Some New Issues on Secret Sharing Schemes In The Name of Allah Some New Issues on Secret Sharing Schemes Mohammad Ehdaie mohammad@ehdaie.com Taraneh Eghlidos teghlidos@sharif.edu Mohammad Reza Aref aref@sharif.edu Sharif University of Technology, Tehran, I.R.Iran

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 … Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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). Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Secret Reconstruction (2) Proof: Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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). Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Simulation Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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. Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Discussions and Results The Scheme DHQ LLY Primary Extended Threshold (2, n) (k, n) Perfect  Without Computation   Ideal Audio Shares Audio Secret Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

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 Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref References Shamir A., "How to share a secret", Comm. ACM 22, pp. 612-613, (1979). Blakley G. R., "Safeguarding cryptographic keys", Proc. of the National Computer Conf., v. 48, pp. 242-268, (1979). Naor M., Shamir A., "Visual cryptography", Eurocrypt 94, pp. 1-12. Desmedt Y., Hou S., Quisquater J., "Audio and optical cryptography", in Advances in Cryptology-Asiacrypt ’98, Springer-Verlag LNCS, pp. 392-404. 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, 605-614 (2003). Tuesday, April 23, 2019Tuesday, April 23, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref