# Simple Soundness Proofs

*2022-10-14 by Alex Kampa*

We present a general method to simplify soundness proofs under certain conditions. Given an adversary \(\mathcal{A}\) able to break a scheme \(S\) with non-negligible probability \(t\), we define the concept of *trace* of a *winning configuration*, which is already implicitly used in soundness proofs. If a scheme can be constructed that (1) takes a random configuration \(e\), being the inputs and execution environment of \(\mathcal{A}\), (2) “guesses” a trace, (3) modifies \(e\) based on its guess so that the modified configuration \(e'\) is statistically indistinguishable from the original one, (4) is then able to execute \(\mathcal{A}\) correctly under the condition that \(e'\) is a winning configuration and that \(B\)’s guess of the trace was correct, and finally (5) that during its execution \(\mathcal{A}\) is unable extract any information about \(B\)’s guess, then the probability of \(B\) winning can be expressed as a simple function of \(t\) and the bit-length of the trace, namely \(\frac{t}{2^m}\). Soundness then results if \(2^m\) is polynomial in the security parameter.

To illustrate the concept, a concrete application of this method to a simple binary voting scheme is then described in detail.

Link to the paper: https://github.com/aragonzkresearch/blog/blob/main/pdf/simple-soundness.pdf

## Introduction

Soundness proofs tend to be quite long and technical. Here, we describe a general method to significantly simplify and shorten such proofs if some specific conditions are met.

## Simple Soundness Proofs

A common method of proving the soundness of a cryptographic scheme S is the following. We first assume that there exists an adversary \(\mathcal{A}\) that can break the scheme with some non-negligible probability. We then construct a scheme B which uses \(\mathcal{A}\) in a simulated environment to break a known-to-be-hard problem P, also with some non-negligible probability.

**Description of \(\mathcal{A}\) in its native environment**

\(\mathcal{A}\) is a deterministic polynomial-time algorithm. There exists a finite set of \textit{execution configurations} \(E(\lambda)\) for \(\mathcal{A}\), where \(\lambda\) is the security parameter. Each such configuration includes inputs to \(\mathcal{A}\) and completely determines the execution of \(\mathcal{A}\). Given a randomly selected execution configuration \(e \in E(\lambda)\), \(\mathcal{A}\) breaks the scheme \(S\) with probability \(t(\lambda)\), where \(t\) is a polynomial function. We then say that \(\mathcal{A}\) wins. In that case, \(e\) is said to be a \textit{winning configuration}.

**The scheme \(B_r\) able to correctly execute \(\mathcal{A}\)**

We denote by \(B_r\) (where “r” stands for “real”) a scheme that simulates the real execution environment of \(\mathcal{A}\). Usually, this will require the knowledge of extra information about some elements of the execution configuration. For example, the execution configuration may include a sequence of group elements, and it may be necessary to know the discrete log of these elements in order to be able to always complete the execution of \(\mathcal{A}\). If \(G\) is a group with \(q\) elements and generator \(g\), a set \(\{ g_i \}_{i \in [n]}\) of random elements of \(G\) can be produced by first sampling \(x_i \leftarrow \mathbb{Z}_q\) and outputting \(\{ g^{x_i} \}\). In general, it is not difficult to generate random execution environments such that \(B_r\) has all the necessary extra information.

**A modified scheme \(B\) to break \(P\)**

To attempt to break \(P\) using \(\mathcal{A}\), the scheme \(B\) will usually need to modify the execution environment slightly. Given the modified environment \(e'\), we say that \(B\) wins if it is able to complete the execution of \(\mathcal{A}\), and \(\mathcal{A}\) wins. In that case, \(B\) also breaks the problem \(P\). In some cases, however, \(B\) will be unable to complete the execution of \(\mathcal{A}\). Running \(\mathcal{A}\) in this simulation is therefore not equivalent of running \(\mathcal{A}\) in its normal execution environment. This fact usually complicates the soundness proof, as conditional probabilities must be introduced to deal with cases where \(B\) has to abort.

**Conditions for a simple soundness proof**

We now describe the conditions necessary for applying our method.

When \(\mathcal{A}\) wins, this is characterised by a unique \textit{trace} of its winning configuration \(e\), denoted \(tr(e)\), which we can think of as a very small subset of its full execution trace. The maximal bit-length m of the trace must be such that \(2^m\) is polynomial in \(\lambda\). The trace will typically be a tuple of numbers, group elements, etc.

