Which type of cryptanalysis method is based on substitution-permutation networks

1 Introduction

Substitution-Permutation Networks. Modern block ciphers are generally constructed using two main paradigms [KL15]: Feistel networks [Fei73] or substitution-permutation networks (SPNs) [Sha49, Fei73]. Examples of block ciphers based on Feistel networks include DES, FEAL, MISTY and KASUMI; block ciphers based on SPNs include AES, Serpent, and PRESENT. These two approaches share the same goal: namely, to extend a “pseudorandom object” on a small domain to a (keyed) pseudorandom permutation on a larger domain by repeating a few, relatively simple operations several times across multiple rounds. Simplifying somewhat, Feistel networks begin with a keyed pseudorandom function on n-bit inputs and extend this to give a keyed pseudorandom permutation on 2n-bit inputs. On the other hand, SPNs start with one or more public “random permutations” on n-bit inputs (called S-boxes) and extend them to give a keyed pseudorandom permutation on wn-bit inputs for some w, by iterating the following steps:

1. 1.

   Substitution step: break down the wn-bit state into w disjoint n-bit blocks, and compute an S-box on each n-bit block;

2. 2.

   Permutation step: apply a non-cryptographic, keyed permutation to the whole wn-bit state (which is also applied to the plaintext before the first round).

Proving the security of a concrete block cipher unconditionally is currently beyond our capabilities. Thus, the usual approach is to prove that the high-level structure is sound in a relevant security model. For Feistel networks, a substantial line of work, starting with Luby and Rackoff’s seminal work [LR88], and culminating with Patarin’s results [Pat03, Pat04], proves optimal security with a sufficient number of rounds. Numerous other articles [Pat10, HR10, HKT11, Tes14, CHK+16] study the security of (variants of) Feistel networks in various security models. In contrast, it is somewhat surprising that there are almost no results about provable security of SPNs (see below.) Here, we address this gap and explore conditions under which SPNs can be proven secure.

Domain Extension of Block Ciphers. Block ciphers following the SPNs typically rely on very small S-boxes (e.g. AES uses an 8-bit S-box). However, it is also possible to use a larger domain block cipher with a fixed key (which has non-trivial efficiency gains and avoid related key attacks) as “S-box” in order to extend the domain of the underlying block cipher, or to use a larger dedicated permutation (e.g., Keccak permutation [BDPA09] or with Gimli [BKL+17]), in order to directly obtain a “wide” block cipher. From this point of view, the substitution-permutation networks can also be viewed as enciphering modes of operation (of a fixed input length), in which the length n of the S-box is not necessarily small. Such enciphering modes of operations have applications to disk encryption that protects the confidentiality of data stored on a sector-addressable device, such as a hard disk. In this scenario, the disk is divided into several sectors, and each sector, viewed as a wide block, should be encrypted and decrypted independently of each other. Non-linear 1-round SPNs with secret S-boxes have already been used to provide domain extension for block ciphers [CS06, Hal07]. These constructions provide the birthday bound security, while this level of security might not be desirable for an environment where stronger security is required. One of our results will address this limitation.

1.1 Our Contribution

We analyze SPNs in the standard sense as a strong pseudorandom permutation [LR88] (i.e., against adaptive chosen-plaintext and chosen-ciphertext attacks).

Linear SPNs. We first characterize the security of linear SPNs, where the permutation layer is a linear function (over \(GF(2^n)\), where n is the size of the S-box) of the current wn-bit round key and the current wn-bit state. Indeed, most current SPN-based block ciphers (e.g., AES, Serpent, PRESENT, etc.) use linear permutation step, which involves a simple key-mixing step followed by an invertible linear transformation. For this widely used setting we give a general against any 2-round linear SPNs with \(w \ge 2\).Footnote 1 Complementing this attack, we show that a 3-round linear SPNs are secure, for any w, if the keyed linear permutations satisfy some very mild technical requirements. This result critically uses the H-coefficients technique [Pat08, CS14].

Non-Linear SPNs. In an effort to reduce the number of rounds (and get other benefits we explain below), we then turn our attention to non-linear SPNs, where the permutation step does not have to be linear (although must remain efficient and “non-cryptographic”). Here we show that even a 1-round SPN can be secure, if appropriate keyed permutations are used. We identify a combinatorial property on the permutations — which we term blockwise universality — that suffices for security in this case, and then study the efficiency of constructing permutations satisfying this property. Specifically, we show a construction of a satisfactory permutation with n-bit keys (but having high degree), and another construction with longer keys but having degree 3.

We then show that, by using such blockwise independent permutations, the security of resulting SPNs increases when we increase the number of rounds: while 1 round already achieves “birthday security”, as our main technical result we show that 2-round non-linear SPNs (with independent S-boxes and keys in different rounds) achieves “beyond-birthday” security (for up to \(2^{2n/3}\) queries). This result uses the refinement of the H-coefficient technique due to [HT16]. We also give an asymptotic analysis of non-linear SPNs built from blockwise universal permutations using the coupling technique of [MRS09, HR10]. In particular, for \(r=2s\) we prove that r-round SPNs are secure as long as the number of adversarial queries is well below \(2^{sn/(s+1)}\). Thus, as r grows, our bounds tend towards optimal \(2^n\) security.

As an additional benefit of this setting, we show that the blockwise universal permutations can be efficiently tweaked, meaning that our non-linear SPN constructions yield tweakable block ciphers [LRW11], which is important for some settings. Finally, we analyze our non-linear SPNs in the multi-user setting using the point-wise proximity technique of [HT16].

Application to Wide Tweakable Block Ciphers. Besides providing theoretical insights on SPN-based block ciphers, our results also have a practical interest in the context of domain extension for block ciphers and permutation-based cryptography. For example, if our construction is instantiated with two n-bit permutations and a tweakable permutation \(\mathsf {TBPE}\) in the permutation layer (as defined in Sect. 2.2), then we can build a wide tweakable block cipher with key space \(\{0,1\}^{6n}\), tweak space \(\{0,1\}^{n}\) and message space \(\{0,1\}^{wn}\) for any integer \(w \ge 2\). This tweakable block cipher requires w calls to each permutation and 3w field multiplications for each encryption/decryption call. The multi-user advantage of any adversary is shown to be small as long as the number of its queries is well below \(2^{2n/3}\). This means that a 192 bit (resp. 384 bit) permutation or block cipher is sufficient to get a provably secure mode of operation as long as the number of adversarial queries is small in front of \(2^{128}\) (resp. \(2^{256}\)). As far as we know, this is the first construction for domain extension of a block cipher/permutation that enjoys beyond birthday-bound security.

Of course, to instantiate this construction we would need a good public permutation with large domain size n. As mentioned earlier, we could either use a larger domain block cipher with a fixed key, or use a larger dedicated permutation on larger domain, such as Keccak permutation [BDPA09] or Gimli [BKL+17].

Open Problems. We conjecture that r-round non-linear SPNs should actually be enough to prove security up to \(\mathcal {O}(2^{rn/(r+1)})\) adversarial queries. Proving it using combinatorial techniques seems very challenging and we leave it as an interesting open problem. It is also interesting if we can prove beyond-birthday security bounds for linear SPNs (with 3 or more rounds), as these SPNs appear to be the ones used in practice. More generally, it would be great to prove tight security bounds and matching attacks for r-round linear and non-linear SPNs.

Implications for small block size. While our results are directly meaningful when the length n of public S-boxes in at least security parameter (e.g., for building wide tweakable block ciphers), our bounds are too weak for regular SPN-based ciphers, such as AES, which use very low values of n for their S-boxes. This “\(2^n\) provable barrier” is inherent using our current modeling, where the S-box of size \(2^n\) is providing the only source of cryptographic hardness. More generally, establishing a sound theory of building block ciphers from small S-boxes is one of the biggest and most important open problems in symmetric-key cryptography. We hope that our structural results for reduced-round SPN ciphers will be useful in establishing such theory, despite not crossing the fundamental “\(2^n\) barrier” mentioned above.

1.2 Related Work

There are only a few prior papers looking at provable security of SPNs. The vast majority of such work analyzes the case of secret, key-dependent S-boxes (rather than public S-boxes as we consider here), and so we survey that work first.

SPNs with secret S-boxes. Naor and Reingold [NR99] prove security for what can be viewed as a non-linear, 1-round SPN. Their ideas were further developed, in the context of domain extension for block ciphers (see further discussion below), by Chakraborty and Sarkar [CS06] and Halevi [Hal07].

Iwata and Kurosawa [IK00] analyze SPNs in which the linear permutation step is based on the specific permutations used in the block cipher Serpent. They show an attack against 2-round SPNs of this form, and prove security for 3-round SPNs against non-adaptive adversaries. In addition to the fact that we consider public S-boxes, our linear SPN model considers generic linear permutations and we prove security against adaptive attackers.

Miles and Viola [MV15] study SPNs from a complexity-theoretic viewpoint. Two of their results are relevant here. First, they analyze the security of linear SPNs using S-boxes that are not necessarily injective (so the resulting keyed functions are not, in general, invertible). They show that r-round SPNs of this type (for \(r \ge 2\)) are secure against chosen-plaintext attacks. (In contrast, our results show that 2-round, linear SPNs are not secure against a combination of chosen-plaintext and chosen-ciphertext attacks when \(w \ge 2\).) They also analyze SPNs based on a concrete set of S-boxes, but in this case they only show security against linear/differential attacks (a form of chosen-plaintext attack), rather than all possible attacks, and only when the number of rounds is \(r = \varTheta (\log n)\).

SPNs with public S-boxes. A difference between our work and all the work discussed above is that we treat the S-boxes as public. We are aware of only one prior work analyzing the provable security of SPNs in this setting. Dodis et al. [DSSL16] recently studied the indifferentiability [MRH04] of confusion-diffusion networks, which can be viewed as unkeyed SPNs. One could translate their results to the keyed setting, but that would require using multiple, key-dependent S-boxes (rather than a fixed, public S-box) and so would not imply our results. We remark further that they show positive results only for 5 rounds and above.

As observed earlier, the Even-Mansour construction [EM97] of a (keyed) pseudorandom permutation from a public random permutation can be viewed as a 1-round, linear SPN in the degenerate case where \(w=1\) (i.e., no domain extension) and all round permutations are instantiated using simple key mixing. Security of the 1-round Even-Mansour construction against adaptive chosen-plaintext/ciphertext attacks, using independent keys for the initial and final key mixing, was shown in the original paper [EM97]. Our positive results imply security of the 1-round Even-Mansour construction (with similar concrete security bounds) as a special case. The r-round generalization of the Even-Mansour cipher has seen a lot of interest over the years, culminating with [CS14, HT16] where it was proved that the r-round Even-Mansour construction is secure up to roughly \(2^{rn/(r+1)}\) adversarial queries, when the public S-boxes are uniformly random and independent permutations and the round keys are independent. Chen et al. [CLL+14] also proved that several minimized variants of the 2-round Even-Mansour construction are also secure up to roughly \(2^{2n/3}\) adversarial queries. None of these results extend to the setting \(w>1\) considered in this work.

Cryptanalysis of SPNs. Researchers have also explored cryptanalytic attacks on generic SPNs [BS10, BBK14, DDKL, BK]. These works generally consider a model of SPNs in which round permutations are secret, random (invertible) linear transformations, and S-boxes may be secret as well; this makes the attacks stronger but positive results weaker. In many cases the complexities of the attacks are exponential in n (though still faster than a brute-force search for the key), and hence do not rule out asymptotic security results. On the positive side, Biryukov et al. [BBK14] show that 2-round SPNs (of the stronger form just mentioned) are secure against some specific types of attacks, but other attacks on such schemes have recently been identified [DDKL].

