Balanced permutations Even-Mansour ciphers

The $r$-rounds Even-Mansour block cipher uses $r$ public permutations of $\{0, 1\}^n$ and $r+1$ secret keys. An attack on this construction was described in \cite{DDKS}, for $r = 2, 3$. Although this attack is only marginally better than brute force, it is based on an interesting observation (due to \cite{NWW}): for a typical permutation $P$, the distribution of $P(x) \oplus x$ is not uniform. To address this, and other potential threats that might stem from this observation in this (or other) context, we introduce the notion of a ``balanced permutation\u27\u27 for which the distribution of $P(x) \oplus x$ is uniform, and show how to generate families of balanced permutations from the Feistel construction. This allows us to define a $2n$-bit block cipher from the $2$-rounds Even-Mansour scheme. The cipher uses public balanced permutations of $\{0, 1\}^{2n}$, which are based on two public permutations of $\{0, 1\}^{n}$. By construction, this cipher is immune against attacks that rely on the non-uniform behavior of $P(x) \oplus x$. We prove that this cipher is indistinguishable from a random permutation of $\{0, 1\}^{2n}$, for any adversary who has oracle access to the public permutations and to an encryption/decryption oracle, as long as the number of queries is $o (2^{n/2})$. As a practical example, we discuss the properties and the performance of a $256$-bit block cipher that is based on AES

Towards Understanding the Known-Key Security of Block Ciphers

Known-key distinguishers for block ciphers were proposed by Knudsen and Rijmen at ASIACRYPT 2007 and have been a major research topic in cryptanalysis since then. A formalization of known-key attacks in general is known to be difficult. In this paper, we tackle this problem for the case of block ciphers based on ideal components such as random permutations and random functions as well as propose new generic known-key attacks on generalized Feistel ciphers. We introduce the notion of known-key indifferentiability to capture the security of such block ciphers under a known key. To show its meaningfulness, we prove that the known-key attacks on block ciphers with ideal primitives to date violate security under known-key indifferentiability. On the other hand, to demonstrate its constructiveness, we prove the balanced Feistel cipher with random functions and the multiple Even-Mansour cipher with random permutations known-key indifferentiable for a sufficient number of rounds. We note that known-key indifferentiability is more quickly and tightly attained by multiple Even-Mansour which puts it forward as a construction provably secure against known-key attacks

An Algebraic System for Constructing Cryptographic Permutations over Finite Fields

In this paper we identify polynomial dynamical systems over finite fields as the central component of almost all iterative block cipher design strategies over finite fields. We propose a generalized triangular polynomial dynamical system (GTDS), and give a generic algebraic definition of iterative (keyed) permutation using GTDS. Our GTDS-based generic definition is able to describe widely used and well-known design strategies such as substitution permutation network (SPN), Feistel network and their variants among others. We show that the Lai-Massey design strategy for (keyed) permutations is also described by the GTDS. Our generic algebraic definition of iterative permutation is particularly useful for instantiating and systematically studying block ciphers and hash functions over $\mathbb{F}_p$ aimed for multiparty computation and zero-knowledge based cryptographic protocols. Finally, we provide the discrepancy analysis a technique used to measure the (pseudo-)randomness of a sequence, for analyzing the randomness of the sequence generated by the generic permutation or block cipher described by GTDS

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

A method for constructing permutations, involutions and orthomorphisms with strong cryptographic properties

S-Boxes are crucial components in the design of many symmetric ciphers. To construct permutations having strong cryptographic properties is not a trivial task. In this work, we propose a new scheme based on the well-known Lai-Massey structure for generating permutations of dimension n = 2к, к 2. The main cores of our constructions are: the inversion in GF(2k), an arbitrary к-bit non-bijective function (which has no pre-image for 0) and any к-bit permutation. Combining these components with the finite field multiplication, we provide new 8-bit permutations without fixed points possessing a very good combination for nonlinearity, differential uniformity and minimum degree — (104; 6; 7) which can be described by a system of polynomial equations with degree 3. Also, we show that our approach can be used for constructing involutions and orthomorphisms with strong cryptographic properties

Introducing two Low-Latency Cipher Families: Sonic and SuperSonic

For many latency-critical operations in computer systems, like memory reads/writes, adding encryption can have a big impact on the performance. Hence, the existence of cryptographic primitives with good security properties and minimal latency is a key element in the wide-spread implementation of such security measures. In this paper, we introduce two new families of low-latency permutations/block ciphers called Sonic and SuperSonic, inspired by the Simon block ciphers

Some properties of the output sequences of combined generator over finite fields

The sequences are an important part of the cryptography and analysis of their properties is of great interest. In this paper, the following characteristics of combined generator are analyzed: period of output sequences and the distribution of elements in the output sequences over finite field

Wide Tweakable Block Ciphers Based on Substitution-Permutation Networks: Security Beyond the Birthday Bound

Substitution-Permutation Networks (SPNs) refer to a family of constructions which build a $wn$-bit (tweakable) block cipher from $n$-bit public permutations. Many widely deployed block ciphers are part of this family and rely on very small public permutations. Surprisingly, this structure has seen little theoretical interest when compared with Feistel networks, another high-level structure for block ciphers. This paper extends the work initiated by Dodis et al. in three directions; first, we make SPNs tweakable by allowing keyed tweakable permutations in the permutation layer, and prove their security as tweakable block ciphers. Second, we prove beyond-the-birthday-bound security for $2$-round non-linear SPNs with independent S-boxes and independent round keys. Our bounds also tend towards optimal security $2^n$ (in terms of the number of threshold queries) as the number of rounds increases. Finally, all our constructions permit their security proofs in the multi-user setting. As an application of our results, SPNs can be used to build provably secure wide tweakable block ciphers from several public permutations, or from a block cipher. More specifically, our construction can turn two strong public $n$-bit permutations into a tweakable block cipher working on $wn$-bit blocks and using a $6n$-bit key and an $n$-bit tweak (for any $w\geq 2$); 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 block

Provably Secure Reflection Ciphers

This paper provides the first analysis of reflection ciphers such as PRINCE from a provable security viewpoint. As a first contribution, we initiate the study of key-alternating reflection ciphers in the ideal permutation model. Specifically, we prove the security of the two-round case and give matching attacks. The resulting security bound takes form $O(qp^2/2^{2n}+q^2/2^n)$, where $q$ is the number of construction evaluations and $p$ is the number of direct adversarial queries to the underlying permutation. Since the two-round construction already achieves an interesting security lower bound, this result can also be of interest for the construction of reflection ciphers based on a single public permutation. Our second contribution is a generic key-length extension method for reflection ciphers. It provides an attractive alternative to the $FX$ construction, which is used by PRINCE and other concrete key-alternating reflection ciphers. We show that our construction leads to better security with minimal changes to existing designs. The security proof is in the ideal cipher model and relies on a reduction to the two-round Even-Mansour cipher with a single round key. In order to obtain the desired result, we sharpen the bad-transcript analysis and consequently improve the best-known bounds for the single-key Even-Mansour cipher with two rounds. This improvement is enabled by a new sum-capture theorem that is of independent interest