There is a well-defined procedure by which, after generating a random configuration \(e\) and “guessing” a trace \(tr'\) by randomly sampling from \(\{0,1\}^m\), \(B\) is able to modify \(e\) based on \(tr'\). This results in a new configuration \(e'\). The following must hold:

the probability space of \(e'\) is the same as the probability space of \(e\);

\(\mathcal{A}\) learns nothing about \(B\)’s guess during the execution, unless \(B\) aborts - at which stage \(B\) cannot win anyway so it has no impact on the result;

if \(e'\) is a winning configuration, and \(B\) guessed the trace \(tr(e')\) correctly, then \(B\) is able to finish the execution of \(\mathcal{A}\) and therefore win.

**Applying the method**

If the above conditions are met, the configuration \(e'\) generated by B is indistinguishable from a uniformly sampled configuration of \(E\). Therefore, with probability \(t\), it will be a winning configuration. The trace \(tr'\) will then be equal to \(tr(e')\) with probability \(\frac{1}{2^m}\), and this will be independent of the probability that \(e'\) is a winning configuration. If \(B\) guessed the trace correctly, it will be able to complete the execution of \(A\) and, as a result, break \(P\), with probability of at least \(\frac{t}{2^m}\).

\(\mathrm{\blacksquare}\)

*The paper then proceeds to show in detail how this method can be applied to prove the soundness of a simple binary voting scheme.*

\(\mathrm{\blacksquare}\)

## Randomly replacing values in vectors

The method described above relies on the following simple fact.

### From one uniformly random configuration to another

Let \(G\) be a finite set and \(n\) a positive integer. Let \(H = G^n\) and \(E = [n] \times G \times H\). The uniform distributions on these sets are denoted \(Pr_H\) and \(Pr_E\). Elements of \(H\) are denoted \(\vec{h}\) while elements of \(E\) are denoted \((k, u, \vec{g})\). Elements of a vector \(\vec{v}\) are denoted \(v(i)\). Given a vector \(\vec{v}\), we denote by \(r(\vec{v}, k, u)\) the vector obtained from \(\vec{v}\) by replacing its \(k\)-th element by \(u\):

\[ r(\vec{v}, k, u)(i) =\begin{cases} v(i) & \text{if $i \neq k$}\\ u & \text{if $i = k$} \end{cases} \]

The set of all elements in \(H\) which are equal to a vector \(\vec{h}\) except at some fixed index \(k \in [n]\) is denoted \(H(\vec{h}, k)\):

\[ H(\vec{h}, k) = \{ \vec{v} \in H \; | \; \forall i \in [n] \setminus \{k\}, \; v(i) = h(i) \} \]

Note that \(\forall (\vec{h}, k), |H(\vec{h}, k)| = |G|\). We further define the function \(\rho\) as follows:

\[ \begin{aligned} \rho : \:\: & E & \longrightarrow \: & H \\ & (k, u, \vec{g}) & \longrightarrow \: & \vec{h} = r(\vec{v}, k, u) \end{aligned} \]

With fixed \(\vec{h} \in H\) and \(\kappa \in [n]\), we have:

\[ Pr_E( \rho = \vec{h} \: | \: k = \kappa) = \frac{|H(\vec{h}, \kappa)|}{|E|} = \frac{|G|}{n|G|^{n+1}} = \frac{1}{n|G|^n} \]

As a result:

\[ Pr_E( \rho = \vec{h}) = \sum_{\kappa \in [n]} Pr_E( \rho = \vec{h} \: | \: k = \kappa) = n \frac{1}{n|G|^n} = \frac{1}{|G|^n} = Pr_H(\vec{h}) \]

Therefore the random variable \(\rho\) is uniform on \(H\). This is not at all surprising: if we take a random vector in \(G^n\), then select a random \(k \in [n]\) and a random \(u \in G\) and replace the \(k\)-th value of the vector with \(u\), we naturally expect the result to also be random.

### Extending the result

We can modify the definitions of \(E\) as follows: instead of \(E = [n] \times G \times G^n\), we define \(E = R \times G \times G^n\) where \(R \subseteq [n]\). It is clear that we will obtain the same result.

Another observation is that if we have for example \(E = [n] \times G \times G^n \times H^m \times ...\), we can extend \(f\) and \(f_{\kappa}\) in an obvious way, resulting in a uniform distribution on \(G^n \times H^m \times ...\).

Finally, it is also clear that the process can be repeated several times. For example, if our initial sample space is \(G^n \times H^m\), we can randomly replace one or more values in \(G^n\), then randomly replace one or more values in \(H^m\), and the resulting distribution will remain uniformly random.

\(\mathrm{\blacksquare}\)

https://github.com/aragonzkresearch/blog/blob/main/pdf/simple-soundness.pdf