Secret Sharing

1
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), 612-613.
• Trappe, W., and Washington, L. Introduction to
  Cryptography with Coding Theory. Prentice Hall,
  Upper Saddle River, NJ, 2002.
• Weisstein, E. Lagrange Interpolating
  Polynomial. From MathWorld-A Wolfram Web
  Resource. http//mathworld.wolfram.com/LangrangeIn
  terpolatingPolynomial.html
• What are some secret sharing schemes. RSA
  Security. http//www.rsasecurity.com/rsalabs/node.
  asp?id2259
• Wikipedia. (2004). Secret Sharing
  (http//en.wikipedia.org/wiki/Secret_Sharing).
  Retrieved Sept. 22, 2005.
• All text is available under the terms of the GNU
  Free Documentation License