Attacks. Attacks due to Joux [Jou03] and to Halevi and Rogaway [HR04], originally developed in the afore-mentioned context of block cipher domain extension (or more exactly, in the construction of tweakable block ciphers with large domains from standard block ciphers with “small” domains) can be translated to the context of linear SPNs as well. Specifically, these attacks imply that linear 2-round SPNs of width \(w \ge 2\) are insecure, as long as the underlying field has characteristic 2.Footnote 2

Domain extension of block ciphers. Non-linear, 1-round SPNs with secret S-boxes have been used for domain extension of block ciphers before [CS06, Hal07]. Other approaches for domain extension, not relying on (pure) SPNs, have also been considered [BD99, HR03, HR04, MF07, CDMS10]. To the best of our knowledge, none of these results achieve beyond-birthday security.

Random Permutation Based Tweakable Block Ciphers. Our tweakable SPNs can be viewed as tweakable block ciphers based on public random permutations. It is easy to see that \(T:(h,t,x)\mapsto x\oplus h(t)\) is \((\delta ,\delta ')\)-blockwise universal (as defined in Sect. 2) if h is chosen from a \(\delta '\)-almost uniform and \(\delta \)-almost XOR-universal hash family. So with this permutation layer (and with \(w=1\)), we obtain the security bound for the Tweakable Even-Mansour constructions [CLS15] in the multi-user setting. In this line of research, a number of efficient constructions have been proposed [GJMN16, Men16].

2 Preliminaries

Throughout this work, we fix positive integers w and n; an element x in \(\{0,1\}^{wn}\) can be viewed as a concatenation of w blocks, each of which is of length n. The i-th block of this representation will be denoted \(x_i\) for \(i=1,\ldots ,w\), so we have

$$x=x_1||x_2||\cdots ||x_w,$$

sometimes written as \(x=(x_1,\ldots ,x_w)\).

For a set R and an integer \(s\ge 1\), \(R^{*s}\) denotes the set of all sequences that consists of s pairwise distinct elements of R. For any integer r such that \(r\ge s\), we will write \((r)_s=r!/(r-s)!\) If \(|R|=r\), then \((r)_s\) becomes the size of \(R^{*s}\). The sets of non-negative integers and non-negative real numbers are denoted \(\mathbb {N}\) and \(\mathbb {R}^{\ge 0}\), respectively. The following inequality will be used in our security proof.

Lemma 1

Let m be an integer and let x be a real number such that \(m\ge 2\) and \(-1 \le x < \frac{1}{m-1}\). Then one has

$$ (1+x)^m \le 1 + \frac{mx}{1-(m-1)x}. $$

2.1 Tweakable Substitution-Permutation Networks

All the notions below are defined for the general tweak set \(\mathcal {T}\); however, the standard “non-tweakable” setting is a special case of the definitions below when \(|\mathcal {T}|=1\).

Tweakable Permutations. For an integer \(m\ge 1\), the set of all permutations on \(\{0,1\}^m\) will be denoted \(\mathsf {Perm}(m)\). A tweakable permutation with tweak space \(\mathcal {T}\) and message space \(\mathcal {X}\) is a mapping \(\widetilde{P}:\mathcal {T}\times \mathcal {X}\rightarrow \mathcal {X}\) such that, for any tweak \(t\in \mathcal {T}\),

$$x\mapsto \widetilde{P}(t,x)$$

is a permutation of \(\mathcal {X}\). The set of all tweakable permutations with tweak space \(\mathcal {T}\) and message space \(\{0,1\}^m\) will be denoted \(\widetilde{\mathsf {Perm}}(\mathcal {T},m)\).

A keyed tweakable permutation with key space \(\mathcal {K}\), tweak space \(\mathcal {T}\) and message space \(\mathcal {X}\) is a mapping \(T:\mathcal {K}\times \mathcal {T}\times \mathcal {X}\rightarrow \mathcal {X}\) such that, for any key \(k\in \mathcal {K}\),

$$(t,x)\mapsto T(k,t,x)$$

is a tweakable permutation with tweak space \(\mathcal {T}\) and message space \(\mathcal {X}\). We will sometimes write T(k, t, x) as \(T_k(t,x)\) or \(T_{k,t}(x)\). For an integer \(s\ge 1\), let \(\mathbf {t}=(t_1,\ldots ,t_s)\in \mathcal {T}^s\), and let \(\mathbf {x}=(x_1,\ldots ,x_s)\in (\mathcal {X})^{*s}\). We will write \(\left( T(k,t_i,x_i)\right) _{1\le i \le s}\) as \(T_{k}(\mathbf {t},\mathbf {x})\) or \(T_{k,\mathbf {t}}(\mathbf {x})\).

Tweakable SPNs. For fixed parameters w and n, let

$$T:\mathcal {K}\times \mathcal {T}\times \{0,1\}^{wn}\longrightarrow \{0,1\}^{wn}$$

be a keyed tweakable permutation with key space \(\mathcal {K}\), tweak space \(\mathcal {T}\) and message space \(\{0,1\}^{wn}\).

For a fixed number of rounds r, an r-round substitution-permutation network (SPN) based on T, denoted \(\mathsf {SP}^T\), takes as input a set of n-bit permutations \( \mathcal {S}=(S_1,\ldots ,S_r)\), and defines a keyed tweakable permutation \(\mathsf {SP}^T[ \mathcal {S}]\) operating on wn-bit blocks with key space \(\mathcal {K}^{r+1}\) and tweak space \(\mathcal {T}\): on input \(x\in \{0,1\}^{wn}\), key \(\mathbf {k}=(k_0,k_1,\ldots ,k_r)\in \mathcal {K}^{r+1}\) and tweak \(t\in \mathcal {T}\), the output of \(\mathsf {SP}^T[ \mathcal {S}]\) is computed as follows (see also Fig. 1).

Remark 1

Both of the permutation layer T and the entire construction \(\mathsf {SP}^T\) can be viewed as keyed tweakable permutations. However, T will typically be built upon non-cryptographic operations such as filed multiplications, while \(\mathsf {SP}^T\) are based on S-boxes which are modeled as public random permutations.

Fig. 1.

A 2-round tweakable SPN with \(w=4\). The input and output blocks of the SPN are represented as \(x=x_1||x_2||x_3||x_4\) and \(y=y_1||y_2||y_3||y_4\), respectively.

Full size image

Blockwise Universal Tweakable Permutations. A keyed tweakable permutation

$$T:\mathcal {K}\times \mathcal {T}\times \{0,1\}^{wn}\longrightarrow \{0,1\}^{wn}$$

is called \((\delta ,\delta ')\)-blockwise universal if the following hold.

1. 1.

   For all distinct (t, x, i), \((t',x',i')\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ k\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:T_{k,t}(x)_i=T_{k,t'}(x')_{i'}\right] \le \delta .$$

2. 2.

   For all \((t,x,i,c)\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\times \{0,1\}^{n}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ k\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:T_{k,t}(x)_i=c\right] \le \delta '.$$

Since each pair of key \(k\in \mathcal {K}\) and tweak \(t\in \mathcal {T}\) defines a permutation \(T_{k,t}\) on \(\{0,1\}^{wn}\), one can define a keyed tweakable permutation

$$T^{-1}:\mathcal {K}\times \mathcal {T}\times \{0,1\}^{wn}\longrightarrow \{0,1\}^{wn}$$

such that \(T^{-1}(k,t,x)=(T_{k,t})^{-1}(x)\). If T and \(T^{-1}\) are both \((\delta ,\delta ')\)-blockwise universal, then T is called \((\delta ,\delta ')\)-super blockwise universal.

2.2 An Efficient Super Blockwise Tweakable Universal Permutation

In this section, we show that an efficient xor-blockwise universal construction, dubbed \(\mathsf {BPE}\), proposed by Halevi [Hal07] can be made tweakable with a slight modification. Other constructions of (tweakable) blockwise universal permutations can be found in [DKS+17] some of which support tweaks. We present \(\mathsf {BPE}\) below and will present the remaining constructions in the full version.

Assuming \(2^n\ge w+3\), let \(\mathbb {F}\) denote a finite field with \(2^n\) elements. For each \(k\in \mathbb {F}\), define a \(w\times w\) matrix over \(\mathbb {F}\), \(M_{k}\mathrel {\mathop =^\mathrm{def}}A_{k}+I\), where I is the identity matrix and

$$A_{k}=\left[ \begin{matrix}k &{} k^2 &{} &{} k^w \\ k &{} k^2 &{} &{} k^w \\ &{} &{} \ddots &{} \\ k &{} k^2 &{} &{} k^w\end{matrix} \right] .$$

Precisely, \((A_k)_{i,j}=k^j\) for \(1\le i,j\le w\). Let z be a primitive element of \(\mathbb {F}\), and let

$$\mathcal {K}=\left\{ k\in \mathbb {F}:\sum _{i=0}^wk^i\ne 0\right\} \times \mathbb {F}.$$

Then \(\mathsf {BPE}\) is defined as follows.

$$\begin{aligned} \mathsf {BPE}:\mathcal {K}\times \{0,1\}^{wn}&\longrightarrow \{0,1\}^{wn}\\ \left( (k,k'),x\right)&\longmapsto M_{k}x\oplus a_{k'}, \end{aligned}$$

where we identify \(x\in \{0,1\}^{wn}\) with a w-dimensional column vector over \(\mathbb {F}\), and

$$a_{k'}=\left[ \begin{matrix}k' \\ zk' \\ \vdots \\ z^{w-1}k' \end{matrix} \right] .$$

It is easy to check that \(M_{k}\) is invertible if \(\sum _{i=0}^wk^i\ne 0\); precisely,

$$M_{k}^{-1}=I\oplus \frac{A_{k}}{k^*},$$

where \(k^*\mathrel {\mathop =^\mathrm{def}}\sum _{i=0}^wk^i\). For any \((k,k')\in \mathcal {K}\), \(\mathsf {BPE}_{k,k'}\) is also invertible with

$$\mathsf {BPE}_{k,k'}^{-1}(x)=M_{k}^{-1}(x\oplus a_{k'})$$

for any \(x\in \{0,1\}^{wn}\). Halevi [Hal07] also proved that for any pair of distinct (x, i), \((x',i')\in \{0,1\}^{wn}\times \{1,\ldots ,w\}\) and \(\varDelta \in \{0,1\}^n\),

$$\begin{aligned} \mathsf {Pr}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {BPE}_{k,k'}(x)_i\oplus \mathsf {BPE}_{k,k'}(x')_{i'}=\varDelta \right]&\le \frac{w}{2^n-w},\nonumber \\ \mathsf {Pr}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {BPE}^{-1}_{k,k'}(x)_i\oplus \mathsf {BPE}^{-1}_{k,k'}(x')_{i'}=\varDelta \right]&\le \frac{w}{2^n-w}. \end{aligned}$$

(1)

For a fixed \((x,i,c)\in \{0,1\}^{wn}\times \{1,\ldots ,w\}\times \{0,1\}^n\), \(\mathsf {BPE}_{k,k'}(x)_i=c\) implies that

$$\sum _{j=1}^wx_jk^j \oplus x_i\oplus z^{i-1}k'=c,$$

which holds with probability \(\frac{1}{2^n}\) over a random choice of \((k,k')\in \mathcal {K}\). On the other hand, \(\mathsf {BPE}^{-1}_{k,k'}(x)_i=c\) implies that

$$\left( z^{i-1}\oplus \frac{1}{k^*}\sum _{j=1}^wz^{j-1}k^j \right) k'\oplus \left( c\oplus x_i\oplus \frac{1}{k^*}\sum _{j=1}^wx_{j}k^j \right) =0.$$

