Probabilistic communication complexity
Journal of Computer and System Sciences
Threshold circuits of bounded depth
Journal of Computer and System Sciences
Paillier's cryptosystem revisited
CCS '01 Proceedings of the 8th ACM conference on Computer and Communications Security
Handbook of Applied Cryptography
Handbook of Applied Cryptography
A linear lower bound on the unbounded error probabilistic communication complexity
Journal of Computer and System Sciences - Complexity 2001
Communication Complexity and Fourier Coefficients of the Diffie-Hellman Key
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
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
Relations Between Communication Complexity, Linear Arrangements, and Computational Complexity
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Towards Proving Strong Direct Product Theorems
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Public-key cryptosystems based on composite degree residuosity classes
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
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
Complexity theoretic aspects of some cryptographic functions
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
New directions in cryptography
IEEE Transactions on Information Theory
Spectral norm in learning theory: some selected topics
ALT'06 Proceedings of the 17th international conference on Algorithmic Learning Theory
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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^x^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 N^1^-^@?, where @?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 fundamental functions. Functions like the discrete multiplication and division, as well as cryptographic functions such as the Diffie-Hellman function (IEEE Trans. Inform. Theory 22(6) (1976) 644-654) and the decryption functions of the Pointcheval (Advances in Cryptology-Proceedings of EUROCRYPT '99, Lecture Notes in Computer Science, Springer, Berlin, 1999, pp. 239-254) and the El Gamal (Advances in Cryptology-CRYPTO '84, 1984, pp. 10-18) 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.