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\section*{Transparent Inner Product}

Bulletproofs\cite{bulletproofs} create arguments of knowledge of the inner product of two vectors $\vec{a}, \vec{b} \in \mathbb{Z}_q^m$ with prime $q$, denoted $\langle \vec{a}, \vec{b} \rangle \in \mathbb{Z}_q$. This is hadamard multiplication, followed by sum, yielding a scalar value. Bulletproofs have $\mathcal{O}(log(m))$ argument size, and $\mathcal{O}(m)$ proving/verification complexity.

However, to achieve hiding, bulletproofs require lightweight (scalar) linear homomorphism. This approach works poorly with post quantum arguments where hidden linear operations are much more expensive. However, a binding only variant is quite cheap using scalar SIS commitments to public coin algebraic structure.

This work proposes using the bulletproof algebraic inner product argument to achieve succinct, \emph{binding only}, arguments of knowledge. This is designed to be composed on top of a module lattice argument of knowledge, which provides \emph{hiding and binding}. Because the inner product argument is stricly logarithmic, scaling to statistical hiding in the module lattice construction has little practical impact on the final argument size.


\subsection{TODO}
Scalar SIS derivation 

\section{Argument}

Given $\vec{a}, \vec{b} \in \mathbb{Z}_q^m$ yield $c \in \mathbb{Z}_q, \pi \in \mathbb{Z}_q^{log(m)}$, arguing $\langle \vec{a}, \vec{b} \rangle = c$. Denote $\mathcal{H}: \mathbb{Z}_q^{\mathbb{Z}^{+}} \rightarrow \mathbb{Z}_q$ as a collision and pre-image resistant hash function. Prover computes:

$$c \leftarrow \langle \vec{a}, \vec{b} \rangle$$
$$transcript: \vec{t} \overset{\$}{\leftarrow} \vec{a}, \vec{b}$$

$$\pi_k \leftarrow \sum_{i = 0}^{log(m)}$$





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\bibitem{bulletproofs} Benedikt Bünz and Jonathan Bootle and Dan Boneh and Andrew Poelstra and Pieter Wuille and Greg Maxwell,
Bulletproofs: Short Proofs for Confidential Transactions and More,
Cryptology {ePrint} Archive, Paper 2017/1066,
2017,
\url{https://eprint.iacr.org/2017/1066}

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