Non-malleable Coding for Arbitrary Varying Channels

TL;DR

This paper introduces non-malleable coding for arbitrary varying channels, achieving near-optimal rates and ensuring message integrity even under adversarial channel conditions, with a focus on cryptographic and communication security.

Contribution

It extends non-malleable coding to arbitrary varying channels, providing a coding scheme that achieves high rates and guarantees message recovery under specific adversarial channel states.

Findings

Non-malleability is achievable at rates approaching 1.

A coding scheme can ensure message recovery even with adversarial channel states.

The approach extends cryptographic protections to more general communication channels.

Abstract

Non-malleable codes protect against an adversary who can tamper with the coded message by using a tampering function in a specified function family, guaranteeing that the tampering result will only depend on the chosen function and not the coded message. The codes have been motivated for providing protection against tampering with hardware that stores the secret cryptographic keys, and have found significant attention in cryptography. Traditional Shannon model of communication systems assumes the communication channel is perfectly known to the transmitter and the receiver. Arbitrary Varying Channels (AVCs) remove this assumption and have been used to model adversarially controlled channels. Transmission over these channels has been originally studied with the goal of recovering the sent message, and more recently with the goal of detecting tampering with the sent messages. In this paper we introduce non-malleability as the protection goal of message transmission over these channels, and study binary (discrete memoryless) AVCs where possible tampering is modelled by the set of channel states. Our main result is that non-malleability for these channels is achievable at a rate asymptotically approaching 1. We also consider the setting of an AVC with a special state s*, and the additional requirement that the message must be recoverable if s* is applied to all the transmitted bits. We give the outline of a message encoding scheme that in addition to non-malleability, can provide recovery for all s* channel.

Equations

$$\mathsf{SD}(\mathsf{X};\mathsf{Y})=\frac{1}{2}\sum _{\omega \in \Omega }|\text{mbox}Pr[\mathsf{X}=\omega ]-\text{mbox}Pr[\mathsf{Y}=\omega ]|.$$

$${\mathsf{W}}_{\mathbf{s}}(\mathbf{x})=({\mathsf{W}}_{s_1}(x_1),\dots ,{\mathsf{W}}_{s_n}(x_n)).$$

$$\mathrm{Tamper}_{\mathbf{m}}^{f}=\left\{\begin{array}[]{c}\mathbf{x}\leftarrow\mathsf{Enc}(\mathbf{m}),\tilde{\mathbf{x}}=f(\mathbf{x}),\tilde{\mathbf{m}}=\mathsf{Dec}(\tilde{\mathbf{x}})\\ \mathrm{Output}\ \tilde{\mathbf{m}}\\ \end{array}\right\},$$

$$\mathrm{Tamper}_{\mathbf{m}}^{f}\stackrel{{\scriptstyle\varepsilon}}{{\approx}}\left\{\begin{array}[]{c}\tilde{\mathbf{m}}\leftarrow\mathcal{D}_{f}\\ \mathrm{Output}\ \mathbf{m}\ \mathrm{if}\ \tilde{\mathbf{m}}=\mathsf{same}^{*};\ \tilde{\mathbf{m}}\ \mathrm{otherwise.}\end{array}\right\},$$

$${\mathrm{Tamper}}_{\mathbf{m}}^f\stackrel{\epsilon }{\approx }\text{mbox}Copy({\mathcal{D}}_f,\mathbf{m}).$$

$$\mathrm{Tamper}_{\mathbf{m}}^{\mathsf{W}_{\mathbf{s}}}=\left\{\begin{array}[]{c}\mathbf{x}\leftarrow\mathsf{Enc}(\mathbf{m}),\mathbf{y}\leftarrow\mathsf{W}_{\mathbf{s}}(\mathbf{x}),\tilde{\mathbf{m}}=\mathsf{Dec}(\mathbf{y})\\ \mathrm{Output}\ \tilde{\mathbf{m}}.\\ \end{array}\right\}.$$

$$\mathsf{SD}({\mathrm{Tamper}}_{\mathbf{m}}^{{\mathsf{W}}_{\mathbf{s}}};\mathsf{Copy}({\mathcal{D}}_{\mathbf{s}},\mathbf{m}))\le \epsilon ,$$

$$\mathsf{Pr}[\mathsf{Dec}({\mathsf{W}}_{\mathbf{s}}(\mathsf{Enc}(\mathbf{m})))=\mathbf{m}]\ge 1-\delta .$$

$$\mathsf{SD}({\mathrm{Tamper}}_{\mathbf{m}}^{{\mathsf{W}}_{\mathbf{s}}};\mathsf{Copy}({\mathcal{D}}_{\mathbf{s}},\mathbf{m}))\le \epsilon ,$$

