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Secret Sharing Schemes and Advanced Encryption Standard

Using a simplified methodology to probe weaknesses in Shamir’s Secret Sharing Scheme.

There are many secret sharing schemes and variations available to hide and reconstruct the given secret. Shamir’s Secret Sharing Scheme, making use of linear Lagrange interpolation on the dealer-generated polynomial, was used to reconstruct the secret from the stipulated threshold number of participants’ shares. Such a scheme had been widely analyzed by mathematicians and computer scientists for potential weaknesses in the reconstruction of the secret by an external eavesdropper.

The objective of this research was to present a variation of Shamir’s threshold secret sharing scheme by manipulating the dealer-generated polynomial into a simplified version such that any eavesdropper can reconstruct the secret by gaining two public shares, instead of the stipulated threshold level. The envisaged improvements would then be evaluated for any impact on side-channel effects on the Advanced Encryption Standards.

Existing and famous mathematical conjectures (including Pillai’s conjecture, the Fermat-Catalan conjecture, and Hall’s conjecture) were built upon to seek a potential weakness in the security of the current secret sharing scheme. Essentially, the analysis aimed to reduce the order of difficulty in reconstructing the secret. Assuming that the dealer-generated polynomial is monic, it is then deconstructed by applying a composite linear function in which two additional variables are introduced.

In general, assuming that the original form of the dealer-generated polynomial is f(x) = a0 +a1x+a2x2 +…+ak-1xk-1, by composing it with the linear function g(x) = x+α, the eventual form of the dealer-generated polynomial can be manipulated to be in the form of f(x) = (x+α)k -b0, where both α and b0 are the two newly introduced variables. The challenge then is reduced to finding the values of both α and b0.

It was postulated that an eavesdropper would be able to recover the secret by simply obtaining two public shares, namely (x1,y1) and (x2,y2), from the multitude of available public shares, and this could be achieved by determining the numerical boundaries for the variable α. Specifically, all encompassing cases, without loss of generality, were considered to ensure that all possibilities were not neglected. The start state would be to take the difference between the two y-values that were easily obtained. From there on, it is just a matter of manipulating the inequalities to screen out the boundaries of α. Once the boundaries of α were found, then it would be trivial to try out the available choices for α, and subsequently b0, and eventually the secret.

While this methodology does not allow for the absolute reconstruction of the secret as compared to Lagrange interpolation, it presents an alternate methodology for an eavesdropper to retrieve the secret using shares that are significantly less than the required threshold number. The boundaries reduced the possibilities of the secret value from a near-infinite number to a manageable cardinality size that could be derived through exhaustive means. The crux is that as long as two shares are gathered together, the value of α can be derived easily through exhaustive means. Once the value of α is found, then it remains trivial to determine b0 through the equation yi = (xi+α)k-b0, where (xi,yi) are known public shares. Subsequently, the secret is reconstructed to be f(0).

This work was done by Bing Yong Lim for the Naval Postgraduate School. NPS-0001

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SECRET SHARING SCHEMES AND ADVANCED ENCRYPTION STANDARD

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    Aerospace & Defense Technology Magazine

    This article first appeared in the October, 2016 issue of Aerospace & Defense Technology Magazine (Vol. 1 No. 7).

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    Overview

    The document is a Master's thesis titled "Secret Sharing Schemes and Advanced Encryption Standard," authored by Lim, Bin Yong, and submitted to the Naval Postgraduate School in September 2015. The thesis addresses the critical area of secret sharing, a cryptographic method that allows a secret to be divided into parts, giving each participant a share. The goal is to ensure that only a specific group of participants can reconstruct the original secret, enhancing security and confidentiality.

    The introduction highlights the challenge of obtaining sensitive information, such as salaries, without compromising individual privacy. This sets the stage for discussing secret sharing as a solution to protect confidential data while still allowing for collective computation or decision-making.

    The thesis likely explores various secret sharing schemes, with a particular focus on Shamir’s Secret Sharing Scheme, which is based on polynomial interpolation. This method allows a secret to be divided into shares such that only a predetermined number of shares (threshold) are needed to reconstruct the secret. The document may also discuss the mathematical principles underlying these schemes, including the use of finite fields and polynomial functions.

    In addition to secret sharing, the thesis examines the Advanced Encryption Standard (AES), a widely used encryption algorithm that secures data through symmetric key cryptography. The relationship between secret sharing and AES is explored, particularly how secret sharing can enhance the security of cryptographic keys used in AES. The author likely discusses the implications of combining these two cryptographic techniques to improve data security in various applications.

    The thesis aims to simplify the methodology for reconstructing secrets and may provide practical examples or case studies demonstrating the effectiveness of secret sharing schemes in real-world scenarios. It emphasizes the importance of maintaining confidentiality in sensitive information sharing while enabling collaborative processes.

    Overall, the document contributes to the field of cryptography by providing insights into secret sharing schemes and their integration with established encryption standards like AES, highlighting their relevance in protecting sensitive data in an increasingly digital world. The views expressed in the thesis are those of the author and do not necessarily reflect the official policy or position of the Department of Defense or the U.S. Government.

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