Analysis of secret sharing schemes based on Nielsen transformations

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Matvei Kotov ◽  
Dmitry Panteleev ◽  
Alexander Ushakov
Keyword(s):  
Computer Algebra ◽  
Secret Sharing ◽  
Computer Algebra System ◽  
Cryptographic Protocols ◽  
Secret Sharing Schemes ◽  
Polynomial Equations ◽  
Polynomial Size ◽  
Security Properties ◽  
System Of Polynomial Equations

Abstract We investigate security properties of two secret-sharing protocols proposed by Fine, Moldenhauer, and Rosenberger in Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger, Cryptographic protocols based on Nielsen transformations, J. Comput. Comm. 4 2016, 63–107] (Protocols I and II resp.). For both protocols, we consider a one missing share challenge. We show that Protocol I can be reduced to a system of polynomial equations and (for most randomly generated instances) solved by the computer algebra system Singular. Protocol II is approached using the technique of Stallings’ graphs. We show that knowledge of {m-1} shares reduces the space of possible values of a secret to a set of polynomial size.

scholarly journals PERMUTATION POLYNOMIALS OF DEGREE 8 OVER FINITE FIELDS OF ODD CHARACTERISTIC

2019 ◽  
Vol 101 (1) ◽  
pp. 40-55 ◽  
Author(s):  
XIANG FAN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Computer Algebra ◽  
Personal Computer ◽  
Computer Algebra System ◽  
Computer Work ◽  
Permutation Polynomials ◽  
Polynomial Equations ◽  
Manual Analysis ◽  
Odd Order

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

A simple proof of Lester’s theorem

2003 ◽  
Vol 87 (510) ◽  
pp. 444-452
Author(s):  
John Rigby
Keyword(s):  
Computer Algebra ◽  
Simple Proof ◽  
Computer Algebra System ◽  
Recent Article ◽  
Diagonal Form ◽  
Computer Assisted ◽  
Polynomial Equations ◽  
Scalene Triangle ◽  
Systems Of Polynomial Equations ◽  
Special Case

Lester’s theorem (1997) states that in any scalene triangle the two Fermat points F and F' (to be defined later), the nine-point centre N, and the circumcentre O, are concyclic, and that the pair of points O,F separates the pair N, F'. (In certain geometrical situations a line is regarded as a circle of infinite radius, so that the word ‘concyclic’ includes ‘collinear’ as a special case, but here ‘concyclic’ means ‘lying on a proper circle of finite radius’.) Previous proofs of Lester’s theorem have involved advanced techniques and/or computer algebra; to quote from Ron Shail’s recent article [1],‘Lester’s original computer-assisted discovery and proof make use of her theory of “complex triangle coordinates” and “complex triangle functions”. ... A proof has also been given by Trott ... using the advanced concept of GrObner bases in the reduction of systems of polynomial equations to “diagonal” form. Trott’s work uses the computer algebra system Mathematica as an essential tool.’

scholarly journals Mind the Gap: Ceremonies for Applied Secret Sharing

2020 ◽  
Vol 2020 (2) ◽  
pp. 397-415
Author(s):  
Bailey Kacsmar ◽  
Chelsea H. Komlo ◽  
Florian Kerschbaum ◽  
Ian Goldberg
Keyword(s):  
Secret Sharing ◽  
Real World ◽  
Use Cases ◽  
Security And Privacy ◽  
Theoretical Research ◽  
Secret Sharing Schemes ◽  
Cryptographic Protocol ◽  
Security Assessment ◽  
Security Properties ◽  
Potential Loss

AbstractSecret sharing schemes are desirable across a variety of real-world settings due to the security and privacy properties they can provide, such as availability and separation of privilege. However, transitioning secret sharing schemes from theoretical research to practical use must account for gaps in achieving these properties that arise due to the realities of concrete implementations, threat models, and use cases. We present a formalization and analysis, using Ellison’s notion of ceremonies, that demonstrates how simple variations in use cases of secret sharing schemes result in the potential loss of some security properties, a result that cannot be derived from the analysis of the underlying cryptographic protocol alone. Our framework accounts for such variations in the design and analysis of secret sharing implementations by presenting a more detailed user-focused process and defining previously overlooked assumptions about user roles and actions within the scheme to support analysis when designing such ceremonies. We identify existing mechanisms that, when applied to an appropriate implementation, close the security gaps we identified. We present our implementation including these mechanisms and a corresponding security assessment using our framework.

scholarly journals A Multisecret-Sharing Scheme Based on LCD Codes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 272 ◽  
Author(s):  
Adel Alahmadi ◽  
Alaa Altassan ◽  
Ahmad AlKenani ◽  
Selda Çalkavur ◽  
Hatoon Shoaib ◽  
...  
Keyword(s):  
Secret Sharing ◽  
Cryptographic Protocols ◽  
Secret Sharing Schemes ◽  
Code Length ◽  
Sharing Scheme ◽  
Lcd Codes

Secret sharing is one of the most important cryptographic protocols. Secret sharing schemes (SSS) have been created to that end. This protocol requires a dealer and several participants. The dealer divides the secret into several pieces ( the shares), and one share is given to each participant. The secret can be recovered once a subset of the participants (a coalition) shares their information. In this paper, we present a new multisecret-sharing scheme inspired by Blakley’s method based on hyperplanes intersection but adapted to a coding theoretic situation. Unique recovery requires the use of linear complementary (LCD) codes, that is, codes in which intersection with their duals is trivial. For a given code length and dimension, our system allows dealing with larger secrets and more users than other code-based schemes.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

A Strengthened Security Notion for Password-Protected Secret Sharing Schemes

2015 ◽  
Vol E98.A (1) ◽  
pp. 203-212 ◽  
Author(s):  
Shingo HASEGAWA ◽  
Shuji ISOBE ◽  
Jun-ya IWAZAKI ◽  
Eisuke KOIZUMI ◽  
Hiroki SHIZUYA
Keyword(s):  
Secret Sharing ◽  
Secret Sharing Schemes ◽  
Security Notion

scholarly journals On the classification of ideal secret sharing schemes

1991 ◽  
Vol 4 (2) ◽  
pp. 123-134 ◽  
Author(s):  
Ernest F. Brickell ◽  
Daniel M. Davenport
Keyword(s):  
Secret Sharing ◽  
Secret Sharing Schemes ◽  
Ideal Secret Sharing Schemes
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