$$\mathsf{W}_{s}=\left[\begin{array}[]{cc}w_{11}&w_{12}\\ w_{21}&w_{22}\\ \end{array}\right],$$

$$0\le w_{ij}\le 1\text{mbox}and,\sum _j{w}_{ij}=1,i=1,2.$$

$${\mathsf{W}}_s={\alpha }_1{\mathsf{W}}_{e_1}+{\alpha }_2{\mathsf{W}}_{e_2}+{\alpha }_3{\mathsf{W}}_{e_3}+{\alpha }_4{\mathsf{W}}_{e_4}$$

$${\alpha }_1=w_{11}-{\alpha }_3,{\alpha }_2=1-w_{22}-{\alpha }_3,$$

$${\alpha }_4={\alpha }_3-(w_{11}-w_{22})$$

$$1-w_{22}\ge {\alpha }_3\ge 0,\text{mbox}and{w}_{11}\ge {\alpha }_3\ge w_{11}-w_{22}.$$

$${\mathsf{W}}_{\mathbf{s}}(\mathbf{x})=({\mathsf{W}}_{s_1}(x_1),\dots ,{\mathsf{W}}_{s_n}(x_n)).$$

$${\mathsf{W}}_{s_i}={\alpha }_{i,1}{\mathsf{W}}_{e_1}+{\alpha }_{i,2}{\mathsf{W}}_{e_2}+{\alpha }_{i,3}{\mathsf{W}}_{e_3}+{\alpha }_{i,4}{\mathsf{W}}_{e_4},$$

$$\sum _{j=1}^4{\alpha }_{i,j}=1.$$

$$\begin{array}[]{ll}\mathsf{W}_{\mathbf{s}}&=(\mathsf{W}_{s_{1}},\ldots,\mathsf{W}_{s_{n}})\\ &=\sum_{(j_{1},\ldots,j_{n})\in[4]^{n}}\left(\prod_{i=1}^{n}\alpha_{i,j_{i}}\right)(\mathsf{W}_{e_{j_{1}}},\ldots,\mathsf{W}_{e_{j_{n}}}).\end{array}$$

$$\sum _{(j_1,\dots ,j_n)\in [4{]}^n}\left(\prod _{i=1}^n{\alpha }_{i,j_i}\right)=\prod _{i=1}^n\left(\sum _{j=1}^4{\alpha }_{i,j}\right)=1.$$

$$\mathsf{SD}({\mathrm{Tamper}}_{\mathbf{m}}^{{\mathsf{W}}_{\mathbf{s}}};\mathsf{Copy}({\mathcal{D}}_{\mathbf{s}},\mathbf{m}))\le \epsilon .$$

$$\begin{array}[]{ll}\Phi_{\mathsf{Keep}}(\mathsf{W}_{s_{i}})=\alpha_{i,1},&\Phi_{\mathsf{Flip}}(\mathsf{W}_{s_{i}})=\alpha_{i,2},\\ \Phi_{\mathsf{Set0}}(\mathsf{W}_{s_{i}})=\alpha_{i,3},&\Phi_{\mathsf{Set1}}(\mathsf{W}_{s_{i}})=\alpha_{i,4},\\ \end{array}$$

$$({\mathsf{W}}_{s_1},\dots ,{\mathsf{W}}_{s_n})=\sum _{f=(f_1,\dots ,f_n)\in {\mathcal{F}}_{\mathsf{BIT}}}\left(\prod _{i\in [n]}{\Phi }_{f_i}({\mathsf{W}}_{s_i})\right)f.$$

$$\mathsf{SD}({\mathrm{Tamper}}_{\mathbf{m}}^f;\mathsf{Copy}({\mathcal{D}}_f,\mathbf{m}))\le \epsilon .$$

$${\mathcal{D}}_{\mathbf{s}}=\sum _{f\in {\mathcal{F}}_{\mathsf{BIT}}}\left(\prod _{i\in [n]}{\Phi }_{f_i}({\mathsf{W}}_{s_i})\right){\mathcal{D}}_f.$$

$$\tilde{\mathbf{m}}={\mathbf{y}}_R{G}_R^{-1}.$$

$$\mathcal{G}:=\{\mathsf{ECCdec}\circ f\circ \mathsf{ECCenc}|f\in \mathcal{F}\},$$

$$\left\{\begin{array}[]{ll}\mathsf{Enc}(\mathbf{m})&=\mathsf{ECCenc}(\mathsf{NMCenc}(\mathbf{m}))\\ \mathsf{Dec}(\mathbf{y})&=\mathsf{NMCdec}(\mathsf{ECCdec}(\mathbf{y})).\end{array}\right.$$

$$g_f(\mathbf{u})={\mathsf{ECCdec}}_R\circ f\circ \mathsf{ECCenc}(\mathbf{u})=(\mathbf{u}G{M}_f+{\Delta }_f{)}_R{G}_R^{-1},$$

