Probabilistic communication complexity
Journal of Computer and System Sciences
A public key cryptosystem and a signature scheme based on discrete logarithms
Proceedings of CRYPTO 84 on Advances in cryptology
Paillier's cryptosystem revisited
CCS '01 Proceedings of the 8th ACM conference on Computer and Communications Security
Polynomial Interpolation of the Discrete Logarithm
Designs, Codes and Cryptography
On the Representation of Boolean Predicates of the Diffie-Hellman Function
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
A Linear Lower Bound on the Unbounded Error Probabilistic Communication Complexity
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
New public key cryptosystems based on the dependent-RSA problems
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
Threshold circuit lower bounds on cryptographic functions
Journal of Computer and System Sciences
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In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from {-1, 1}N × N. It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N1-Ɛ, where Ɛ 0 denotes a fixed constant. For instance, the size of a two-layer threshold circuit (with polynomially bounded weights for the gates in the hidden layer, but unbounded weights for the output gate) grows exponentially fast with n:= logN. We prove sufficient conditions on M that imply small spectral norms (and thus high computational complexity in restricted models). Our general results cover specific cases, where the matrix M represents a bit (the least significant bit or other fixed bits) of a cryptographic decoding function. For instance, the decoding functions of the Pointcheval [9], the El Gamal [6], and the RSA-Paillier [2] cryptosystems can be addressed by our technique. In order to obtain our results, we make a detour on exponential sums and on spectral norms of matrices with complex entries. This method might be considered interesting in its own right.