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Threshold Secret Sharing with Geometric Algebra
  • David William Honorio Araujo da Silva,
  • Luke Harmon,
  • Gaetan Delavignette
David William Honorio Araujo da Silva
Algemetric, LLC

Corresponding Author:contact@davidwsilva.com

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Luke Harmon
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Gaetan Delavignette
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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.
01 Feb 2022Submitted to Mathematical Methods in the Applied Sciences
01 Feb 2022Submission Checks Completed
01 Feb 2022Assigned to Editor
07 Feb 2022Reviewer(s) Assigned
20 May 2022Review(s) Completed, Editorial Evaluation Pending
22 May 2022Editorial Decision: Revise Minor
12 Aug 20221st Revision Received
13 Aug 2022Submission Checks Completed
13 Aug 2022Assigned to Editor
16 Aug 2022Reviewer(s) Assigned
29 Aug 2022Review(s) Completed, Editorial Evaluation Pending
19 Oct 2022Editorial Decision: Revise Minor
03 Dec 20222nd Revision Received
05 Dec 2022Submission Checks Completed
05 Dec 2022Assigned to Editor
05 Dec 2022Review(s) Completed, Editorial Evaluation Pending
06 Dec 2022Reviewer(s) Assigned
14 Feb 2023Editorial Decision: Revise Minor
06 Apr 20233rd Revision Received
07 Apr 2023Assigned to Editor
07 Apr 2023Submission Checks Completed
07 Apr 2023Review(s) Completed, Editorial Evaluation Pending
08 Apr 2023Reviewer(s) Assigned
17 May 2023Editorial Decision: Revise Minor
28 Jun 20234th Revision Received
10 Jul 2023Submission Checks Completed
10 Jul 2023Assigned to Editor
10 Jul 2023Review(s) Completed, Editorial Evaluation Pending
28 Jul 2023Editorial Decision: Accept