Full text

I Introduction

Reliable message transmission over a noisy channel is a central problem in communication theory. Shannon [12] modelled a message transmission system and, using random coding argument, showed that reliable communication in the sense of message recovery is possible as long as the information transmission rate is below the channel capacity. A fundamental assumption in Shannon’s model is that the channel is known to the communicants. For Discrete Memoryless Channels (DMC) the probability law for $n$ times applications of the channel is given by, $\mathsf{W}^{n}(\mathbf{y}|\mathbf{x})=\Pi_{t=1}^{n}W(y_{t}|x_{t})$ where $W(y_{t}|x_{t})$ is specified by a probability stochastic matrix $W$ labeled by the elements of the input alphabet ${\cal X}$ (rows) and the output alphabet ${\cal Y}$ (columns) , and $\mathbf{x}=(x_{1}\cdots x_{n})\in{\cal X}^{n}$ and $\mathbf{y}=(y_{1}\cdots y_{n})\in{\cal Y}^{n}$.

Blackwell, Brieman and Thomasian [2] initiated the study of Arbitrarily Varying Channels (AVC’s): communication channels with unknown parameters that can vary with time and in an unknown way. A discrete memoryless AVC (referred to simply as AVC in the sequel), $\mathsf{W}:{\cal X}\times{\cal S}\rightarrow{\cal Y}$, is specified by a set of stochastic matrices $\{\mathsf{W}_{s}\colon{\cal X}\rightarrow{\cal Y}|s\in{\cal S}\}$, and ${\cal S}$ is called the set of states of the channel. For an input sequence $\mathbf{x}=(x_{1},\ldots,x_{n})\in{\cal X}^{n}$ to the AVC, the output distribution is given by $\mathsf{W}^{n}(\mathbf{y}|\mathbf{x},\mathbf{s})=\Pi_{t=1}^{n}W(y_{t}|x_{t},s_{t})$, where $s_{t}\in{\cal S},t=1\cdots n$. AVCs capture adversarially controlled channels, in particular, the set of channel states and their choice for each symbol captures possible adversary’s influence on transmitted symbols. The traditional goal of communication over AVC’s is message recovery by the receiver. For powerful adversaries however one cannot expect the message to be recoverable, and so less demanding goals such as detection of tempering, have been considered. Recently, Kosut and Kliewer [9] studied message authentication in a setting where the set of states includes a “no adversary" state in which transmission will not be influenced by the adversary. This captures a real life situation where the adversary may or may not be present. They considered two communication goals: when the adversary is absent, the error due to the stochastic matrix of the channel that corresponds to the no adversary state must be corrected; otherwise, the presence of the adversary must be detected. Codes for detection of tampering in shared secret key setting was first considered by Gilbert, MacWilliams and Sloane [8]. They introduced Message Authentication Codes (MAC) that can detect arbitrary tampering in the form of message substitution or injection of fruadulant messages. MAC is one of the most widely used cryptographic primitives for protection of communication.

Kosut et al.’s model however does not assume shared key.

In this paper we consider a yet weaker goal for communication for AVCs called non-malleability, that prevents the adversary from tampering with the communication such that the decoded message be “related" to the sent one. Non-malleability has been a widely studied goal in cryptography [6]. More recently Dziembowski, Pietrzak and Wichs [7] introduced non-malleable codes (NM codes), where the adversary is defined by a family of tampering functions, $\cal F$, and non-malleable is defined as a relaxation of error correction and error detection. NM codes are binary stochastic codes with randomized encoders and deterministic decoders. Non-malleability is defined by the following experiment: a message $\mathbf{m}$ is encoded to a (randomized) codeword; the adversary chooses a function $f\in\cal F$ and tampers with the codeword; and finally the (deterministic) decoder is used to decode the tampered codeword. Non-malleability requires that the decoded message is either the same as the encoded one, or results in a random message which is distributed according to a distribution that is determined by the tampering function $f$ only (and is independent of the encoded message).

