Secret Sharing

1
Secret Sharing
Based on Adi Shamirs How to Share a Secret
Sachin K Agarwal
SC 700 Internet information Protocols
2
Inspiration

• 4 Colonels are to have the code key to arm a
  nuclear missile. But the Government wants to make
  sure that at least any 3 of them agree to launch
  before the missile is dispatched.
• So the Government wants to break the key code
  into 4 pieces so that at least 3 are needed to
  put the key code back together.
• Generally, assume we want to divide the
  information among n people such that at least
  k are needed to unlock the information. Such a
  scheme is called a (k, n) threshold scheme.

3
How ???

• Consider a polynomial
• P (Z) Ak-1Zk-1 Ak-2Zk-2 Ak-3Zk-3 ... A1Z
  A0
• P (Z) is a k-1 degree polynomial with the
  constant term A0
• The information D to be distributed is set to A0
• P (Z) is constructed randomly ( which means that
  the coefficients Ak-1, Ak-2, Ak-3, , A1 are
  selected at random).
• P (Z) is then evaluated at n points. We
  evaluate P at Z 1,2,3n for example to get
  P(1), P(2) P(n). Each one of the keys are
  distributed to n people.

4
Example Lets help the Colonels!

• Consider again example where we have 4 colonels
  who are to be given a key and at least 3 of them
  have to agree in order to unlock the data. Let
  the information data be 5.
• Here n 4, k 3, A0 5 (Information Data)
• Select P(Z) 9Z2 6Z 5. (Coefficients 9 and 6
  are randomly selected).
• P(1) 20, P(2) 53, P(3) 104, P(4) 173.
  Note that P(0) 5, which is the data itself.
• P(1), P(2), P(3) and P(4) are distributed to the
  4 colonels.
• So now we have set up a (3,4) Threshold Scheme.

5
Putting it Back Again (Decoding)

• So 3 colonels get together and decide to blow
  away AybiL by launching the Nuclear missile.
• When they start decoding, they have the following
  information
• They know that they are using a (3, 4) threshold
  scheme.
• They know their keys. for example lets assume
  the 3 colonels know P(1), P(2) and P(4)
  respectively.
• So the Colonels know that they need to determine
• A0 from some equation of the type
• P(Z) A2Z2 A1Z A0 .
• They Interpolate to determine P(Z)

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Interpolation ???

• Interpolation is the opposite of Evaluation of P
  (Z) at a given point.
• In interpolation, we are given some points and
  the corresponding values the polynomial P(Z)
  calculates to at these points.
• We determine the coefficients of P(Z) from this
  information, and hence P(Z).
• And A0.

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Colonel Example continued

• P(1) 20, P(2) 53, P(4) 173
• Using a (3, 4) threshold scheme gt k 3.
• So P(Z) A2Z2 A1Z A0
• Solve
• A2 A1 A0 20 (Putting Z 1)
• 4A2 2A1 A0 53 (Putting Z 2)
• 16A2 4A1 A0 173 (Putting Z 4)
• A0 5, A1 6, A2 9
• Information A0 5

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Issues

• Randomization of the Coefficients while
  generating the keys.
• P (Z) Ak-1Zk-1 Ak-2Zk-2 Ak-3Zk-3 ... A1Z
  A0
• Re using the scheme.
• Choosing a safely large prime field.

9
References

• A. Shamir How to Share a Secret Communications
  of the ACM v. 24, n. 11, Nov 1979
• G.R.Blakely, Safeguarding Cryptographic Keys,
  proceedings of the National Computer Conference,
  1979, American Federation of Information
  Processing Societies.
• Bruce Schneier Applied Cryptography Wiley
  publishers