The study, coding and experimental results of secret sharing schemes (SSS) along with a proposed
method are presented in this book. It is very important and essential in any security application,
because none of the security techniques can be developed without pre-negotiation of security keys
or values. For instance, different key exchange protocols are used in IPSec/SSL for pre-establishment
of secret keys. Also, for symmetric encryption, which is much faster than public-key encryption, a
mutually known pre-secret value is used for encryption and decryption of sensitive information to be
exchanged between entities. In 1979, a perfect 𝑡/𝑛 threshold SSS was introduced by Shamir, where
1 < 𝑡 ≤ 𝑛 and any group with 𝑡more participants can reconstruct the secret selected by a trusted
third party (TTP) known as Dealer 𝐷 , however, any group with less than 𝑡 participants cannot get
the secret. This scheme is perfectly secure; however, it has a flaw as one or at most 𝑡 − 1
dishonest participants can exchange with their fake shares (instead of their own genuine shares as
received from 𝐷 secretly), with other group members and obtain the correct secret only for
themselves. It was first noticed and shown by Tompa in 1998 and proposed a simple method for
reducing the cheating probability. In his method, a prime parameter 𝑝 ≥ 𝑚𝑎𝑥 {(𝑠−1)(𝑡−1)/ɛ
+ 𝑡, 𝑛} is taken such that if cheating is occurred, then the secret reconstructed would be out of the secret set
𝑠 = {0, 1, 2, … , 𝑠 − 1} considered. Here ɛ > 0 is a very small number. In this thrilling work, we develop algorithms and coding in Python for Shamir’s SSS, Tompa’s cheating and Harn-Lin’s SSS for detection of cheating. Some experimental results for each of them are also presented for better understanding of Shamir’s method and cheating prevention. We also present an improvement over the method proposed by Harn-Lin in areas of cheating detection.
Linear (t,n) Secret Sharing Scheme with Reduced Number of Polynomials