The following example elucidates the role of the function family in achieving non-malleability. It is easy to see that linear stochastic codes cannot provide non-malleability with respect to the family ${\cal F}=\{f(\mathbf{x})=\mathbf{x}+\Delta|\Delta\in\{0,1\}^{n}\}$. This is because for an encoding $\mathbf{c}$ of a message $\mathbf{m}$, one can choose $\Delta$ to be the codeword corresponding to the message $\mathbf{1}$, consisting of all 1’s. By applying the tampering function $f(\mathbf{x})=\mathbf{x}+\Delta$ to $\mathbf{c}$, we have $\mathbf{c}+\Delta$ which because of the linearity of the code, is the encoding of the message $\mathbf{m}+\mathbf{1}$, which is equal to $\mathbf{m}$ with every bit flipped, contradicting the definition of non-malleability.

It has been shown that non-malleable codes exist if $\log\log|{\cal F}|\leq\alpha n$, for any $\alpha<1$ [7]. Using a random coding argument, Cheraghchi and Guruswami [4] derived the lower bound $1-\log\log|{\cal F}|/n$, on the achievable rate of NM codes for a function family ${\cal F}$. This bound depends only on the size of the function family. An important family of tampering functions is the Bit-wise Independent Tampering (BIT) family in which a function $f\colon\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ can be written as $f=(f_{1},\ldots,f_{n})$, where $f_{i}\colon\{0,1\}\rightarrow\{0,1\}$ for $i=1,\ldots,n$. Explicit constructions of non-malleable codes for the BIT family that achieves information rate $1$, were given in [5, 1]. We will also use non-malleable codes for the family of affine tampering functions $f\colon\{0,1\}^{n}\rightarrow\{0,1\}^{n}$, where each output bit of a tampering function can be written as an affine function of the $n$ input bits. Non-malleable codes for this family were explicitly constructed in [3].

Our work. In this paper, we introduce non-malleability as the communication goal of (discrete memoryless) AVCs. That is, we require that the adversary cannot modify the communication in a way that the decoded message by the receiver be related to the sent one: any tampering through the AVC will result in either the sent message to be correctly received, or completely lose its informational value. We model the adversarial channel by an AVC $\{\mathsf{W}_{s}\colon{\cal X}\rightarrow{\cal Y}|s\in{\cal S}\}$, and require that for any state sequence $\mathbf{s}\in\mathcal{S}^{n}$, the non-malleability guarantee should hold. We focus on AVCs where the input and output alphabets are binary.

We show that for such AVCs non-malleable communication is always possible at rate $1$. We show this by proving that non-malleable codes for the family bitwise independent tampering can be used for non-malleable coding over these channels, achieving rate $1$.

We also consider a setting similar to Kosut and Kliewer [9] where there is a special state $s^{*}$, specified by the binary channel ${\mathsf{W}}_{s^{*}}$, for which we require the property that if used on all bits of the sent codeword, allow the original message to be recovered. The special state that we consider is defined by an erasure channel where each binary input symbol is erased with a constant probability $p^{*}$. We thus expand the set of possible states of the AVC by the set of erasure channels.

With this AVC with the new set of states, we require the communication to be non-malleable. We outline a construction that uses a non-malleable code with respect to the family affine tampering functions and a linear erasure correcting code, and provide the combined requirements.

Discussion and future work. We consider non-malleable coding for AVCs with binary alphabets (input and output). To our knowledge non-malleability is the least demanding goal for protecting a communication channel against tampering. Binary discrete memoryless AVCs that are considered in this paper are a natural starting point for the study of NM channel codes.This is inline with the study of bitwise tampering function family in [7] that first introduced NM codes. The power of the adversary in (discrete memoryless) AVCs is modelled by the set of possible states. An interesting research question is to capture realistic tampering adversaries for communication channels as AVCs, and design non-malleable channel codes for them.

In [11], non-malleable codes for a family functions that is inspired by the adversarial tampering of a communication channel is considered: the adversary chooses the tampering functions after observing some of the codeword bits (adversary chooses which bit to observe). Considering a similar function family in AVC setting requires more general AVCs that allow the state of the channel for a particular symbol to depend on the adversary’s observation of the channel.

Other function families that have been considered for non-malleable codes could also be considered in AVC setting.

II Models and main results

We use the following notations. A sequence (vector) of $n$ elements $x_{i}\in{\cal X}$ is denoted by a bold symbol: for example ${\bf x}=(x_{1}\cdots x_{n})$. The statistical distance (total variation distance) between two random variables (their corresponding distributions) $\mathsf{X}$ and $\mathsf{Y}$ that are defined over the set $\Omega$, is defined as follows.

[TABLE]

We say $\mathsf{X}$ and $\mathsf{Y}$ are $\varepsilon$-close (denoted $\mathsf{X}\stackrel{{\scriptstyle\varepsilon}}{{\approx}}\mathsf{Y}$) if $\mathsf{SD}(\mathsf{X};\mathsf{Y})\leq\varepsilon$.

