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\begin{document}

\title{Threshold Secret Sharing with Geometric Algebra}



\author[1]{David William Honorio Araujo da Silva}%
\author[2]{Luke Harmon}%
\author[2]{Gaetan Delavignette}%
\affil[1]{Algemetric, LLC}%
\affil[2]{Algemetric}%


\vspace{-1em}



  
  \date{May 25, 2024}


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\begin{abstract}
As establishing a foundation for a new line of investigations on
threshold secret sharing schemes with geometric algebra (GA), we propose
a variation of a well-known threshold secret sharing scheme introduced
by Adi Shamir in 1979, a cryptographic solution that allows a secret
input to be divided into multiple random shares which are then sent,
each one, to distinct parties. The computation of these shares is done
so that there are proper subsets of these shares that allow
reconstructing the secret input using polynomial interpolation. To
reconstruct the secret input, Shamir's scheme requires a minimum number
of shares, smaller than the total number of shares, referred to as a
threshold. Any number of shares smaller than the threshold reveals
nothing about the input secret. The random shares are generated such
that each party can perform computations, generating a new set of shares
that, when reconstructed, are equivalent to performing those exact
computations directly on the secret input data. Our variant replaces the
algebra in which the original secrets lie from integers to GA while
preserving fundamental properties in Shamir's scheme, such as perfect
secrecy and idealness (both secret and random shares are members of the
same space). As a direct result, any application in GA dealing with
multivectors can immediately add threshold security using our scheme.
Non-GA applications can also benefit from our results by using
multivectors as a vessel for sharing multiple secrets at once.%
\end{abstract}%



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