, Peggy sends c A as the characteristic polynomial

, Peggy and Victor enter a determinant certificate for rI ? A

, Once convinced, Victor checks that det(rI ? A) ? == c A (r)

, For a random r ? S, in the determinant sub-certificate, rI ? A will be nonsingular if c A (r) = 0, hence with probability ? 1 ? n/|S|. Then det(rI ? A) is certified using the certificate of Figure 5. Best known algorithms for computing the characteristic polynomial, using either quadratic space and fast matrix multiplication or linear space

B. Beckerman and G. Labahn, A uniform approach for the fast computation of matrix-type Padé approximants, SIAM Journal on Matrix Analysis and Applications, vol.15, issue.3, pp.804-823, 1994.

G. Bertoni, J. Daemen, M. Peeters, and G. Assche, Sponge-based pseudo-random number generators, CHES 2010, pp.33-47, 2010.

L. Chen, W. Eberly, E. L. Kaltofen, B. D. Saunders, W. J. Turner et al., Efficient matrix preconditioners for black box linear algebra, Linear Algebra Appl, pp.119-146, 2002.
URL : https://hal.archives-ouvertes.fr/hal-02101893

K. Chung, Y. T. Kalai, and S. P. Vadhan, Improved delegation of computation using fully homomorphic encryption, CRYPTO 2010, vol.6223, pp.483-501, 2010.

D. Coppersmith, Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm, Mathematics of Computation, vol.62, issue.205, pp.333-350, 1994.

R. J. Cramer, Modular Design Of Secure Yet Practical Cryptographic Protocols, 1997.

R. A. Demillo and R. J. Lipton, A probabilistic remark on algebraic program testing, Inf. Process. Letters, vol.7, issue.4, pp.193-195, 1978.

J. Dumas, E. Kaltofen, and E. Thomé, Certificates for the verification of Wiedemann's Krylov sequence, 2016.

J. Dumas and E. L. Kaltofen, Essentially optimal interactive certificates in linear algebra, pp.146-153, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00932846

W. Eberly and E. Kaltofen, On randomized Lanczos algorithms, pp.176-183

A. Fiat and A. Shamir, How to prove yourself: Practical solutions to identification and signature problems, Advances in Cryptology -CRYPTO'86, vol.263, pp.11-15, 1986.

D. Fiore and R. Gennaro, Publicly verifiable delegation of large polynomials and matrix computations, with applications, ACM CCS '12, pp.501-512, 2012.

R. Freivalds, Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, vol.74, pp.57-69, 1979.

C. Gentry, J. Groth, Y. Ishai, C. Peikert, A. Sahai et al., Using fully homomorphic hybrid encryption to minimize non-interative zero-knowledge proofs, Journal of Cryptology, pp.1-24, 2014.

S. Goldwasser, Y. T. Kalai, and G. N. Rothblum, Delegating computation: interactive proofs for muggles, pp.113-122, 2008.

E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Mathematics of Computation, vol.64, issue.210, pp.777-806, 1995.

E. Kaltofen and V. Pan, Processor efficient parallel solution of linear systems over an abstract field, ACM SPAA '91, pp.180-191, 1991.

E. Kaltofen and G. Villard, On the complexity of computing determinants, Computational Complexity, vol.13, issue.3, pp.91-130, 2005.
URL : https://hal.archives-ouvertes.fr/hal-02102099

E. L. Kaltofen, B. Li, Z. Yang, and L. Zhi, Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, Journal of Symbolic Computation, vol.47, issue.1, pp.1-15, 2012.

E. L. Kaltofen, M. Nehring, and B. D. Saunders, Quadratic-time certificates in linear algebra, pp.171-176, 2011.

, ISSAC'97, 1997.

, EUROSAM '79, International Symposium on Symbolic and Algebraic Computation, vol.72, 1979.

, SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions, NIST. FIPS Publication, vol.202, 2015.

B. Parno, J. Howell, C. Gentry, and M. Raykova, Pinocchio: Nearly practical verifiable computation, IEEE SP '13, pp.238-252, 2013.

J. T. Schwartz, Probabilistic algorithms for verification of polynomial identities, pp.200-215

J. Thaler, Time-optimal interactive proofs for circuit evaluation, Lecture Notes in Computer Science, vol.8043, issue.13, pp.71-89, 2013.

G. Villard, Further analysis of Coppersmith's block Wiedemann algorithm for the solution of sparse linear systems, pp.32-39

G. Villard, Computing the Frobenius normal form of a sparse matrix, CASC'00, pp.395-407, 2000.

D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Transactions on Information Theory, vol.32, issue.1, pp.54-62, 1986.

R. Zippel, Probabilistic algorithms for sparse polynomials, pp.216-226