II-A Non-malleable coding for AVC

We consider a discrete memoryless AVC, $\mathsf{W}:{\cal X}\times{\cal S}\rightarrow{\cal Y}$, with (discrete) input and output alphabet sets $\mathcal{X}$ and $\mathcal{Y}$, respectively, and $\mathcal{S}$ denoting the set of possible states.

The set of states is corresponding to the set of stochastic matrices $\{\mathsf{W}_{s}\colon{\cal X}\rightarrow{\cal Y}|s\in{\cal S}\}$.

The application of $\mathsf{W}$ to an input sequence $\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathcal{X}^{n}$ for a state sequence is given by,

[TABLE]

In this paper we consider AVC with binary input and output alphabet $\mathcal{X}=\mathcal{Y}=\{0,1\}$.

Let $\bot$ be a special symbol that denotes detection of tampering.

Definition 1** ([7])**

A $(k,n)$-coding scheme consists of a randomized encoding function $\mathsf{Enc}:\{0,1\}^{k}\rightarrow\{0,1\}^{n}$ (randomness is implicit), and a deterministic decoding function $\mathsf{Dec}:\{0,1\}^{n}\rightarrow\{0,1\}^{k}\cup\{\perp\}$ such that, for each $\mathbf{m}\in\{0,1\}^{k}$, $\mathsf{Pr}[\mathsf{Dec}(\mathsf{Enc}(\mathbf{m}))=\mathbf{m}]=1$, and the probability is over the randomness of encoding.

A tampering function for a $(k,n)$-coding scheme is a function $f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ that is applied to a codeword, and results in a binary $n$-vector. Consider an experiment where a message $\mathbf{m}$ is encoded by the encoder, the codeword is tampered by a function $f$, the decoder is applied on the tampered codeword outputting a message $\tilde{\mathbf{m}}$. Let $\mathsf{same}^{*}$ be a special symbol which means that $\tilde{\mathbf{m}}$ is the same as $\mathbf{m}$.

Definition 2** ([7])**

Let $\mathcal{F}$ be a family of tampering functions. For each $f\in\mathcal{F}$ and $\mathbf{m}\in\{0,1\}^{k}$, define the tampering-experiment

[TABLE]

which is a random variable over the set $\{0,1\}^{k}\cup\{\bot\}$, and the randomness is due to the randomized encoding. A coding scheme $(\mathsf{Enc},\mathsf{Dec})$ is non-malleable with respect to $\mathcal{F}$ if for any $f\in\mathcal{F}$, there exists a distribution $\mathcal{D}_{f}$ over the set $\{0,1\}^{k}\bigcup\{\perp,\mathsf{same}^{*}\}$ such that, for all $\mathbf{m}\in\{0,1\}^{k}$, we have:

[TABLE]

*and $\mathcal{D}_{f}$ is efficiently samplable. *

That is, $\mathrm{Tamper}_{\mathbf{m}}^{f}$ is a random variable that is $\epsilon$-close to a distribution that is defined by the right hand side above, which is commonly denoted by $\mbox{Copy}(\mathcal{D}_{f},\mathbf{m})$. Non-malleability is then rewritten as follows.

[TABLE]

Our definition of *non-malleable codes for AVCs * can be seen as a natural extension of non-malleable codes to include probabilistic tampering.

Definition 3

Let $\mathsf{W}:{\cal X}\times{\cal S}\rightarrow{\cal Y}$ be a binary AVC. For a state sequence $\mathbf{s}\in\mathcal{S}^{n}$ and message $\mathbf{m}\in\{0,1\}^{k}$, consider the random variable $\mathrm{Tamper}_{\mathbf{m}}^{\mathsf{W}_{\mathbf{s}}}$ that is defined by the following tampering-experiment:

[TABLE]

This random variable is over the set $\{0,1\}^{k}\cup\{\bot\}$, and the randomness is from the encoder and also the application of channel $\mathsf{W}_{\mathbf{s}}(\cdot)$. The coding scheme is $\varepsilon$-non-malleable for the AVC, if for any message $\mathbf{m}\in\{0,1\}^{k}$ and any state sequence $\mathbf{s}\in\mathcal{S}^{n}$, there exists a distribution $\mathcal{D}_{\mathbf{s}}$ over the set $\{0,1\}^{k}\cup\{\bot,\mathsf{same}^{*}\}$ satisfying

[TABLE]

where the distribution $\mathcal{D}_{\mathbf{s}}$ is independent of the message $\mathbf{m}$.

Our main result is to show that non-malleability for binary AVCs is always achievable. We show this by proving that a non-malleable code with respect to the BIT function family, provides non-malleability for transmission over binary AVCs.

