Secure and Effective Key Management Using Secret Sharing Schemes in Cloud Computing

Secret Sharing Scheme

Adi-Shamir & George-Blakeley (Shamir, 1979; Blakely, 1979) developed Secret-Sharing Scheme’s. The Shamir- Secret-sharing schemes makes use of polynomial interpolation. Shamir (k, n) threshold scheme states that ‘k’-points are required to create a polynomial of [k-1] degree. In this scheme Lagrange’ Interpolating polynomial is used. The Secret is partitioned into several shares(n) and minimum k threshold value is required to re-assemble the original secret data. Each user will have a share of secret. To re assemble the secret data, a sufficient number of shares should be collected. This sufficient number of shares are determined by threshold value k. If users can have k number of shares, they can re-assemble the secret data. The Secret-Sharing Schemes (SSS) can also be used to secure the data in ‘multi-clouds’ (Muhil et al., 2015). The encrypted data is stored on multiple clouds.

Definition

The given method is called ‘k-n’ threshold scheme. The algorithm partitions the data into n number of shares share1, share2, -----, share n such that:

• 1.

  Collecting k pieces of share will retrieve the original data;

• 2.

  Collecting k-1 or lesser pieces of share will not be sufficient to re-assemble the original data;

• 3.

  Collecting k=n pieces of share will retrieve the original data.

The aim of Shamir-SSS is to define that (k) points are necessary to describe a polynomial of degree k-1 i.e., a line is constructed using 2 points and a parabola is constructed using 3 points.

The data is divided into shares by defining polynomial of appropriate degree. The data/secret (S) is divided into ‘n’ number of shares. ‘P’ is the size of finite field ‘F’ and it is a prime number. ‘k-1’ positive integers are chosen randomly as a₁, -----, a_k-1, where ‘a_i’< ‘P’:

In which [a ₀ =secret ‘S’], and a₀ is the constant term.