A Novel Secret Sharing Scheme from Audio Perspective In The Name of Allah A Novel Secret Sharing Scheme from Audio Perspective Mohammad Ehdaie mohammad@ehdaie.com Taraneh Eghlidos teghlidos@sharif.edu Mohammad Reza Aref aref@sharif.edu Sharif University of Technology, Tehran, Iran

M. Ehdaie, T. Eghlidos, M.R. Aref Contents Introduction Preliminaries Shares Generation Secret Reconstruction Security Simulation Ideality & Practical Issues Discussions and Results Conclusions & Further Research References Q & A Wednesday, April 10, 2019Wednesday, April 10, 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 Wednesday, April 10, 2019Wednesday, April 10, 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. Wednesday, April 10, 2019Wednesday, April 10, 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 Wednesday, April 10, 2019Wednesday, April 10, 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 … Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Previous Works - Ultimate Goal The Scheme DHQ LLY Goal Threshold (2, n) (k, n) Perfect  Without Computation Ideal  Audio Shares Audio Secret Wednesday, April 10, 2019Wednesday, April 10, 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. Vector c*A corresponds to the amplified audio file F with the coefficient c. Playing two audio files simultaneously, leads to an audio file which corresponds to the sum of their corresponding vectors. If a*A + b*B = c*C, then we can hear the amplified C by playing simultaneously amplified A & B with the coefficients a & b respectively. Wednesday, April 10, 2019Wednesday, April 10, 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. Let A be the amplitude of the secret file in a particular interval. Let S1, S2, …, Sn be the amplitudes of the n share files in the same interval. Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Shares Generation Generate S1, S2, …, Sk-1 Randomly. Generate an (n-k+1)×k matrix with non-zero random elements: Return to step 2, if C contains any singular submatrix. For , compute Si from the equation below: Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Secret Reconstruction (1) Lemma: Assume are k participants who want to reconstruct the secret. There is a linear combination of equations ek, ek+1, …, en leading to an equation of the form below: Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Secret Reconstruction (2) Proof: Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Security (1) Lemma: There is no linear combination of equations ek, ek+1, …, en, such that the coefficients of only k-1 or fewer shares are nonzero. Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Security (2) Proof: Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Simulation The audio secret The noisy reconstructed secret The reconstructed secret One of the shares Another share Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Ideality & Practical Issues The file size of the secret = The file size of each share Practical Issues: Physical properties of amplifiers … Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

Discussions and Results The Scheme DHQ LLY New Threshold (2, n) (k, n) Perfect  Secure Without Computation Ideal  It seems to be Ideal Audio Shares Audio Secret Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Conclusions A new audio secret sharing scheme Secure and it seems to be ideal (k, n) threshold scheme Audio shares and audio secret Secret reconstruction without any computation Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref

M. Ehdaie, T. Eghlidos, M.R. Aref Further Research Generation of non-suspicious shares Implementation and summing up practical issues Extending the new scheme to the visual case Perfectness & Ideality proof from Information Theoretical view Wednesday, April 10, 2019Wednesday, April 10, 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). Wednesday, April 10, 2019Wednesday, April 10, 2019 M. Ehdaie, T. Eghlidos, M.R. Aref