Theorem 1

*A $\varepsilon$-non-malleable coding scheme for BIT function family is a $\varepsilon$-non-malleable coding scheme for a binary AVC $\mathsf{W}:\{0,1\}\times\mathcal{S}\rightarrow\{0,1\}$. *

II-B Application to AVC with a special state

We next consider a setting where one of the states $s^{*}\in\cal S$ is a special state, in the sense that if that state is chosen for all $1\leq i\leq n$, the receiver will be able to recover the message. That is in addition to non-malleability that is guaranteed for any state sequence, we require the additional guarantee of message recovery for a special state sequence $(s^{*})^{n}$. This special sequence captures a known state of the channel by the sender and the receiver and their goal of providing reliable communication for that.

Definition 4

Let $\mathsf{W}:{\cal X}\times{\cal S}\rightarrow{\cal Y}$ be a binary AVC with a special state $s^{*}\in\mathcal{S}$. A $(k,n)$-coding scheme is $(\delta,\varepsilon)$-non-malleable with respect to $\mathsf{W}$, if the following properties hold.

For any message $\mathbf{m}\in\{0,1\}^{k}$, when the state sequence $\mathbf{s}=(s^{*})^{n}$,

[TABLE] 2. 2.

For any message $\mathbf{m}\in\{0,1\}^{k}$ and any state sequence $\mathbf{s}\in(\mathcal{S}^{n}\setminus\{(s^{*})^{n}\})$, there exists a distribution $\mathcal{D}_{\mathbf{s}}$ over the set $\{0,1\}^{k}\cup\{\bot,\mathsf{same}^{*}\}$ satisfying

[TABLE]

where the distribution $\mathcal{D}_{\mathbf{s}}$ is independent of $\mathbf{m}$.

The special state that we consider corresponds to a Binary Erasure Channel (BEC). We thus extend the set of states of the AVC $\mathsf{W}:\{0,1\}\times{\cal S}\rightarrow\{0,1\}\cup\{\bot\}$ to include BEC’s, and use a specific BEC $\mathsf{W}_{s^{*}}$ for the special state $s^{*}\in\mathcal{S}$.

We sketch a generic construction that uses a non-malleable code for the family of affine tampering functions and a linear erasure correcting code. The construction first encodes a $k$ bit message using a $(k,m)$ non-malleable coding scheme for the family of affine functions on $m$ bits, and then encodes the resulting $m$-bit codeword using an erasure correcting code into a $n$-bit final codeword for the BEC $\mathsf{W}_{s^{*}}$. We use a decoder that correctly decodes the message if there are up to $p^{*}n$ erasures, where $p^{*}$ is erasure probability of $\mathsf{W}_{s^{*}}$, and declares failure (error detection) for more erasures. When the state sequence is $\mathbf{s}=(s^{*})^{n}$, the erasure correcting code guarantees that the message is correctly recovered with probability at least $1-\delta$, where $\delta$ can be calculated for the code. For an arbitrary state sequence for the AVC that includes the erasure channels and $\mathbf{s}\neq(s^{*})^{n}$, the proof intuitively works as follows. If the received word has too many erasures, the decoder will flag detection. When the number of erasures is less than $p^{*}n$, the decoder will recover an $n$ bit string. As will be shown in Section IV, the effect of this decoding is that tampering on the $m$-bit NM codeword will be an affine function, that can be protected against because of the property of the NM code with protection against affine tampering.

III Proofs of theorems

Let $\mathcal{F}_{\mathrm{BIT}}$ denote the family of Bitwise Independent Tampering (BIT) functions. For a binary string of length $n$, a BIT function $f\in\mathcal{F}_{\mathrm{BIT}}$ is written as $f=(f_{1},\ldots,f_{n})\in\{\mathsf{Keep},\mathsf{Flip},\mathsf{Set0},\mathsf{Set1}\}^{n}$, where $\mathsf{Set0}$ and $\mathsf{Set1}$ overwrite the input with [math] and $1$, respectively, and $\mathsf{Keep}$ and $\mathsf{Flip}$ are the identity (no change) and the flip functions.

Non-malleable coding schemes for the function family $\mathcal{F}_{\mathsf{BIT}}$ has been widely studied. In fact, the fist construction of non-malleable codes [7] was given for this family. The following theorem shows that an NM code that provides protection against the function family $\mathcal{F}_{\mathsf{BIT}}$ will provide protection against any binary AVC.