This equation holds with probability at most \(\frac{w}{2^n-w}+\frac{1}{2^n}\). To summarize, we have

$$\begin{aligned} \mathsf {Pr}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {BPE}_{k,k'}(x)_i=c \right]&\le \frac{1}{2^n},\nonumber \\ \mathsf {Pr}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {BPE}^{-1}_{k,k'}(x)_i=c \right]&\le \frac{w+1}{2^n-w}. \end{aligned}$$

(2)

Now we define a tweakable variant of \(\mathsf {BPE}\), dubbed \(\mathsf {TBPE}\) (for Tweakable Blockwise Polynomial-Evaluation), with tweak space \(\mathcal {T}=\{0,1\}^n\) as follows.

$$\begin{aligned} \mathsf {TBPE}:\mathcal {K}\times \mathcal {T}\times \{0,1\}^{wn}&\longrightarrow \{0,1\}^{wn}\\ \left( (k,k'),t,x\right)&\longmapsto M_{k}(x\oplus b_t)\oplus a_{k'}\oplus b_t, \end{aligned}$$

where \(b_t\) is the column vector whose entries are all t, namely,

$$b_{t}=\left[ \begin{matrix}t \\ t \\ \vdots \\ t \end{matrix} \right] .$$

Since each pair of key \((k,k')\in \mathcal {K}\) and tweak \(t\in \mathcal {T}\) defines a permutation \(\mathsf {TBPE}_{k,k',t}\) on \(\{0,1\}^{wn}\), one can define a keyed tweakable permutation

$$\mathsf {TBPE}^{-1}:\mathcal {K}\times \mathcal {T}\times \{0,1\}^{wn}\longrightarrow \{0,1\}^{wn}.$$

Then we can prove the following lemma.

Lemma 2

Let \(\mathsf {TBPE}\) be the keyed tweakable permutation as defined above, and let \(\mathsf {TBPE}^{-1}\) be its inverse.

1. 1.

   For all distinct (t, x, i), \((t',x',i')\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {TBPE}_{k,k',t}(x)_i=\mathsf {TBPE}_{k,k',t'}(x')_{i'}\right] \le \frac{w}{2^n-w}.$$

2. 2.

   For all \((t,x,i,c)\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\times \{0,1\}^{n}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {TBPE}_{k,k',t}(x)_i=c\right] \le \frac{1}{2^n}.$$

3. 3.

   For all distinct (t, x, i), \((t',x',i')\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {TBPE}^{-1}_{k,k',t}(x)_i=\mathsf {TBPE}^{-1}_{k,k',t'}(x')_{i'}\right] \le \frac{w}{2^n-w}.$$

4. 4.

   For all \((t,x,i,c)\in \mathcal {T}\times \{0,1\}^{wn}\times \{1,\ldots ,w\}\times \{0,1\}^{n}\), we have

   $${{\mathrm{\mathsf {Pr}}}}\left[ (k,k')\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}:\mathsf {TBPE}^{-1}_{k,k',t}(x)_i=c\right] \le \frac{w+1}{2^n-w}.$$

Proof

For distinct (t, x, i) and \((t',x',i')\), we have

$$\mathsf {TBPE}_{k,k',t}(x)_i\oplus \mathsf {TBPE}_{k,k',t'}(x')_{i'}=\mathsf {BPE}_{k,k'}(x\oplus b_t)_i\oplus \mathsf {BPE}_{k,k'}(x'\oplus b_{t'})_{i'}\oplus t\oplus t'.$$

If \((x\oplus b_t,i)\ne (x'\oplus b_{t'},i')\), then \(\mathsf {BPE}_{k,k'}(x\oplus b_t)_i\oplus \mathsf {BPE}_{k,k'}(x'\oplus b_{t'})_{i'}\oplus t\oplus t'=0\) with probability at most \(\frac{w}{2^n-w}\) by (1). If \((x\oplus b_t,i)=(x'\oplus b_{t'},i')\), then it implies \(t\ne t'\), and hence \(\mathsf {BPE}_{k,k'}(x\oplus b_t)_i\oplus \mathsf {BPE}_{k,k'}(x'\oplus b_{t'})_{i'}\oplus t\oplus t'=t\oplus t'\ne 0\).

For a fixed (t, x, i, c), \(\mathsf {TBPE}_{k,k',t}(x)_i=c\) if and only if \(\mathsf {BPE}_{k,k'}(x\oplus b_t)_i=c\oplus t\), and this equation holds with probability at most \(\frac{1}{2^n}\). The remaining properties are proved similarly.   \(\square \)

From Lemma 2, it follows that \(\mathsf {TBPE}\) is \(\left( \frac{w}{2^n-w},\frac{w+1}{2^n-w}\right) \)-super blockwise universal. Except constant multiplications \(z^ik'\), \(i=1,\ldots , w-1\), (which also can be precomputed), each evaluation of \(\mathsf {TBPE}_{k,k',t}(x)\) requires w field multiplications.

2.3 Indistinguishability in the Multi-user Setting

Let \(\mathsf {SP}^T[ \mathcal {S}]\) be an r-round SPN based on a set of S-boxes \( \mathcal {S}=(S_1,\ldots ,S_r)\) and a keyed tweakable permutation T with key space \(\mathcal {K}\) and tweak space \(\mathcal {T}\). So \(\mathsf {SP}^T[ \mathcal {S}]\) becomes a keyed tweakable permutation on \(\{0,1\}^{wn}\) with key space \(\mathcal {K}^{r+1}\) and tweak space \(\mathcal {T}\).

In the multi-user setting, let \(\ell \) denote the number of users. In the real world, \(\ell \) secret keys \(\mathbf {k}_1,\ldots \mathbf {k}_{\ell }\in \mathcal {K}^{r+1}\) are chosen independently at random. A set of independent S-boxes \( \mathcal {S}=(S_1,\ldots ,S_r)\) is also randomly chosen from \(\mathsf {Perm}(n)^r\). A distinguisher \(\mathcal {D}\) is given oracle access to \((\mathsf {SP}^T_{\mathbf {k}_1}[ \mathcal {S}],\ldots , \mathsf {SP}^T_{\mathbf {k}_{\ell }}[ \mathcal {S}])\) as well as \( \mathcal {S}=(S_1,\ldots ,S_r)\). In the ideal world, \(\mathcal {D}\) is given a set of independent random tweakable permutations \(\widetilde{\mathcal {P}}=(\widetilde{P}_1,\ldots ,\widetilde{P}_{\ell })\in \widetilde{\mathsf {Perm}}(\mathcal {T},wn)^{\ell }\) instead of \((\mathsf {SP}^T_{\mathbf {k}_1}[ \mathcal {S}],\ldots , \mathsf {SP}^T_{\mathbf {k}_{\ell }}[ \mathcal {S}])\). However, oracle access to \( \mathcal {S}=(S_1,\ldots ,S_r)\) is still allowed in this world.

The adversarial goal is to tell apart the two worlds \((\mathsf {SP}^T_{\mathbf {k}_1}[ \mathcal {S}],\ldots , \mathsf {SP}^T_{\mathbf {k}_{\ell }}[ \mathcal {S}], \mathcal {S})\) and \((\widetilde{P}_1,\ldots ,\widetilde{P}_{\ell }, \mathcal {S})\) by adaptively making forward and backward queries to each of the constructions and the S-boxes. Formally, \(\mathcal {D}\)’s distinguishing advantage is defined by

$$\begin{aligned} {{\mathrm{Adv}}}^{\mathrm {mu}}_{\mathsf {SP}^T}(\mathcal {D})&={{\mathrm{\mathsf {Pr}}}}\left[ \widetilde{P}_1,\ldots ,\widetilde{P}_{\ell }\overset{{\tiny {\$}}}{\leftarrow }\widetilde{\mathsf {Perm}}(\mathcal {T},wn), \mathcal {S}\overset{{\tiny {\$}}}{\leftarrow }\mathsf {Perm}(n)^r:1\leftarrow \mathcal {D}^{ \mathcal {S}, \widetilde{P}_1,\ldots ,\widetilde{P}_{\ell }}\right] \\ {}&- \, {{\mathrm{\mathsf {Pr}}}}\left[ \mathbf {k}_1,\ldots ,\mathbf {k}_{\ell }\overset{{\tiny {\$}}}{\leftarrow }\mathcal {K}^{r+1}, \mathcal {S}\overset{{\tiny {\$}}}{\leftarrow }\mathsf {Perm}(n)^r:1\leftarrow \mathcal {D}^{ \mathcal {S},\mathsf {SP}^T_{\mathbf {k}_1}[ \mathcal {S}],\ldots ,\mathsf {SP}^T_{\mathbf {k}_{\ell }}[ \mathcal {S}]}\right] . \end{aligned}$$

For p, \(q>0\), we define

$${{\mathrm{Adv}}}_{\mathsf {SP}^T}(p,q)=\max _{\mathcal {D}}{{\mathrm{Adv}}}_{\mathsf {SP}^T}(\mathcal {D})$$

where the maximum is taken over all adversaries \(\mathcal {D}\) making at most p queries to each of the S-boxes and at most q queries to the outer tweakable permutations. In the single-user setting with \(\ell =1\), \({{\mathrm{Adv}}}^{\mathrm {mu}}_{\mathsf {SP}^T}(\mathcal {D})\) and \({{\mathrm{Adv}}}^{\mathrm {mu}}_{\mathsf {SP}^T}(p,q)\) will also be written as \({{\mathrm{Adv}}}^{\mathrm {su}}_{\mathsf {SP}^T}(\mathcal {D})\) and \({{\mathrm{Adv}}}^{\mathrm {su}}_{\mathsf {SP}^T}(p,q)\), respectively.

H-coefficient Technique. Suppose that a distinguisher \(\mathcal {D}\) makes p queries to each of the S-boxes, and total q queries to the construction oracles. The queries made to the j-th construction oracle, denoted \(C_j\), are recorded in a query history

$$\mathcal {Q}_{C_j}=(j,t_{j,i},x_{j,i},y_{j,i})_{1\le i\le q_j}$$

for \(j=1,\ldots ,\ell \), where \(q_j\) is the number of queries made to \(C_j\) and \((j,t_{j,i},x_{j,i},y_{j,i})\) represents the evaluation obtained by the i-th query to \(C_j\).Footnote 3 So according to the instantiation, it implies either \(\mathsf {SP}^T_{\mathbf {k}_j}[ \mathcal {S}](t_{j,i},x_{j,i})=y_{j,i}\) or \(\widetilde{P}_j(t_{j,i},x_{j,i})=y_{j,i}\). Let

$$\mathcal {Q}_C=\mathcal {Q}_{C_1}\cup \cdots \cup \mathcal {Q}_{C_{\ell }}.$$

For \(j=1,\ldots ,r\), the queries made to \(S_j\) are recorded in a query history

$$\mathcal {Q}_{S_j}=(j,u_{j,i},v_{j,i})_{1\le i\le p},$$

where \((j,u_{j,i},v_{j,i})\) represents the evaluation \(S_{j}(u_{j,i})=v_{j,i}\) obtained by the i-th query to \(S_j\). Let

$$\mathcal {Q}_S=\mathcal {Q}_{S_1}\cup \cdots \cup \mathcal {Q}_{S_r}.$$

Then the pair of query histories

$$\tau =(\mathcal {Q}_C,\mathcal {Q}_S)$$

will be called the transcript of the attack: it contains all the information that \(\mathcal {D}\) has obtained at the end of the attack. In this work, we will only consider information theoretic distinguishers. Therefore we can assume that a distinguisher is deterministic without making any redundant query, and hence the output of \(\mathcal {D}\) can be regarded as a function of \(\tau \), denoted \(\mathcal {D}(\tau )\) or \(\mathcal {D}(\mathcal {Q}_C,\mathcal {Q}_S)\).

