28 September 2005 Secret Sharing Amin Y. Teymorian Department of Computer Science The George Washington University

2 What is it? a method of distributing a secret amongst a group of people by giving each person a share (or shadow) of the secret Schemes  Splitting  Threshold

3 Splitting split a secret amongst people such that all people are needed to reconstruct the secret choose random numbers, and give them to of the people and give xxxx to the remaining person special case of...

4 Threshold split a secret into pieces such that any group of people can reconstruct the secret, but a group of less than people cannot called a threshold scheme

5 Why do we want ? security vs. reliability  why not keep the key in one very safe place?  why not have multiples copies? safety vs. convenience  copies are convenient but easier to misuse  requiring all parties is inconvenient

6 Combinatorial Approach need locks each of the people need keys secret still exists in one place... very impractical!

7 Shamir Approach invented by Adi Shamir in 1979 based on points uniquely determining a polynomial of degree we divide our secret into pieces by picking a random degree polynomial xxxxxxxxxxxx in which xxxx, and

8 Shamir Approach (continued) represent each share as a point all arithmetic done modulo a prime number x that is greater than both and - why? coefficients of are randomly chosen from a uniform distribution over the integers in

9 Shamir Approach (continued) uses Lagrange Interpolation

10 Example 1: even with a single share (point), the secret can still be any value in with equal probability

11 Example 2: threshold scheme give each of the people a share

12 Example 2 (continued) suppose people with shares,, and decide to reconstruct the secret

13 Remarks using requires adversaries to acquire more than shares Claude Shannon would be space-efficient shares (same size as secret)

14 Remarks (continued) add or delete shares without affecting others easy to create new shares without changing secret easy to create hierarchical schemes can be used for secrets of secrets

15 Blakley Approach invented by George Blakley in 1979 based on any -dimensional hyperplanes intersecting at one point each share defines a hyperplane secret is the point of intersection of any hyperplanes

16 Example 1: the secret is at the intersection of of the x -dimensional hyperplanes

17 Example 2: the secret is at the intersection of of the x -dimensional hyperplanes what about out of shares?

18 Remarks Claude Shannon would be less space-efficient shares ( times the size of secret)

19 Other uses Verifiable Secret Sharing  allows the people to be certain up to some iprobability of error that no other people are ilying about the value of their shares Visual Secret Sharing  each share is a transparency and can be icombined without computation to obtain secret

20 References Shamir, A. How to Share a Secret. Comm. ACM 22, 11 (Nov. 1979), Trappe, W., and Washington, L. Introduction to Cryptography with Coding Theory. Prentice Hall, Upper Saddle River, NJ, Weisstein, E. “Lagrange Interpolating Polynomial.” From MathWorld-A Wolfram Web Resource. “What are some secret sharing schemes.” RSA Security. Wikipedia. (2004). Secret Sharing ( Retrieved Sept. 22, All text is available under the terms of the GNU Free Documentation LicenseGNU Free Documentation License