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

Decorrelation: A Theory for Block Cipher Security

Pseudorandomness is a classical model for the security of block ciphers. In this paper we propose convenient tools in order to study it in connection with the Shannon Theory, the Carter-Wegman universal hash functions paradigm, and the Luby-Rackoff approach. This enables the construction of new ciphers with security proofs under specific models. We show how to ensure security against basic differential and linear cryptanalysis and even more general attacks. We propose practical construction scheme

Pseudorandom Functions: Three Decades Later

In 1984, Goldreich, Goldwasser and Micali formalized the concept of pseudorandom functions and proposed a construction based on any length-doubling pseudorandom generator. Since then, pseudorandom functions have turned out to be an extremely influential abstraction, with applications ranging from message authentication to barriers in proving computational complexity lower bounds. In this tutorial we survey various incarnations of pseudorandom functions, giving self-contained proofs of key results from the literature. Our main focus is on feasibility results and constructions, as well as on limitations of (and induced by) pseudorandom functions. Along the way we point out some open questions that we believe to be within reach of current techniques

Provable Security of Substitution-Permutation Networks

Many modern block ciphers are constructed based on the paradigm of substitution-permutation networks (SPNs). But, somewhat surprisingly---especially in comparison with Feistel networks, which have been analyzed by dozens of papers going back to the seminal work of Luby and Rackoff---there are essentially no provable-security results about SPNs. In this work, we initiate a comprehensive study of the security of SPNs as strong pseudorandom permutations when the underlying $S$-box is modeled as a public random permutation. We show that 3~rounds of S-boxes are necessary and sufficient for secure linear SPNs, but that even 1-round SPNs can be secure when non-linearity is allowed. Additionally, our results imply security in settings where an SPN structure is used for domain extension of a block cipher, even when the attacker has direct access to the small-domain block cipher

Security Evaluation of MISTY Structure with SPN Round Function

This paper deals with the security of MISTY structure with SPN round function. We study the lower bound of the number of active s-boxes for differential and linear characteristics of such block cipher construction. Previous result shows that the differential bound is consistent with the case of Feistel structure with SPN round function, yet the situation changes when considering the linear bound. We carefully revisit such issue, and prove that the same bound in fact could be obtained for linear characteristic. This result combined with the previous one thus demonstrates a similar practical secure level for both Feistel and MISTY structures. Besides, we also discuss the resistance of MISTY structure with SPN round function against other kinds of cryptanalytic approaches including the integral cryptanalysis and impossible differential cryptanalysis. We confirm the existence of 6-round integral distinguishers when the linear transformation of the round function employs a binary matrix (i.e., the element in the matrix is either 0 or 1), and briefly describe how to characterize 5/6/7-round impossible differentials through the matrix-based method

Towards Minimizing Non-linearity in Type-II Generalized Feistel Networks

Recent works have revisited blockcipher structures to achieve MPC- and ZKP-friendly designs. In particular, Albrecht et al. (EUROCRYPT 2015) first pioneered using a novel structure SP networks with partial non-linear layers (P-SPNs) and then (ESORICS 2019) repopularized using multi-line generalized Feistel networks (GFNs). In this paper, we persist in exploring symmetric cryptographic constructions that are conducive to the applications such as MPC. In order to study the minimization of non-linearity in Type-II Generalized Feistel Networks, we generalize the (extended) GFN by replacing the bit-wise shuffle in a GFN with the stronger linear layer in P-SPN and introducing the key in each round. We call this scheme Generalized Extended Generalized Feistel Network (GEGFN). When the block-functions (or S-boxes) are public random permutations or (domain-preserving) functions, we prove CCA security for the 5-round GEGFN. Our results also hold when the block-functions are over the prime fields F_p, yielding blockcipher constructions over (F_p)^*

Decorrelation: a theory for block cipher security

Pseudorandomness is a classical model for the security of block ciphers. In this paper we propose convenient tools in order to study it in connection with the Shannon Theory, the Carter-Wegman universal hash functions paradigm, and the Luby-Rackoff approach. This enables the construction of new ciphers with security proofs under specific models. We show how to ensure security against basic differential and linear cryptanalysis and even more general attacks. We propose practical construction scheme