Fix a transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\), a key \(\mathbf {k}\in \mathcal {K}^{r+1}\), a tweakable permutation \(\widetilde{P}\in \widetilde{\mathsf {Perm}}(\mathcal {T},wn)\), a set of S-boxes \( \mathcal {S}=(S_1,\ldots ,S_r)\in \mathsf {Perm}(n)^r\) and \(j\in \{1,\ldots ,\ell \}\): if \(S_j(u_{j,i})=v_{j,i}\) for every \(i=1,\ldots ,p\), then we will write \(S_j\vdash \mathcal {Q}_{S_j}\). We will write \( \mathcal {S}\vdash \mathcal {Q}_S\) if \( \mathcal {S}_j\vdash \mathcal {Q}_{S_j}\) for every \(j=1,\ldots ,r\). Similarly, if \(\mathsf {SP}^T_{\mathbf {k}}[ \mathcal {S}](t_{j,i},x_{j,i})=y_{j,i}\) (resp. \(\widetilde{P}(t_{j,i},x_{j,i})=y_{j,i}\)) for every \(i=1,\ldots ,q_j\), then we will write \(\mathsf {SP}^T_{\mathbf {k}}[ \mathcal {S}]\vdash \mathcal {Q}_{C_j}\) (resp. \(\widetilde{P}\vdash \mathcal {Q}_{C_j}\)).

Let \(\mathbf {k}_1,\ldots ,\mathbf {k}_{\ell }\in \mathcal {K}^{r+1}\) and \(\widetilde{\mathcal {P}}=(\widetilde{P}_1,\ldots \widetilde{P}_{\ell })\in \widetilde{\mathsf {Perm}}(\mathcal {T},wn)^{\ell }\). If \(\mathsf {SP}^T_{\mathbf {k}_j}[ \mathcal {S}]\vdash \mathcal {Q}_{C_j}\) (resp. \(\widetilde{P}_j\vdash \mathcal {Q}_{C_j}\)) for every \(j=1,\ldots ,\ell \), then we will write \((\mathsf {SP}^T_{\mathbf {k}_j}[ \mathcal {S}])_{j=1,\ldots ,\ell }\vdash \mathcal {Q}_{C}\) (resp. \(\widetilde{\mathcal {P}}\vdash \mathcal {Q}_{C}\)).

If there exist \(\widetilde{\mathcal {P}}\in \widetilde{\mathsf {Perm}}(\mathcal {T},wn)^{\ell }\) and \( \mathcal {S}\in \mathsf {Perm}(n)^w\) that outputs \(\tau \) at the end of the interaction with \(\mathcal {D}\), then we will call the transcript \(\tau \) attainable. So for any attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\), there exist \(\widetilde{\mathcal {P}}\in \widetilde{\mathsf {Perm}}(\mathcal {T},wn)^{\ell }\) and \( \mathcal {S}\in \mathsf {Perm}(n)^w\) such that \(\widetilde{\mathcal {P}}\vdash \mathcal {Q}_C\) and \( \mathcal {S}\vdash \mathcal {Q}_S\). For an attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\), let

With these definitions, the following lemma, the core of the H-coefficients technique (without defining “bad” transcripts), will be also used in our security proof.

Lemma 3

Let \(\varepsilon >0\). Suppose that for any attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\),

$$\begin{aligned} \mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)\ge (1-\varepsilon )\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S). \end{aligned}$$

(3)

Then one has

$$\begin{aligned} {{\mathrm{Adv}}}^{\mathrm {mu}}_{\mathsf {SP}^T}(\mathcal {D})\le \varepsilon . \end{aligned}$$

The lower bound (3) is called \(\varepsilon \)-point-wise proximity of the transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\). The point-wise proximity of a transcript in the multi-user setting is guaranteed by the point-wise proximity of \((\mathcal {Q}_{C_j},\mathcal {Q}_S)\) for each \(j=1,\ldots ,\ell \) in the single-user setting. The following lemma is a restatement of Lemma 3 in [HT16].

Lemma 4

Let \(\varepsilon :\mathbb {N}\times \mathbb {N}\rightarrow \mathbb {R}^{\ge 0}\) be a function such that

1. 1.

   \(\varepsilon (x,y)+\varepsilon (x,z)\le \varepsilon (x,y+z)\) for every x, y, \(z\in \mathbb {N}\),

2. 2.

   \(\varepsilon (\cdot ,z)\) and \(\varepsilon (z,\cdot )\) are non-decreasing functions on \(\mathbb {N}\) for every \(z\in \mathbb {N}\).

Suppose that for any distinguisher \(\mathcal {D}\) in the single-user setting that makes p primitive queries to each of the underlying S-boxes and makes q construction queries, and for any attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\) obtained by \(\mathcal {D}\), one has

$$\begin{aligned} \mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)\ge (1-\varepsilon (p,q))\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S). \end{aligned}$$

Then for any distinguisher \(\mathcal {D}\) in the multi-user setting that makes p primitive queries to each of the underlying S-boxes and makes total q construction queries, and for any attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\) obtained by \(\mathcal {D}\), one has

$$\begin{aligned} \mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)\ge (1-\varepsilon (p+wq,q))\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S). \end{aligned}$$

2.4 Coupling Technique

Given a finite event space \(\varOmega \) and two probability distributions \(\mu \) and \(\nu \) defined on \(\varOmega \), the total variation distance between \(\mu \) and \(\nu \), denoted \(\Vert \mu -\nu \Vert \), is defined as

$$\Vert \mu -\nu \Vert =\frac{1}{2}\sum _{x\in \varOmega }|\mu (x)-\nu (x)|.$$

The following definitions are also all equivalent.

$$ \Vert \mu -\nu \Vert =\max _{Z\subset \varOmega }\{\mu (Z)-\nu (Z)\}=\max _{Z\subset \varOmega }\{\nu (Z)-\mu (Z)\}=\max _{Z\subset \varOmega }\{|\mu (Z)-\nu (Z)|\}. $$

A coupling of \(\mu \) and \(\nu \) is a distribution \(\tau \) on \(\varOmega \times \varOmega \) such that for all \(x\in \varOmega \), \(\sum _{y\in \varOmega }\tau (x,y)=\mu (x)\) and for all \(y\in \varOmega \), \(\sum _{x\in \varOmega }\tau (x,y)=\nu (x)\). In other words, \(\tau \) is a joint distribution whose marginal distributions are respectively \(\mu \) and \(\nu \). We will use the following two lemmas in our security proof.

Lemma 5

Let \(\mu \) and \(\nu \) be probability distributions on a finite event space \(\varOmega \), let \(\tau \) be a coupling of \(\mu \) and \(\nu \), and let (X, Y) be a random variable sampled according to distribution \(\tau \). Then \(\Vert \mu -\nu \Vert \le {{\mathrm{\mathsf {Pr}}}}[X\ne Y]\).

Lemma 6

Let \(\varOmega \) be some finite event space and \(\nu \) be the uniform probability distribution on \(\varOmega \). Let \(\mu \) be a probability distribution on \(\varOmega \) such that \(\Vert \mu -\nu \Vert \le \varepsilon \). Then there is a set \(Z\subset \varOmega \) such that

1. 1.

   \(|Z|\ge (1-\sqrt{\varepsilon })|\varOmega |\),

2. 2.

   \(\mu (x)\ge (1-\sqrt{\varepsilon })\nu (x)\) for every \(x\in Z\).

We refer to [LPS12] for the proof of the above two lemmas.

3 Security of Linear SPNs

All the results in this section are for the “non-tweakable” setting (\(|\mathcal {T}|=1\)) Hence, we do not explicitly refer to the tweak in the notation. Further, the results in this section hold even when a single n-bit permutation S is used, i.e., even when \(S_1 = \ldots = S_r = S\) and are presented as such. We start by defining linear (non-tweakable) SPNs.

Definition 1

Keyed permutation \(T : \mathcal {K}\times \{0,1\}^{wn}\longrightarrow \{0,1\}^{wn}\) is linear if

$$\begin{aligned} T(k,x) = (T_k \cdot k) + (T_x \cdot x) + \varDelta , \end{aligned}$$

where \(T_k, T_x \in \mathcal {K}\times \{0,1\}^{wn}\) are linear transformations, \(T_x\) is invertible, and \(\varDelta \in \{0,1\}^{wn}\). An SPN is linear if all its round permutations \(\{T_{k_i}\}_{i=0}^r\) are linear.

We present an attack showing that 2-round, linear SPNs cannot be secure for \(w \ge 2\). The attack is based on one shown by Halevi and Rogaway [HR04] in a different context (and is a simple application of the boomerang technique [Wag99]); our contribution here is to observe that the attack is applicable to any 2-round, linear SPN. The attack relies on the fact that the field \(\mathbb {F}= \text {GF}(2^n)\) is of characteristic 2. This attack and an attack that works for fields of arbitrary characteristic can be found in [DKS+17].

3.1 Security of 3-Round, Linear (non-tweakable) SPNs

We now explore conditions under which 3-round, linear SPNs are secure. Recall that a 3-round SPN has four round permutations \(\{T_i\}_{i=0}^3\), and without loss of generality we may assume

$$\begin{aligned} T_i(k_i, x) = {\left\{ \begin{array}{ll} x \oplus k_i &{} i \in \{0, 3\}\\ T'_i\cdot (x \oplus k_i) &{} i \in \{1, 2\} \end{array}\right. }, \end{aligned}$$

(4)

where \(T'_1, T'_2 \in \mathbb {F}^{w \times w}\) are invertible linear transformations. We prove that a 3-round, linear SPN is secure so long as (i) \(T'_1\) and \({T'_2}^{-1}\) contain no zero entries (Miles and Viola [MV15] show that matrices with maximal branch number [Dae95] satisfy this property), and (ii) round keys \(k_0\) and \(k_3\) are (individually) uniform. The proof of this theorem can be found in [DKS+17].

Theorem 1

Assume \(w > 1\). Let \(\mathsf {SP}^T\) be a 3-round, linear (non-tweakable) SPN with round permutations as in (4) and with distribution \(\mathcal {K}\) over keys \(k_0, k_1, k_2, k_3\). If \(k_0\) and \(k_3\) are uniformly distributed and the matrices \(T'_1\), \({T'_2}^{-1}\) contain no zero entries, then

$$\begin{aligned} \,\, {{\mathrm{Adv}}}^{\mathrm {su}}_{\mathsf {SP}^T}(p,q) \le \frac{5w^2q^2 + 4wpq}{2^n - p - 2w} + \frac{q^2}{2^{wn}}\,. \end{aligned}$$

A minimal secure (linear) SPN. We proved that a 3-round, linear SPN is secure if the keys \(k_0\) and \(k_3\) are individually uniform and \(T'_1, {T'_2}^{-1}\) contain no 0-entries. No assumptions were made about independence of \(k_0, k_3\), nor were any assumptions made about the distributions of \(k_1, k_2\). So the theorem implies security for the following “minimal” 3-round, linear SPN: Let \(k_0 = k_3 = k\), where k is uniform, set \(k_1 = k_2 = 0^{wn}\), and let \(T'_1 = {T'_2}^{-1}=T\) be invertible with no 0-entries. Define keyed permutations

$$\begin{aligned} \pi _i(k, x) = {\left\{ \begin{array}{ll} x \oplus k &{} i \in \{0, 3\}\\ T' x &{} i = 1\\ {T'}^{-1} x &{} i = 2.\\ \end{array}\right. } \end{aligned}$$

(5)

We have:

Corollary 1

Assume \(w>1\). Let \(\mathsf {SP}^T\) be a 3-round, linear SPN with round permutations as in (5) and \(\mathcal {K}\) choosing uniform \(k_0=k_3\) and \(k_1=k_2=0^{wn}\). Then

$$\begin{aligned} \,\, {{\mathrm{Adv}}}^{\mathrm {su}}_{\mathsf {SP}^T}(p,q) \le \frac{5w^2q^2 + 4wpq}{2^n - p - 2w} + \frac{q^2}{2^{wn}}\,. \end{aligned}$$

Reducing key-length. It is in fact sufficient for the wn-bit key k \((= k_0 = k_3)\) in Corollary 1 to satisfy the following conditions: informally, for any n-bit constant c and distinct indices \(i, i'\), (a) k[i] equals c with negligible probability, and (b) the sum of k[i] and \(k[i']\) equals c with negligible probability. This can be achieved by choosing a uniform n-bit key \(k'\) and letting \(k[i] = a_i \cdot k'\) where \(a_i\) are distinct non-zero elements of \(\mathbb {F}\). Thus, one can make do with a “master key” of only n bits, while preserving the same security as in Corollary 1.

4 Security of Non-Linear SPNs

In this section, we first show that keyed tweakable blockwise universal permutations help construct (non-linear) tweakable SPNs. As a preliminary step, we show that 1 round is sufficient to obtain this result. However, the security of the SPN is only up to the birthday attack in this case. Towards obtaining a better security bound, we show that 2 rounds suffice to go beyond the birthday bound and in addition, also present multi-user security beyond the birthday bound for the 2-round tweakable SPN. Finally, we show that if T is a super blockwise tweakable universal permutation, then the security of \(\mathsf {SP}^T\) converges to \(2^n\) as the number of rounds r increases.

4.1 Birthday Security of 1-Round SPNs

We show that a tweakable blockwise-universal permutation is useful in constructing non-linear tweakable SPNs. The proof of the theorem is a straightforward extension of the non-tweakable version found in [DKS+17]. Consider the 1-round SPN \(\mathsf {SP}^T\) with \(T_{k_1} := T^{-1}_{k_0}\) where T is a keyed blockwise universal tweakable permutation.

Theorem 2

Let T be a \((\delta ,\delta ')\)-blockwise universal tweakable permutation. Then for any integers p and q, one has \( \mathbf{Adv }^{\mathrm {su}}_{\mathsf {SP}^{T}}(p,q) \le {q}^2w^2\delta + {p}{q}w\delta '\).

4.2 Beyond-Birthday Security of 2-Round SPNs

In this section, we will prove the following theorem.

Theorem 3

Let \(\delta \), \(\delta '>0\), and let n and w be positive integers such that \(w\ge 2\). Let T be a \((\delta ,\delta ')\)-super blockwise universal tweakable permutation. Then for any integers p and q such that \(wp + 3w^2q< 2^n/2\), one has

$$\begin{aligned} \mathbf{Adv }^{\mathrm {su}}_{\mathsf {SP}^{T}}(p,q)&\le w^2 q(\delta ' p+\delta w q)(3\delta ' p+3\delta wq + 2\delta 'w q)+\frac{q^2}{2^{wn}}+\frac{q(2wp\!+\!6w^2q)^2}{2^{2n}},\\ \mathbf{Adv }^{\mathrm {mu}}_{\mathsf {SP}^{T}}(p,q)&\le w^2 q(\delta ' p+(\delta +\delta ') w q)(3\delta ' p+3\delta wq + 5\delta 'w q)\\&+\frac{q^2}{2^{wn}}+\frac{q(2wp+8w^2q)^2}{2^{2n}}. \end{aligned}$$

Remark 2

For the sake of simplicity, we assume that the three keyed layers are actually the same, which is why we require T to be \((\delta ,\delta ')\)-super blockwise tweakable universal. However, if one looks closely at the proof, only the middle layer has to be super-blockwise-universal. The first and the last layer only need to be \((\delta ,\delta ')\)-blockwise universal.

Remark 3

When the S-boxes are modeled as block ciphers using secret keys, the security bound (in the standard model) is obtained by setting \(p=0\).

The proof of Theorem 3 relies on the following lemma (with the lower bound simplified) and on Lemma 3 and Lemma 4.

Lemma 7

Let p and q be positive integers such that \(wp + 3w^2q< 2^n/2\), and let \(\mathcal {D}\) be a distinguisher in the single-user setting that makes p primitive queries to each of \(S_1\) and \(S_2\) and makes q construction queries. Then for any attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\), one has

$$ \frac{\mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)}{\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S)}\ge 1-w^2 q(\delta ' p+\delta w q)(3\delta ' p+3\delta wq + 2\delta 'w q)-\frac{q^2}{2^{wn}}-\frac{q(2wp+6w^2q)^2}{2^{2n}}. $$

Outline of Proof of Lemma 7. Throughout the proof, we will write a 2-round \(\mathsf {SP}\) construction as

$$ \mathsf {SP}^{T}[ \mathcal {S}]_{\mathbf {k}}(t,x)=T_{k_2,t}\left( S_2^{||}\left( T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x)\right) \right) \right) \right) , $$

where \( \mathcal {S}=(S_1,S_2)\) is a pair of two public random permutations of \(\{0,1\}^n\), \(\mathbf {k}=(k_0,k_1,k_2)\in \mathcal {K}^3\) is the key, \(x\in \{0,1\}^{wn}\) is the plaintext, and, for \(i=1,2\),

$$\begin{aligned} S_i^{||}\,:\, \{0,1\}^{wn}&\rightarrow \{0,1\}^{wn}\\ x=x_1||x_2||\ldots ||x_w&\longmapsto S_i(x_1)||S_i(x_2)||\ldots ||S_i(x_w). \end{aligned}$$

We also fix a distinguisher \(\mathcal {D}\) as described in the statement and fix an attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\) obtained by \(\mathcal {D}\). Let

$$\begin{aligned} \mathcal {Q}^{(0)}_{S_1}&=\{(u,v)\in \{0,1\}^{n}\times \{0,1\}^{n}: (1,u,v)\in \mathcal {Q}_S\},\\ \mathcal {Q}^{(0)}_{S_2}&=\{(u,v)\in \{0,1\}^{n}\times \{0,1\}^{n}: (2,u,v)\in \mathcal {Q}_S\} \end{aligned}$$

and let

$$\begin{aligned} U^{(0)}_1&=\{u_1\in \{0,1\}^n: (u_1,v_1)\in \mathcal {Q}^{(0)}_{S_1}\},&V^{(0)}_1&=\{v_1\in \{0,1\}^n: (u_1,v_1)\in \mathcal {Q}^{(0)}_{S_1}\},\\ U^{(0)}_2&=\{u_2\in \{0,1\}^n: (u_2,v_2)\in \mathcal {Q}^{(0)}_{S_2}\},&V^{(0)}_2&=\{v_2\in \{0,1\}^n: (u_2,v_2)\in \mathcal {Q}^{(0)}_{S_2}\} \end{aligned}$$

denote the domains and ranges of \(\mathcal {Q}^{(0)}_{S_1}\) and \(\mathcal {Q}^{(0)}_{S_2}\), respectively.

This type of lemma is usually proved by defining a large enough set of “good” keys, and then, for each choice of a good key, lower bounding the probability of observing this transcript, again by lower bounding the number of possible “intermediate” values. A key is usually said to be good if the adversary cannot use the transcript to follow the path of computation of the encryption/decryption of a query up to a contradiction. However, since the S-boxes are used several times in each round, there will not be enough information in the transcript to allow such a naive definition. Therefore, instead of summing over the choice of the key, we will define an extension of the transcript, that will provide the necessary information, and then sum over every possible good extension.

We will first define what we mean by an extension of the transcript \(\tau \). Then we will define bad extensions and explain the link between good extended transcripts and the ratio \(\frac{\mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)}{\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S)}\). Finally, we will show that the number of bad extended transcripts is small enough in Lemma 8, and then show that the probability to obtain any good extension in the real world is sufficiently close to the probability to obtain \(\tau \) the ideal world in Lemma 9. We stress that extended transcripts are completely virtual and are not disclosed to the adversary. They are just an artificial intermediate step to lower bound the probability to observe transcript \(\tau \) in the real world.

Extension of a transcript. We will extend the transcript \(\tau \) of the attack via a certain randomized process. We begin with choosing a pair of keys \((k_0,k_2)\in \mathcal {K}^2\) uniformly at random. Once these keys have been chosen, some construction queries will become involved in collisions. A colliding query is defined as a construction query \((t,x,y)\in \mathcal {Q}_C\) such that one of the following conditions holds:

1. 1.

   there exist an S-box query \((1,u,v)\in \mathcal {Q}_S\) and an integer \(i\in \{1,\ldots ,w\}\) such that \(T_{k_0,t}(x)_i=u\);

2. 2.

   there exist an S-box query \((2,u,v)\in \mathcal {Q}_S\) and an integer \(i\in \{1,\ldots ,w\}\) such that \(T_{k_2,t}^{-1}(y)_i=v\);

3. 3.

   there exist a construction query \((t',x',y')\in \mathcal {Q}_C\) and integers \(i,j\in \{1,\ldots ,w\}\) such that \((t,x,y,i)\ne (t',x',y',j)\) and \(T_{k_0,t}(x)_i=T_{k_0,t'}(x')_j\);

4. 4.

   there exist a construction query \((t',x',y')\in \mathcal {Q}_C\) and integers \(i,j\in \{1,\ldots ,w\}\) such that \((t,x,y,i)\ne (t',x',y',j)\) and \(T_{k_2,t}^{-1}(y)_i=T_{k_2,t'}^{-1}(y')_j\).

We are now going to build a new set \(\mathcal {Q}'_S\) of S-box evaluations that will play the role of an extension of \(\mathcal {Q}_S\). For each colliding query \((t,x,y)\in \mathcal {Q}_C\), we will add tuples \((1,T_{k_0}(t,x)_i,v')_{1\le i \le w}\) (if (t, x, y) collides at the input of \(S_1\)) or \((2,u',T_{k_2,t}^{-1}(y)_i)_{1\le i \le w}\) (if (t, x, y) collides at the output of \(S_2\)) by lazy sampling \(v'=S_1(T_{k_0,t}(x)_i)\) or \(u'=S_2^{-1}(T_{k_2,t}^{-1}(y)_i)\), as long as it has not been determined by any existing query in \(\mathcal {Q}_S\). We finally choose a key \(k_1\) uniformly at random. An extended transcript of \(\tau \) will be defined as a tuple \(\tau '=(\mathcal {Q}_C,\mathcal {Q}_S,\mathcal {Q}'_S,\mathbf {k})\) where \(\mathbf {k}=(k_0,k_1,k_2)\). For each collision between a construction query and a primitive query, or between two construction queries, the extended transcript will contain enough information to compute a complete round of the evaluation of the SPN. This will be useful to lower bound the probability to get the transcript \(\tau \) in the real world.

Definition of Bad Transcript Extensions. Let

$$\begin{aligned} \mathcal {Q}^{(1)}_{S_1}&=\{(u,v)\in \{0,1\}^{n}\times \{0,1\}^{n}: (1,u,v)\in \mathcal {Q}_S\cup \mathcal {Q}'_S\}\\ \mathcal {Q}^{(1)}_{S_2}&=\{(u,v)\in \{0,1\}^{n}\times \{0,1\}^{n}: (2,u,v)\in \mathcal {Q}_S\cup \mathcal {Q}'_S\}. \end{aligned}$$

In words, \(\mathcal {Q}^{(1)}_{S_i}\) summarizes each constraint that is forced on \(S_i\) by \(\mathcal {Q}_S\) and \(\mathcal {Q}'_S\). Let

$$\begin{aligned} U_1&=\{u_1\in \{0,1\}^n: (1,u_1,v_1)\in \mathcal {Q}^{(1)}_{S_1}\},&V_1&=\{v_1\in \{0,1\}^n: (1,u_1,v_1)\!\in \mathcal {Q}^{(1)}_{S_1}\},\\ U_2&=\{u_2\in \{0,1\}^n: (2,u_2,v_2)\in \mathcal {Q}^{(1)}_{S_2}\},&V_2&=\{v_2\in \{0,1\}^n: (2,u_2,v_2)\in \mathcal {Q}^{(1)}_{S_2}\} \end{aligned}$$

be the domains and ranges of \(\mathcal {Q}^{(1)}_{S_1}\) and \(\mathcal {Q}^{(1)}_{S_2}\), respectively. We define two quantities characterizing an extended transcript \(\tau '\), namely

$$\begin{aligned} \alpha _1&\mathrel {\mathop =^\mathrm{def}}\left| \{ (x,y)\in \mathcal {Q}_C : T_{k_0}(x)_i\in U_1 \text { for some }i\in \{1,\ldots ,w\}\}\right| ,\\ \alpha _2&\mathrel {\mathop =^\mathrm{def}}\left| \{ (x,y)\in \mathcal {Q}_C : T_{k_2}^{-1}(y)_i\in V_2 \text { for some }i\in \{1,\ldots ,w\}\}\right| . \end{aligned}$$

In words, \(\alpha _1\) (resp. \(\alpha _2\)) is the number of queries \((t,x,y)\in \mathcal {Q}_C\) which collide with a query \((u_1,v_1)\in \mathcal {Q}^{(1)}_{S_1}\) (resp. which collide with a query \((u_2,v_2)\in \mathcal {Q}^{(1)}_{S_2}\)) in the extended transcript. This corresponds to the number of queries \((t,x,y)\in \mathcal {Q}_C\) which collide with either an original query \((u_1,v_1)\in \mathcal {Q}^{(0)}_{S_1}\) (resp. \((u_2,v_2)\in \mathcal {Q}^{(0)}_{S_2}\)) or with a query \((t',x',y')\in \mathcal {Q}_C\) at an input of \(S_1\) (resp. at the output of \(S_2\)), once the choice of \((k_0,k_2)\) has been made. We will also denote

$$ \beta _i=|\mathcal {Q}^{(1)}_{S_i}|-|\mathcal {Q}^{(0)}_{S_i}|=|\mathcal {Q}^{(1)}_{S_i}|-p $$

for \(i=1,2\), the number of additional queries included in the extended transcript.

We say an extended transcript \(\tau '\) is bad if at least one of the following conditions is fulfilled:

1. (C-1)

   there exist \((t,x,y)\in \mathcal {Q}_C\), \(i,j\in \{1,\ldots ,w\}\), \(u_1 \in U_1\), and \(v_2 \in V_2\) such that \(T_{k_0,t}(x)_i=u_1\) and \(T^{-1}_{k_2,t}(y)_j=v_2\);

2. (C-2)

   there exist \((t,x,y)\in \mathcal {Q}_C\), \(i,j\in \{1,\ldots ,w\}\), \(u_1 \in U_1\), and \(u_2 \in U_2\) such that \(T_{k_0,t}(x)_i=u_1\) and \(T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x)\right) \right) _j=u_2\)Footnote 4;

3. (C-3)

   there exist \((t,x,y)\in \mathcal {Q}_C\), \(i,j\in \{1,\ldots ,w\}\), \(v_1 \in V_1\), and \(v_2 \in V_2\) such that \(T_{k_2,t}^{-1}(y)_i=v_2\) and \(T_{k_1,t}^{-1}\left( \left( S_2^{-1}\right) ^{||}\left( T_{k_2,t}^{-1}(y)\right) \right) _j=v_1\);

4. (C-4)

   there exist \((t,x,y),(t',x',y')\in \mathcal {Q}_C\), \(i,i',j,j'\in \{1,\ldots ,w\}\) with \((t,x,j)\ne (t',x',j')\), \(u_1,u'_1 \in U_1\) such that \(T_{k_0,t}(x)_i=u_1\), \(T_{k_0,t'}(x')_{i'}=u'_1\) and

   $$\begin{aligned} T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x)\right) \right) _j=T_{k_1,t'}\left( S_1^{||}\left( T_{k_0,t'}(x')\right) \right) _{j'}; \end{aligned}$$

5. (C-5)

   there exist \((t,x,y),(t',x',y')\in \mathcal {Q}_C\), \(i,i',j,j'\in \{1,\ldots ,w\}\) with \((y,j)\ne (y',j')\), \(v_2,v'_2 \in V_2\) such that \(T^{-1}_{k_2,t}(y)_i=v_2\), \(T^{-1}_{k_2,t'}(y')_{i'}=v'_2\) and

   $$\begin{aligned} T_{k_1,t}^{-1}\left( \left( S_2^{-1}\right) ^{||}\left( T^{-1}_{k_2,t}(y)\right) \right) _{j}=T_{k_1,t'}^{-1}\left( \left( S_2^{-1}\right) ^{||}\left( T^{-1}_{k_2,t'}(y')\right) \right) _{j'}. \end{aligned}$$

Any extended transcript that is not bad will be called good. Given an original transcript \(\tau \), we denote \(\varTheta _\mathrm{good}(\tau )\) (resp. \(\varTheta _\mathrm{bad}(\tau )\)) the set of good (resp. bad) extended transcripts of \(\tau \) and \(\varTheta '(\tau )\) the set of all extended transcripts of \(\tau \).

From Attainable Transcripts to Good Extended Transcripts. We are now going to justify the usefulness of extended transcripts. For any extended transcript \(\tau '=(\mathcal {Q}_C,\mathcal {Q}_S,\mathcal {Q}'_S,\mathbf {k})\), let us denote

Note that one has

which gives

$$\begin{aligned} \mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S)&\le \frac{1}{(2^{wn})_{q}},\\ \mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)&\ge \sum _{\tau '\in \varTheta _\mathrm{good}(\tau )}\frac{1}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}\mathsf {p}(\tau '). \end{aligned}$$

Thus one has

$$\begin{aligned} \frac{\mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)}{\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S)}&\ge \sum _{\tau '\in \varTheta _\mathrm{good}(\tau )}\frac{(2^{wn})_{q}}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}\mathsf {p}(\tau ')\\\nonumber&\ge \min _{\tau '\in \varTheta _\mathrm{good}(\tau )}((2^{wn})_{q}\mathsf {p}(\tau '))\sum _{\tau '\in \varTheta _\mathrm{good}(\tau )}\frac{1}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}. \end{aligned}$$

Note that the weighted sum \(\sum _{\tau '\in \varTheta _\mathrm{good}(\tau )}\frac{1}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}\) corresponds exactly to the probability that a random extended transcript is good when it is sampled as follows:

1. 1.

   choose keys \(k_0,k_2\in \mathcal {K}\) uniformly and independently at random;

2. 2.

   choose the partial extension of the S-box queries based on the new collisions \(\mathcal {Q}'_S\) uniformly at random (meaning that each possible u or v is chosen uniformly at random in the set of its authorized values);

3. 3.

   finally choose \(k_1\) uniformly at random, independently from everything else.

Thus, the exact probability of observing the extended transcript \(\tau '\) is

$$ \frac{1}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}, $$

and we have

$$ \sum _{\tau '\in \varTheta _\mathrm{good}(\tau )}\frac{1}{|\mathcal {K}|^3(2^n-p)_{\beta _1}(2^n-p)_{\beta _2}}=\mathsf {Pr}\left[ \tau '\in \varTheta _\mathrm{good}(\tau ) \right] . $$

One finally gets

$$\begin{aligned} \frac{\mathsf {p}_2(\mathcal {Q}_C|\mathcal {Q}_S)}{\mathsf {p}_1(\mathcal {Q}_C|\mathcal {Q}_S)}&\ge \mathsf {Pr}\left[ \tau '\in \varTheta _\mathrm{good}(\tau ) \right] \cdot \min _{\tau '\in \varTheta _\mathrm{good}(\tau )}((2^{wn})_{q}\mathsf {p}(\tau ')). \end{aligned}$$

(6)

Lemma 8 and Lemma 9 lower bound \(\mathsf {Pr}\left[ \tau '\in \varTheta _\mathrm{good}(\tau ) \right] \) (by upper bounding \(\mathsf {Pr}\left[ \tau '\in \varTheta _\mathrm{bad}(\tau ) \right] \)) and \(\min _{\tau '\in \varTheta _\mathrm{good}(\tau )}((2^{wn})_{q}\mathsf {p}(\tau '))\), respectively. Then combining (6) with Lemma 8 and Lemma 9 will complete the proof of Lemma 7.

Lemma 8

One has

$$\begin{aligned} \mathsf {Pr}\left[ \tau '\in \varTheta _\mathrm{bad}(\tau ) \right]&\le w^2 q(\delta ' p+\delta w q)(3\delta ' p+3\delta wq + 2\delta 'w q). \end{aligned}$$

Proof

We fix any attainable transcript, denoted \((\mathcal {Q}_C,\mathcal {Q}^{(0)}_{S_1},\mathcal {Q}^{(0)}_{S_2})\). For any fixed construction query \((t,x,y)\in \mathcal {Q}_C\), define event

$$\mathsf {Coll}_1(t,x,y)\Leftrightarrow \text {there exist }i\in \{1,\ldots ,w\}\text { and }u_1 \in U_1\text { such that }T_{k_0,t}(x)_i=u_1.$$

This event can be broken down into the following two subevents:

• there exist \(i\in \{1,\ldots ,w\}\), \(j\in \{1,\ldots ,p\}\) such that \(T_{k_0,t}(x)_i=u_j\),

• there exist \((t',x',y')\in \mathcal {Q}_C\), \(i,j\in \{1,\ldots ,w\}\) such that \((t,x,y,i)\ne (t',x',y',j)\) and \(T_{k_0,t}(x)_i=T_{k_0,t'}(x')_{j}\).

Note that these events only involve queries from the original transcript, which means that the choice of the key is actually independent from these values. By the blockwise uniformity of T, one has

$$\begin{aligned} \mathsf {Pr}\left[ k_0\in \mathcal {K}: \mathsf {Coll}_1(t,x,y) \right] \le \delta ' w p + \delta w^2 q. \end{aligned}$$

(7)

Similarly, let

$$\mathsf {Coll}_2(t,x,y)\Leftrightarrow \text {there exist }i\in \{1,\ldots ,w\}\text { and }v_2 \in V_2\text { such that }T^{-1}_{k_2,t}(y)_i=v_2.$$

Then one has

$$\begin{aligned} \mathsf {Pr}\left[ k_2\in \mathcal {K}: \mathsf {Coll}_2(x,y) \right] \le \delta ' w p + \delta w^2 q. \end{aligned}$$

(8)

Also note that one has \(|\mathcal {Q}^{(1)}_{S_1}|,|\mathcal {Q}^{(1)}_{S_2}|\le p+wq\), as additional tuples in \(\mathcal {Q}'_S\) come from the completion of partial information about a construction query.

