for the resulting ciphers. We also apply our framework to the design of stream ciphers. Here, however, we focus on the simplicity of the resulting construction, for which we managed to find a direct "big-box"-style security justification, under a well studied and widely believed eXact Linear Parity with Noise (XLPN) assumption. Overall, we hope that our work will initiate many follow-up results in the area of small-box cryptography

Permutation Based EDM: An Inverse Free BBB Secure PRF

In CRYPTO 2019, Chen et al. have initiated an interesting research direction in designing PRF based on public permutations. They have proposed two beyond the birthday bound secure n-bit to n-bit PRF constructions, i.e., SoEM22 and SoKAC21, which are built on public permutations, where n is the size of the permutation. However, both of their constructions require two independent instances of public permutations. In FSE 2020, Chakraborti et al. have proposed a single public permutation based n-bit to n-bit beyond the birthday bound secure PRF, which they refer to as PDMMAC. Although the construction is minimal in the number of permutations, it requires the inverse call of its underlying permutation in their design. Coming up with a beyond the birthday bound secure public permutation based n-bit to n-bit PRF with a single permutation and two forward calls was left as an open problem in their paper. In this work, we propose pEDM, a single permutation based n-bit to n-bit PRF with two calls that do not require invertibility of the permutation. We have shown that our construction is secured against all adaptive information-theoretic distinguishers that make roughly up to 22n/3 construction and primitive queries. Moreover, we have also shown a matching attack with similar query complexity that establishes the tightness of our security bound

BBB Secure Nonce Based MAC Using Public Permutations

In the recent trend of CAESAR competition and NIST light-weight competition, cryptographic community have witnessed the submissions of several cryptographic schemes that are build on public random permutations. Recently, in CRYPTO 2019, Chen et al. have initiated an interesting research direction in designing beyond birthday bound PRFs from public random permutations and they proposed two instances of such PRFs. In this work, we extend this research direction by proposing a nonce-based MAC build from public random permutations. We show that our proposed MAC achieves $2n/3$ bit security (with respect to the state size of the permutation) and the bound is essentially tight. Moreover, the security of the MAC degrades gracefully with the repetition of the nonce

Tight Security Analysis of the Public Permutation-Based PMAC_Plus

Yasuda proposed a variable input-length PRF in CRYPTO 2011, called \textsf{PMAC_Plus}, based on an $n$-bit block cipher. \textsf{PMAC_Plus} is a rate-$1$ construction and inherits the well-known $\textsf{PMAC}$ parallel network with a low additional cost. However, unlike $\textsf{PMAC}$, \textsf{PMAC_Plus} is secure roughly up to $2^{2n/3}$ queries. Zhang et al. proposed \textsf{3kf9} in ASIACRYPT 2012, Naito proposed \textsf{LightMAC_Plus} in ASIACRYPT 2017, and Iwata et al. proposed \textsf{GCM-SIV2} in FSE 2017 -- all of them secure up to around $2^{2n/3}$ queries. Their structural designs and corresponding security proofs were unified by Datta et al. in their framework {\em Double-block Hash-then-Sum} (\textsf{DbHtS}). Leurent et al. in CRYPTO 2018 and then Lee et al. in EUROCRYPT 2020 established a tight security bound of $2^{3n/4}$ on \textsf{DbHtS}. That \textsf{PMAC_Plus} provides security for roughly up to $2^{3n/4}$ queries is a consequence of this result. In this paper, we propose a public permutation-based variable input-length PRF called {\textsf{pPMAC_Plus}}. We show that {\textsf{pPMAC_Plus}} is secure against all adversaries that make at most $2^{2n/3}$ queries. We also show that the bound is essentially tight. It is of note here that instantiation of each block cipher of {\textsf{pPMAC_Plus}} with the two-round iterated Even-Mansour cipher can yield a beyond the birthday bound secure PRF based on public permutations. Altogether, the solution incurs $(2\ell + 4)$ permutation calls, whereas our proposal requires only $(\ell+2)$ permutation calls, $\ell$ being the maximum number of message blocks

Multi-User BBB Security of Public Permutations Based MAC

At CRYPTO 2019, Chen et al. have shown a beyond the birthday bound secure $n$-bit to $n$-bit PRF based on public random permutations. Followed by the work, Dutta and Nandi have proposed a beyond the birthday bound secure nonce based MAC $\textsf{nEHtM}_p$ based on public random permutation. In particular, the authors have shown that $\textsf{nEHtM}_p$ achieves tight $2n/3$-bit security ({\em with respect to the state size of the permutation}) in the single-user setting, and their proven bound gracefully degrades with the repetition of the nonces. However, we have pointed out that their security proof is not complete (albeit it does not invalidate their security claim). In this paper, we propose a minor variant of $\textsf{nEHtM}_p$ construction, called $\textsf{nEHtM}^*_p$ and show that it achieves a tight $2n/3$ bit security in the multi-user setting. Moreover, the security bound of our construction also degrades gracefully with the repetition of nonces. Finally, we have instantiated our construction with the PolyHash function to realize a concrete beyond the birthday bound secure public permutation-based MAC, $\textsf{nEHtM}_p^+$ in the multi-user setting

Tweakable Blockciphers for Efficient Authenticated Encryptions with Beyond the Birthday-Bound Security

Modular design via a tweakable blockcipher (TBC) offers efficient authenticated encryption (AE) schemes (with associated data) that call a blockcipher once for each data block (of associated data or a plaintext). However, the existing efficient blockcipher-based TBCs are secure up to the birthday bound, where the underlying keyed blockcipher is a secure strong pseudorandom permutation. Existing blockcipher-based AE schemes with beyond-birthday-bound (BBB) security are not efficient, that is, a blockcipher is called twice or more for each data block. In this paper, we present a TBC, XKX, that offers efficient blockcipher-based AE schemes with BBB security, by combining with efficient TBC-based AE schemes such as ΘCB3 an

Provable security for lightweight message authentication and encryption

The birthday bound often limits the security of a cryptographic scheme to half of the block size or internal state size. This implies that cryptographic schemes require a block size or internal state size that is twice the security level, resulting in larger and more resource-intensive designs. In this thesis, we introduce abstract constructions for message authentication codes and stream ciphers that we demonstrate to be secure beyond the birthday bound. Our message authentication codes were inspired by previous work, specifically the message authentication code EWCDM by Cogliati and Seurin, as well as the work by Mennink and Neves, which demonstrates easy proofs of security for the sum of permutations and an improved bound for EWCDM. We enhance the sum of permutations by incorporating a hash value and a nonce in our stateful design, and in our stateless design, we utilize two hash values. One advantage over EWCDM is that the permutation calls, or block cipher calls, can be parallelized, whereas in EWCDM they must be performed sequentially. We demonstrate that our constructions provide a security level of 2n/3 bits in the nonce-respecting setting. Subsequently, this bound was further improved to 3n/4 bits of security. Additionally, it was later discovered that security degrades gracefully with nonce repetitions, unlike EWCDM, where the security drops to the birthday bound with a single nonce repetition. Contemporary stream cipher designs aim to minimize the hardware module's resource requirements by incorporating an externally available resource, all while maintaining a high level of security. The security level is typically measured in relation to the size of the volatile internal state, i.e., the state cells within the cipher's hardware module. Several designs have been proposed that continuously access the externally available non-volatile secret key during keystream generation. However, there exists a generic distinguishing attack with birthday bound complexity. We propose schemes that continuously access the externally available non-volatile initial value. For all constructions, conventional or contemporary, we provide proofs of security against generic attacks in the random oracle model. Notably, stream ciphers that use the non-volatile initial value during keystream generation offer security beyond the birthday bound. Based on these findings, we propose a new stream cipher design called DRACO