1. Some New Issues on Secret Sharing Schemes
In The Name of Allah
Some New Issues on Secret Sharing Schemes
Mohammad Ehdaie
Taraneh Eghlidos
Mohammad Reza Aref
Sharif University of Technology, Tehran, I.R.Iran

2. M. Ehdaie, T. Eghlidos, M.R. Aref
Contents
Introduction
Preliminaries
A New (2, n) Audio Secret Sharing Scheme
Extension to a (2m, 4m) Scheme
Extension to a (k, n) Scheme
Discussions and Results
Conclusions
References
Q & A
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3. M. Ehdaie, T. Eghlidos, M.R. Aref
Secret Sharing
Definition:
Secret sharing is a method to distribute a secret between some participants such that particular subsets (i.e. authorized subsets) could obtain the secret, whereas unauthorized subsets could not.
Significance:
The risk of fully authorizing a person
The risk of Information Disruption
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4. M. Ehdaie, T. Eghlidos, M.R. Aref
Terminology
Threshold Scheme
Cardinality of all minimum allowed subsets, k, is constant.
Perfect Scheme
Unauthorized subsets could not get any information about the secret by pooling their shares together.
Ideal Scheme
The size of each share is equal to the size of the secret.
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5. M. Ehdaie, T. Eghlidos, M.R. Aref
The History
Geometric Scheme, Blakley, 1979
Interpolation Scheme, Shamir, 1979 …
Visual Secret Sharing, Naor & Shamir, 1994
Audio Secret Sharing, Desmedt et al., 1998
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6. M. Ehdaie, T. Eghlidos, M.R. Aref
Motivation
What is an Audio Secret Sharing Scheme (ASSS)?
Audio Shares and/or Audio Secret
Why?
Audio Applications …
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7. M. Ehdaie, T. Eghlidos, M.R. Aref
Preliminaries
The audio file F corresponds to an Amplitude Vector A.
The human audio system is insensitive to the phase of the signal (e.g. amplitude vector –A corresponds to an audio file which is same to F).
Playing two audio files simultaneously, leads to an audio file which corresponds to the sum of their corresponding vectors (The Interference Property).
The Sign Properties:
The amplitude vector Sign(A) corresponds to an audio file which is the noisy version of F.
The amplitude vector B, where Sign(B) = Sign(A), corresponds to an audio file which is the noisy version of F.
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8. M. Ehdaie, T. Eghlidos, M.R. Aref
Setup
Divide the audio secret file into some small intervals.
Generate the shares in each interval independently.
Put all share parts of each participant together and generate an audio share.
A: The amplitude vector of the secret file.
S1, S2, …, Sn: The amplitude vectors of the n share files.
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9. M. Ehdaie, T. Eghlidos, M.R. Aref
Shares Generation
For all elements of the secret amplitude vector, A:
Select a random real number, xt, secretly, such that xt>>n.
For i, 1 ≤ i ≤ n, generate Si,t as:
Finally, distribute the files correspond to S1,S2, ... , Sn to n participants.
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10. Secret Reconstruction (1)
Lemma: Assume Pi and Pj, j > i, are two participants who want to reconstruct the secret. Set B equal to Sj-Si. Then we have Sign(B) = Sign(A).
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11. Secret Reconstruction (2)
Proof:
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12. M. Ehdaie, T. Eghlidos, M.R. Aref
Security
For every t, xt is a private number.
Each participant Pi does not know whether his share in the tth interval is xt + i or xt − i.
The sign of At is unknown to him.
He could not get any information about the Sign(A).
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13. M. Ehdaie, T. Eghlidos, M.R. Aref
Simulation
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14. Extension to a (2m, 4m) Scheme (1)
A (2, 4) scheme.
A secret bit, b.
Eight participants P1,P2,…,P8.
Extension to a (4, 8) scheme.
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15. Extension to a (2m, 4m) Scheme (2)
Q1 = {P1, P2}, Q2 = {P3, P4}, Q3 = {P5, P6}, Q4 = {P7, P8}
{Q1,Q2} , {Q3,Q4}
{Q1,Q3} , {Q2,Q4}
{Q1,Q4} , {Q2,Q4}
For every four participants, there is a row i, 1 ≤ i ≤ 3, such that two participants could obtain bi,1 and two of them could get bi,2.
P1, P6, P7 and P8 :
P1 & P6  b2,1 P7 & P8  b2,2
b = b2,1 × b2,2 is revealed.
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16. Extension to a (k, n) Scheme (1)
A (3, 7) Scheme:
A (4, 8) Scheme
Pass 7 Shares to 7 Participants
Announce the 8th Share
A (4, 7) Scheme:
Destroy the 8th Share
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17. Extension to a (k, n) Scheme (2)
Lemma: Assume we have a (t, m) threshold secret sharing scheme. Then, we could have a (k, n) threshold secret sharing scheme if the following assertions hold:
i) k ≤ t
ii) n ≤ m
iii) t − k ≤ m − n
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18. Discussions and Results
The Scheme
DHQ
LLY
Primary
Extended
Threshold
(2, n)
(k, n)
Perfect 
Without Computation  
Ideal
Audio Shares
Audio Secret
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19. M. Ehdaie, T. Eghlidos, M.R. Aref
Conclusions
Contributions of Authors :
A new audio secret sharing scheme
Perfect
(2, n) threshold scheme
Audio shares and audio secret
Ideal or Secret reconstruction without any computation
Extending any (2, n) scheme to a (k, n) one
Continuation of this research:
A (k, n) scheme with all of the good properties
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20. M. Ehdaie, T. Eghlidos, M.R. Aref
References
Shamir A., "How to share a secret", Comm. ACM 22, pp , (1979).
Blakley G. R., "Safeguarding cryptographic keys", Proc. of the National Computer Conf., v. 48, pp , (1979).
Naor M., Shamir A., "Visual cryptography", Eurocrypt 94, pp
Desmedt Y., Hou S., Quisquater J., "Audio and optical cryptography", in Advances in Cryptology-Asiacrypt ’98, Springer-Verlag LNCS, pp
Lin C. C., Laih C. S., Yang C. N., "New Audio Secret Sharing Schemes With Time Division Technique", J. of Information Science and Engineering 19, (2003).
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