Proof of Theorem 1. Define the set of elementary binary channels to be $\{\mathsf{W}_{e_{1}},\mathsf{W}_{e_{2}},\mathsf{W}_{e_{3}},\mathsf{W}_{e_{4}}\}$. corresponding to the bit functions $\{\mathsf{Keep},\mathsf{Flip},\mathsf{Set0},\mathsf{Set1}\}$. The proof has the following steps. First, we show that a binary channel can be written as a convex combination of the four elementary binary channels. Second, using the above result we show that a sequence of binary channels of length $n$ (that is applied to an input sequence of length $n$) can be written as a convex combination of the $4^{n}$ channel sequences of length $n$ over elementary channels. Finally, we use the properties of non-malleable codes for $\mathcal{F}_{\mathsf{BIT}}$ to show non-malleabilityis achievable for any binary AVC. More details are given below.

Claim 1: *A binary channel can be written as a convex combination of four elementary channels.

*We use $\mathsf{W}_{s}$ to denote a binary channel and its two-by-two channel transition matrix. First, consider a binary DMC, with channel transition matrix

[TABLE]

where rows and columns are labeled by possible inputs and outputs, respectively. The matrix entries $w_{ij}$’s are $\mathsf{W}_{s}(j-1|i-1),i,j\in\{1,2\}$ and satisfy the following:

[TABLE]

We next show the matrix $\mathsf{W}_{s}$ can be written as a sum of the transition matrices of the four elementary channels:

[TABLE]

where the coefficients $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are non-negative real numbers that satisfy $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=1$.

By solving linear equations, we have the following relations,

[TABLE]

Note that we do have $\alpha_{1}+\alpha_{3}+\alpha_{2}+\alpha_{4}=1$. What is left to show is that these coefficients can be chosen to be all non-negative. This can be satisfied if the following holds,

[TABLE]

When $w_{11}\geq w_{22}$, we can let $\min\{w_{11},1-w_{22}\}\geq\alpha_{3}\geq w_{11}-w_{22}$. When $w_{11}<w_{22}$, we let $\min\{w_{11},1-w_{22}\}\geq\alpha_{3}\geq 0$. Thus such coefficients always exist, and so choosing the value of $\alpha_{3}$ allows computation of $\alpha_{1},\alpha_{2}$ and $\alpha_{4}$.

**Claim 2: ** For a state sequence of length $n$, the corresponding application of the The output of a binary AVC of length $n$ with a state sequence $\mathbf{s}$ on an input sequence $\mathbf{x}$ of length $n$, can be written as a convex combination of the outputs of $4^{n}$ AVCs, each applied on the same input $\mathbf{x}$. The AVCs correspond to the $4^{n}$ possible state sequences of length $n$ over elementary channels.

Using the notations in Section II-A, for a state sequence $(s_{1},\ldots,s_{n})\in\mathcal{S}^{n}$, we have

[TABLE]

Using the results of Claim 1, we have the following decomposition for $i=1,\ldots,n$,

[TABLE]

where $\alpha_{i,j}$ are non-negative real numbers and

[TABLE]

This means that applying channel $\mathsf{W}_{s_{i}}$ to $x_{i}$ results in the same output distribution as applying the elementary channels $\mathsf{W}_{e_{j}}$ with associated probabilities $\alpha_{i,j}$. For the channel sequence $(\mathsf{W}_{s_{1}},\ldots,\mathsf{W}_{s_{n}})$, the probability that $\mathsf{W}_{e_{j_{i}}}$ is applied to $x_{i}$ for all $i=1,\ldots,n$ is then $\prod_{i=1}^{n}\alpha_{i,j_{i}}$. Using $\mathsf{W}_{\mathbf{s}}$ to denote the channel sequence, and its corresponding sequence of channel transition matrices, we have:

[TABLE]

The coefficients in the above decomposition are non-negative and we have

[TABLE]

**Claim 3: ** *An $\epsilon$-non-malleable code for function family $\mathcal{F}_{\mathrm{BIT}}$ provides $\epsilon$-non-malleable protection for an AVC $\{0,1\}\times\mathcal{S}\rightarrow\{0,1\}$.

*Given a $(k,n)$-coding scheme that is $\varepsilon$-non-malleable with respect to the function family $f\in\{\mathsf{Keep},\mathsf{Flip},\mathsf{Set0},\mathsf{Set1}\}^{n}$, we want to show that for any state sequence $\mathbf{s}=(s_{1},\ldots,s_{n})\in\mathcal{S}^{n}$ there is a distribution $\mathcal{D}_{\mathbf{s}}$ that satisfies

[TABLE]

Define the following projection maps for binary channels $\mathsf{W}_{s_{i}}$.