We now upper bound the probabilities of the five conditions in turn. The sets of attainable transcripts fulfilling condition (C-1), (C-2), (C-3), (C-4), (C-5) will be denoted \(\varTheta _1\), \(\varTheta _2\), \(\varTheta _3\), \(\varTheta _4\), \(\varTheta _5\), respectively.

Condition (C-1). One has

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _1 \right] \le \sum _{(t,x,y)\in \mathcal {Q}_C}\mathsf {Pr}\left[ \mathsf {Coll}_1(t,x,y) \wedge \mathsf {Coll}_2(t,x,y) \right] . $$

Since the random choice of \(k_0\) and \(k_2\) are independent, and by (7) and (8), one has

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _1 \right] \le q (\delta ' w p + \delta w^2 q)^2. $$

Condition (C-2) and (C-3). Fix any query \((t,x,y)\in \mathcal {Q}_C\). Since the random choice of \(k_1\) is independent from the queries transcript and from the choice of \(k_0\), the probability, over the random choice of \(k_1\), that there exist \(i\in \{1,\ldots ,w\}\) and \(u_2\in U_2\) such that \(T_{k_1,t}\left( S_1^{||} \left( T_{k_0,t}(x) \right) \right) _i=u_2\), conditioned on \(\mathsf {Coll}_1(t,x,y)\), is upper bounded by \(\delta ' w (p+wq)\). Thus, by summing over every construction query and using (7), one has

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _2 \right] \le \delta ' w q(p+wq) (\delta ' w p + \delta w^2 q). $$

Similarly, one has

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _3 \right] \le \delta ' w q(p+wq) (\delta ' w p + \delta w^2 q). $$

Conditions (C-4), and (C-5). Given two distinct pairs \((i,(t,x,y)),(i',(t',x',y'))\in \{1,\ldots ,w\}\times \mathcal {Q}_C\) such that (t, x, y) and \((t',x',y')\) are both colliding queries, let us define event

$$\mathsf {Coll}(t,x,y,t',x',y')_{i,i'} \Leftrightarrow T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x)\right) \right) _i=T_{k_1,t'}\left( S_1^{||}\left( T_{k_0,t'}(x')\right) \right) _{i'}.$$

Then for any distinct pairs \((i,(t,x,y)),(i',(t',x',y'))\in \{1,\ldots ,w\}\times \mathcal {Q}_C\), one has

where, for the last inequality, we used the \((\delta ,\delta ')\)-blockwise uniformity of T and the fact that the event \(\mathsf {Coll}_1(t,x,y) \wedge \mathsf {Coll}_1(t',x',y')\) only depends on the choice of \(k_0\) whereas \(\mathsf {Coll}(t,x,y,t',x',y')_{i,i'}\) involves the choice of \(k_1\). Thus, by summing over every such pair, one obtains

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _4 \right] \le \delta w^2 q^2(\delta ' w p + \delta w^2 q). $$

Similarly, one has

$$ \mathsf {Pr}\left[ \tau '\in \varTheta _5 \right] \le \delta w^2 q^2(\delta ' w p + \delta w^2 q). $$

The lemma follows by taking a union bound over all the conditions.   \(\square \)

Our next step is to study good extended transcripts.

Lemma 9

For any good extended transcript \(\tau '\), one has

$$ (2^{wn})_{q}\mathsf {p}(\tau ')\ge 1-\frac{q^2}{2^{wn}}-\frac{q(2wp+6w^2q)^2}{2^{2n}}. $$

Proof

Fix any good extended transcript \(\tau '=(\mathcal {Q}_C,\mathcal {Q}_{S},\mathcal {Q}'_{S},(k_0,k_1,k_2))\). Let us denote \(p_1=|\mathcal {Q}^{(1)}_{S_1}|\) and \(p_2=|\mathcal {Q}^{(1)}_{S_2}|\).

Our goal is then to prove that \(\mathsf {p}(\tau ')\) is close enough to \(1/(2^{wn})_{q}\). In order to do so, we are going to group the construction queries according to the type of collision they are involved in:

$$\begin{aligned} \mathcal {Q}_{U_1}&=\{(t,x,y)\in \mathcal {Q}_C: T_{k_0,t}(x)_i\in U_1 \text { for }i=1,\ldots ,w\}\\ \mathcal {Q}_{V_2}&=\{(t,x,y)\in \mathcal {Q}_C: T_{k_2,t}^{-1}(y)_i\in V_2 \text { for }i=1,\ldots ,w\}\\ \mathcal {Q}_{0}&=\mathcal {Q}_C\setminus \left( \mathcal {Q}_{U_1}\cup \mathcal {Q}_{V_2}\right) . \end{aligned}$$

Note that, thanks to the additional queries from \(\mathcal {Q}'_S\), there is an equivalence between the events “\(T_{k_0,t}(x)_i\in U_1\) for each \(i=1,\ldots ,w\)” and “there exists \(i\in \{1,\ldots ,w\}\) such that \(T_{k_0,t}(x)_i\in U_1\)”. Thus, one has by definition \(|\mathcal {Q}_{U_1}|=\alpha _1\). Similarly, one has \(|\mathcal {Q}_{V_2}|=\alpha _2\). Also note that these sets form a partition of \(\mathcal {Q}_C\):

• \(\mathcal {Q}_0 \cap \mathcal {Q}_{U_1}=\emptyset \) by definition;

• \(\mathcal {Q}_0 \cap \mathcal {Q}_{V_2}=\emptyset \) by definition;

• \(\mathcal {Q}_{U_1} \cap \mathcal {Q}_{V_2}=\emptyset \) since otherwise \(\tau '\) would satisfy (C-1).

If we denote respectively \(\mathsf {E}_{U_1},\mathsf {E}_{V_2}\) and \(\mathsf {E}_0\) the event \(\mathsf {SP}^{T}[ \mathcal {S}]_\mathbf {k}\vdash \mathcal {Q}_{U_1},\mathcal {Q}_{V_2},\mathcal {Q}_0\), the event \(\mathsf {SP}^{T}[ \mathcal {S}]_\mathbf {k}\vdash \mathcal {Q}_{C}\) is equivalent to \(\mathsf {E}_{U_1} \wedge \mathsf {E}_{V_2} \wedge \mathsf {E}_0\). Note that, by definition of \(\mathcal {Q}_{U_1}\), each \((t,x,y)\in \mathcal {Q}_{U_1}\) is such that \(T_{k_0,t}(x)_i\in U_1\) for each \(i=1,\ldots ,w\); this means that the output of \(S_1\) is already fixed by \(\mathcal {Q}^{(1)}_{S_1}\) and \(\mathsf {E}_{U_1}\) actually only involves \(S_2\). A similar reasoning can be made for \(\mathsf {E}_{V_2}\). Thus we have

(9)

where

In order to evaluate

$$ S_2\left( T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x) \right) \right) _i \right) =T_{k_2,t}^{-1}(y)_i $$

for every \((t,x,y)\in \mathcal {Q}_{U_1}\) and \(i\in \{1,\ldots ,w\}\). All the values \(T_{k_1,t}\left( S_1^{||}\left( T_{k_0,t}(x) \right) \right) _i\) are actually pairwise distinct and outside \(U_2\) since otherwise (C-2) or (C-4) would be satisfied. Similarly, the values \(T_{k_2,t}^{-1}(y)_i\) are pairwise distinct and outside \(V_2\) since otherwise (C-1) would be satisfied. Indeed, if a collision between two values \(T_{k_2,t}^{-1}(y)_i\) had occurred, then these values would also appear in \(V_2\). Hence the event \(\mathsf {E}_{U_1}\) is actually equivalent to \(w\alpha _1\) new and distinct equations on \(S_2\), so that

(10)

By a similar reasoning, one has

(11)

The next step is to lower bound

$$ \mathcal {Q}_0=\{(t_1,x_1,y_1),\ldots ,(t_{q_0},x_{q_0},y_{q_0})\}, $$

using an arbitrary ordering of the queries.

Our goal is now to compute a lower bound on the number of possible “intermediate values” such that the event \(\mathsf {E}_0\) is equivalent to new and distinct equations on \(S_1\) and \(S_2\). First note that the values \(T_{k_0,t}(x)_i\) for each \((t,x,y)\in \mathcal {Q}_0,i\in \{1,\ldots ,w\}\) are pairwise distinct and outside \(U'_1\). Indeed, if this were not the case, then at least one query in \(\mathcal {Q}_0\) would be a colliding query. By definition of our security experiment, this means that this query would either be in \(\mathsf {E}_{U_1}\) or \(\mathsf {E}_{V_2}\), depending on the type of collision it is involved in. Similarly, the values \(T_{k_2,t}^{-1}(y)_i\) for each \((t,x,y)\in \mathcal {Q}_0,i\in \{1,\ldots ,w\}\) are pairwise distinct and outside \(V'_2\).

Let \(N_0\) be the number of tuples of distinct values \((v_{1,i,j})_{1\le i \le q_0,1\le j\le w }\) in \(\{0,1\}^n\!\setminus \!V'_1\) such that the values \((T_{k_1,t_{i}}\left( ||_{k=1}^wv_{1,i,k}\right) _j)_{1\le i \le q_0,1\le j\le w }\) are also pairwise distinct and outside \(U'_2\). Let \(i\in \{1,\ldots ,q_0\}\). There are exactly \((2^n-|V'_1|-w(i-1))_w\) possible tuples of distinct values \((v_{1,i,j})_{1\le j\le w }\) in \(\{0,1\}^n\!\setminus \!V'_1\) that will also be different from the previous values \(v_{1,i,j}\) for \(i<q_0\) and \(j\in \{1,\ldots ,w\}\). Similarly, there are exactly \((2^n-|U'_2|-w(i-1))_w\) possible tuples of distinct values for \((T_{k_1,t_i}(||_{k=1}^wv_{1,i,k}))_{1\le j\le w }\) in \(\{0,1\}^n\!\setminus \!U'_2\) that will also be different from the previous values \(T_{k_1,t_i}(||_{k=1}^wv_{1,i,k})\) for \(i<q_0\) and \(j\in \{1,\ldots ,w\}\). This removes at most \(2^{wn}-(2^n-|U'_2|-w(i-1))_w\) tuples of values for \((T_{k_1,t_i}(||_{k=1}^wv_{1,i,k}))_{1\le j\le w }\). Since \(T_{k_1,t_i}\) is a permutation, we have to remove at most \(2^{wn}-(2^n-|U'_2|-w(i-1))_w\) possible tuples of values for \((v_{1,i,j})_{1\le j\le w }\). Thus

$$\begin{aligned} N_0\ge \prod _{i=1}^{q_0}\left( (2^n-|V'_1|-w(i-1))_w + (2^n-|U'_2|-w(i-1))_w - 2^{wn} \right) . \end{aligned}$$

(12)

For any tuple of values \((v_{1,i,j})\) fulfilling the previous conditions, then, conditioned on \(S_1\) satisfying \(S_1(T_{k_0,t_i}(x_i))_j=v_{1,i,j}\), the event \(\mathsf {E}_0\) is equivalent to \(wq_0\) distinct and new equations on \(S_2\). Hence, it follows that

(13)

