Rounding in lattices and its cryptographic applications
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
On the distribution of the power generation
Mathematics of Computation
On the Generalised Hidden Number Problem and Bit Security of XTR
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Security of the most significant bits of the Shamir message passing scheme
Mathematics of Computation
Hidden number problem with hidden multipliers, timed-release crypto, and noisy exponentiation
Mathematics of Computation
The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces
Designs, Codes and Cryptography
Proving Hard-Core Predicates Using List Decoding
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Security preserving amplification of hardness
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Do all elliptic curves of the same order have the same difficulty of discrete log?
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
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We study the security of elliptic curve Diffie-Hellman secret keys in the presence of oracles that provide partial information on the value of the key. Unlike the corresponding problem for finite fields, little is known about this problem, and in the case of elliptic curves the difficulty of representing large point multiplications in an algebraic manner leads to new obstacles that are not present in the case of finite fields. To circumvent this obstruction, we introduce a small multiplier version of the hidden number problem, and we use its properties to analyze the security of certain Diffie-Hellman bits. We suggest new character sum conjectures that guarantee the uniqueness of solutions to the hidden number problem, and provide some evidence in support of the conjectures by showing that they hold on average in certain cases. We also present a Gröbner basis algorithm for solving the hidden number problem and recovering the Diffie-Hellman secret key when the elliptic curve is defined over a constant degree extension field and the oracle is a coordinate function in the polynomial basis.