[TABLE]

where $\alpha_{i,1},\alpha_{i,2},\alpha_{i,3},\alpha_{i,4}$ are given in (4). According to Claim 2, we have that

[TABLE]

Recall that the coding scheme is $\varepsilon$-non-malleable with respect to functions $f\in\{\mathsf{Keep},\mathsf{Flip},\mathsf{Set0},\mathsf{Set1}\}^{n}$. That is for the function $f$, there exists a distribution $\mathcal{D}_{f}$ that satisfies

[TABLE]

We define

[TABLE]

Then $\mathcal{D}_{\mathbf{s}}$ is the corresponding distribution for the $\mathsf{W}_{\mathbf{s}}$ that satisfies Definition 3.

There are explicit constructions of non-malleable codes with respect to bit-wise independent tampering that achieves information rate $1$ [5, 1]. According to Theorem 1, we then have rate $1$ non-malleable codes for any AVC.

Corollary 1

There exist non-malleable coding schemes for any AVC $\mathsf{W}:\{0,1\}\times{\cal S}\rightarrow\{0,1\}$ achieving rate $1$.

IV A coding scheme for AVC with special state

We consider a linear erasure correcting code for the BEC $\mathsf{W}_{s^{*}}$, with encoder $\mathsf{ECCenc}\colon\{0,1\}^{m}\rightarrow\{0,1\}^{n}$ and the following decoder $\mathsf{ECCdec}\colon\{0,1\}^{n}\rightarrow\{0,1\}^{m}\cup\{\bot\}$.

$\mathsf{ECCdec}$: * Let $G_{m\times n}$ be the generator matrix of the code. Consider a received word $\mathbf{y}$ with erased bits given by $E\subset[n]$. The decoder $\mathsf{ECCdec}$ finds a subset $R\subset[n]\backslash E$ with $|R|=m$, such that $G_{R}$ is an invertible a submatrix of $G_{m\times n}$ with columns corresponding to $R$. Such submatrix will exist with overwhelming probability because of erasure correction property of the code. The decoder’s output is,*

[TABLE]

*If the set $E$ of erased bits is too large such that no reconstruction set $R$ exists, the decoder simply outputs $\bot$. *

As described in Section II-B, the construction uses a two step coding. We first define the notion of induced tampering.

Definition 5** ([10])**

Let $\mathsf{ECC}$ be an erasure correcting code with an encoder $\mathsf{ECCenc}\colon\{0,1\}^{m}\rightarrow\{0,1\}^{n}$ and a decoder $\mathsf{ECCdec}$. Let $\mathcal{F}$ be a family of tampering functions over $\{0,1\}^{n}$. The tampering family $\mathcal{G}$ over $\{0,1\}^{m}$ induced by $\mathcal{F}$ and the ECC is defined as follows.

[TABLE]

where “$\circ$” denotes the composition of functions.

Let $(\mathsf{NMCenc},\mathsf{NMCdec})$ be a coding scheme that is non-malleable with respect to the family of affine tampering functions over $\{0,1\}^{m}$. Consider the $(k,n)$-coding scheme

[TABLE]

We will use an erasure correcting code with detection (failure) error $\delta$ that can recover up to $p^{*}n$ errors. Consider the case where $\mathbf{s}\neq(s^{*})^{n}$. If the number of erased symbols less than $p^{*}n$ (i.e., the adversary has not erased too many symbols), the decoder will have an output.

From the proof of Theorem 1, we only need to consider the state sequences that consists of bit-wise independent tampering functions. For any bit-wise independent tampering function $f$ over $\{0,1\}^{n}$, let $g_{f}=\mathsf{ECCdec}\circ f\circ\mathsf{ECCenc}$ be the tampering function over $\{0,1\}^{m}$ induced by $f$, through the encoder and decoder of the erasure code.

We now argue that $g_{f}$ is an affine function. This follows by noting that, (i) from equation 5 the decoder output $\tilde{\mathbf{m}}=\mathbf{y}_{R}G_{R}^{-1}$, and (ii) a bitwise independent function $f=(f_{1},\ldots,f_{n})$ can be expressed as an affine function $f(\mathbf{x})=\mathbf{x}M_{f}+\Delta_{f}.$

The combination of these result in the induced function on the recovered word to be given by,

[TABLE]

where $g_{f}$ is an affine function given any choice of $R\subset[n]$ with $|R|=m$ such that $G_{R}$ is invertible.

Acknowledgment

The work of Safavi-Naini is supported in part by Natural Sciences and Engineering Research Council of Canada, Discovery Grant Program. The work of Lin, Ling, Wang is supported by Singapore Ministry of Education under Research Grant MOE2016-T2-2-014(S) and RG133/17 (S).

The authors list is alphabetical.