Combining (9), (10), (11), (12) (13), we obtain

$$\begin{aligned} (2^{wn})_q\mathsf {p}(\tau ')&\ge \frac{(2^{wn})_{q}\prod \limits _{i=0}^{q_0-1}\left( \begin{array}{l}(2^n-p_1-w(\alpha _2+i))_w\\ \qquad + (2^n-p_2-w(\alpha _1+i))_w - 2^{wn}\end{array} \right) }{(2^n-p_1)_{wq_0+w\alpha _2}(2^n-p_2)_{wq_0+w\alpha _1}}\\&=\frac{(2^{wn})_{q}}{2^{q_0wn}(2^n-p_1)_{w\alpha _2}(2^n-p_2)_{w\alpha _1}}\\&\times \prod \limits _{i=0}^{q_0-1}\frac{2^{wn}\left( \begin{array}{l}(2^n-p_1-w(\alpha _2+i))_w\\ \qquad + (2^n-p_2-w(\alpha _1+i))_w - 2^{wn}\end{array} \right) }{(2^n-p_1-w\alpha _2-wi)_{w}(2^n-p_2-w\alpha _1-wi)_{w}}\\&\ge \frac{(2^{wn})_{q}}{2^{q_0wn}(2^n-p_1)_{w\alpha _2}(2^n-p_2)_{w\alpha _1}}\cdot \prod \limits _{i=0}^{q_0-1}\varDelta _i \end{aligned}$$

where

$$ \varDelta _i=1-\left( \frac{2^{wn}}{(2^n-p_2-w\alpha _1-wi)_w}-1\right) \left( \frac{2^{wn}}{(2^n-p_1-w\alpha _2-wi)_w}-1\right) $$

for \(i=0,\ldots ,q_0-1\). We also have \(\alpha _1 \le q\) and \(p_2\le p + wq\), which gives

$$ \frac{2^{wn}}{(2^n-p_2-w\alpha _1-wi)_w}\le \left( \frac{2^{n}}{2^n-p-3wq} \right) ^w\le \left( 1+ \frac{p+3wq}{2^n-p-3wq} \right) ^w. $$

Then, since \(wp + 3w^2q< 2^n/2\), we can apply Lemma 1 and we get

$$ \frac{2^{wn}}{(2^n-p_2-w\alpha _1-wi)_w}\le 1+\frac{wp+3w^2q}{2^n-wp-3w^2q}\le 1+\frac{2wp+6w^2q}{2^n}. $$

Similarly, one has

$$ \frac{2^{wn}}{(2^n-p_1-w\alpha _2-wi)_w}\le 1+\frac{2wp+6w^2q}{2^n}. $$

Thus one has

$$ \varDelta _i\ge 1-\left( \frac{2wp+6w^2q}{2^n}\right) ^2. $$

Moreover, one has

$$ \frac{(2^{wn})_{q}}{2^{q_0wn}(2^n-p_1)_{w\alpha _2}(2^n-p_2)_{w\alpha _1}}\ge \frac{(2^{wn}-q)^{q}}{2^{qwn}}\ge \left( 1-\frac{q}{2^{wn}}\right) ^{q}\ge 1-\frac{q^2}{2^{wn}}. $$

Finally, we get

$$\begin{aligned} (2^{wn})_{q}\mathsf {p}(\tau ')&\ge \left( 1-\frac{q^2}{2^{wn}}\right) \left( 1-\left( \frac{2wp+6w^2q}{2^n}\right) ^2\right) ^{q_0}\\&\ge \left( 1-\frac{q^2}{2^{wn}}\right) \left( 1-\frac{q(2wp+6w^2q)^2}{2^{2n}}\right) \\&\ge 1-\frac{q^2}{2^{wn}}-\frac{q(2wp+6w^2q)^2}{2^{2n}}. \end{aligned}$$

\(\square \)

4.3 Asymptotically Optimal Security of SPNs

In this section, we will prove that if T is a super blockwise tweakable universal permutation, then the security of \(\mathsf {SP}^T\) converges to \(2^n\) (in terms of the threshold number of queries) as the number of rounds r increases.

Theorem 4

For an even integer r, let \(\mathsf {SP}^{T}\) be an r-round substitution-permutation network based on a \((\delta ,\delta ')\)-super blockwise tweakable universal permutation T. Then one has

$$ {{\mathrm{Adv}}}^{\mathrm {mu}}_{\mathsf {SP}^T}(p,q)\le 4\sqrt{q}\left( 2wp\delta '+2w^2q(\delta '+\delta )+w^2\delta \right) ^{\frac{r}{4}}. $$

Hence, assuming \(\delta ,\delta ' \simeq 2^{-n}\) and \(p=q\), an r-round \(\mathsf {SP}^{T}\) is secure up to \(2^{\frac{rn}{r+2}}\) queries.

Proof of Theorem 4. We assume that \(r=2s\) for a positive integer s. Let \(\overline{\mathsf {SP}}^T[ \mathcal {S}]\) denote a variant of \(\mathsf {SP}^T[ \mathcal {S}]\) without the last permutation layer. Then one has

$$\mathsf {SP}^T[ \mathcal {S}]=\left( \overline{\mathsf {SP}}^{T^{-1}}[ \mathcal {S}^{(2)}]\right) ^{-1}\circ T\circ \overline{\mathsf {SP}}^T[ \mathcal {S}^{(1)}]$$

for \( \mathcal {S}^{(1)}=(S_1,\ldots ,S_{s})\) and \( \mathcal {S}^{(2)}=(S^{-1}_{2s},\ldots ,S^{-1}_{s+1})\). Our proof strategy is to first prove NCPA-security of \(\overline{\mathsf {SP}}\) in the multi-user setting and lift it to CCA-security by doubling the number of rounds.

Suppose that a distinguisher \(\mathcal {D}\) makes p primitive queries to each of the underlying S-boxes and makes q construction queries in the multi-user setting, obtaining an attainable transcript \(\tau =(\mathcal {Q}_C,\mathcal {Q}_S)\). We can partition \(\mathcal {Q}_C\) and \(\mathcal {Q}_S\) as follows.

$$\begin{aligned} \mathcal {Q}_C&=\mathcal {Q}_{C_1}\cup \cdots \cup \mathcal {Q}_{C_{\ell }},\\ \mathcal {Q}_S&=\mathcal {Q}_{S_1}\cup \cdots \cup \mathcal {Q}_{S_{s}}\cup \mathcal {Q}_{S_{s+1}}\cup \cdots \cup \mathcal {Q}_{S_{2s}}, \end{aligned}$$

where we will write

$$\begin{aligned} \mathcal {Q}^{(1)}_S&=\mathcal {Q}_{S_1}\cup \cdots \cup \mathcal {Q}_{S_{s}},\\ \mathcal {Q}^{(2)}_S&=\mathcal {Q}_{S_{s+1}}\cup \cdots \cup \mathcal {Q}_{S_{2s}}. \end{aligned}$$

Throughout the proof, we will write \(\mathcal {Q}_{C_j}=(t_{j,i},x_{j,i},y_{j,i})_{1\le i\le q_j}\) for \(j=1,\ldots ,\ell \). So \(q_j\) denotes the number of queries made to the j-th construction oracle \(C_j\), and \((t_{j,i},x_{j,i},y_{j,i})\) represents the evaluation obtained by the i-th query to \(C_j\). We will also write \(\mathbf {t}=(\mathbf {t}_j)_{1\le j\le \ell }\), \(\mathbf {x}=(\mathbf {x}_j)_{1\le j\le \ell }\), \(\mathbf {y}=(\mathbf {y}_j)_{1\le j\le \ell }\), where

$$\begin{aligned} \mathbf {t}_j=(t_{j,1},\ldots ,t_{j,q_j}),\qquad \mathbf {x}_j&=(x_{j,1},\ldots ,x_{j,q_j}),\qquad \mathbf {y}_j=(y_{j,1},\ldots ,y_{j,q_j}), \end{aligned}$$

for \(j=1,\ldots ,\ell \). Without loss of generality, we can assume that the indices (j, i) have been grouped by their tweaks \(t_{j,i}\); suppose that \(\mathbf {t}_{j}\) consists of d different tweaks, \(t^*_1,\ldots ,t^*_d\in \mathcal {T}\). Then by dropping j for simplicity (when it will be clear from the context), we can write

$$\mathbf {x}_{j}=(\mathbf {x}^*_1,\ldots ,\mathbf {x}^*_d),$$

so that \(\mathbf {x}^*_{i}=(x^*_{i,1},\ldots ,x^*_{i,q'_{i}})\) corresponds to \(t^*_{i}\) for \(i=1,\ldots ,d\), where \(q'_{i}\) is the multiplicity of \(t^*_{i}\) in \(\mathbf {t}_{j}\) (satisfying \(q'_1+\ldots +q'_d=q_{\beta }\)). Let

$$\begin{aligned} \varOmega _{\mathbf {t}_j}&=\left\{ (u_1,\ldots ,u_{q_j})\in (\{0,1\}^n)^{q_j}:\forall i\ne i', (t_{j,i},u_i)\ne (t_{j,i'},u_{i'})\right\} ,\\ \varOmega _{\mathbf {t}}&=\varOmega _{\mathbf {t}_1}\times \ldots \times \varOmega _{\mathbf {t}_{\ell }}. \end{aligned}$$

With these notations, we define probability distributions \(\mu _1\) and \(\mu _2\) on \(\varOmega _{\mathbf {t}}\); for each \(\mathbf {z}=(\mathbf {z}_1,\ldots ,\mathbf {z}_{\ell })\in \varOmega _{\mathbf {t}}\),

where we write \(\mathbf {z}_j=(z_{j,i})_{1\le i\le q_j}\) for \(j=1,\ldots ,\ell \). Using the coupling technique, we can upper bound the statistical distance between \(\mu _{c}\) and the uniform probability distribution for \(c=1,2\). The proof of the following lemma can be found in [CL18].

Lemma 10

For \(c=1,2\), let \(\mu _{c}\) be the probability distribution defined as above, and let \(\nu \) be the uniform probability distribution on \(\varOmega _{\mathbf {t}}\). Then for \(c=1,2\), one has \(\Vert \mu _{c}-\nu \Vert \le \varepsilon \), where

$$\varepsilon =\varepsilon (p,q)\mathrel {\mathop =^\mathrm{def}}q\left( 2wp\delta '+2 w^2q(\delta '+\delta )+w^2\delta \right) ^{s}.$$

By Lemma 6 and Lemma 10, we have a subset \(Z_1\subset \varOmega _{\mathbf {t}}\) such that \(|Z_1|\ge (1-\sqrt{\varepsilon })|\varOmega _{\mathbf {t}}|\) and

$$\begin{aligned} \mu _{1}(\mathbf {z})\ge (1-\sqrt{\varepsilon })\nu (\mathbf {z})=\frac{1-\sqrt{\varepsilon }}{|\varOmega _{\mathbf {t}}|} \end{aligned}$$

for every \(\mathbf {z}\in Z_1\). Similarly, we also have a subset \(Z_2\subset \varOmega _{\mathbf {t}}\) such that \(|Z_2|\ge (1-\sqrt{\varepsilon })|\varOmega _{\mathbf {t}}|\) and

$$\begin{aligned} \mu _{2}(\mathbf {z})\ge (1-\sqrt{\varepsilon })\nu (\mathbf {z})=\frac{1-\sqrt{\varepsilon }}{|\varOmega _{\mathbf {t}}|} \end{aligned}$$

for every \(\mathbf {z}\in Z_2\). For a fixed key \((k_1,\ldots ,k_{\ell })\in \mathcal {K}^{\ell }\), let

$$Z'_2=\{(T_{k_1,\mathbf {t}_1}^{-1}(\mathbf {z}_1),\ldots ,T_{,k_{\ell },\mathbf {t}_{\ell }}^{-1}(\mathbf {z}_{\ell })):(\mathbf {z}_1,\ldots ,\mathbf {z}_{\ell })\in Z_2\},$$

and let \(Z=Z_1\cap Z'_2\). Then it follows that

since \(|Z|\ge (1-2\sqrt{\varepsilon })|\varOmega _{\mathbf {t}}|\). By Lemma 3, we complete the proof of Theorem 4.

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