Untraceable off-line cash in wallet with observers
CRYPTO '93 Proceedings of the 13th annual international cryptology conference on Advances in cryptology
Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
On sharing secrets and Reed-Solomon codes
Communications of the ACM
Communications of the ACM
An Efficient Public Key Traitor Tracing Scheme
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
A Practical Public Key Cryptosystem Provably Secure Against Adaptive Chosen Ciphertext Attack
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Efficient Construction of (Distributed) Verifiable Random Functions
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Improved Decoding of Reed-Solomon and Algebraic-Geometric Codes
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Robust threshold DSS signatures
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Lower bounds for discrete logarithms and related problems
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
An efficient threshold public key cryptosystem secure against adaptive chosen ciphertext attack
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
Distributed Pseudo-random functions and KDCs
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
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Given a corrupted word w=(w1,...wn) from a Reed-Solomon code of distance d, there are many ways to efficiently find and correct its errors. But what if we are instead given $(g^{w_1},...g^{w_n})$ where g generates some large cyclic group — can the errors still be corrected efficiently? This problem is called error correction in the exponent, and though it arises naturally in many areas of cryptography, it has received little attention. We first show that unique decoding and list decoding in the exponent are no harder than the computational Diffie-Hellman (CDH) problem in the same group. The remainder of our results are negative: – Under mild assumptions on the parameters, we show that bounded-distance decoding in the exponent, under e = d - k1−ε errors for any ε 0, is as hard as the discrete logarithm problem in the same group. – For generic algorithms (as defined by Shoup, Eurocrypt 1997) that treat the group as a “black-box,” we show lower bounds for decoding that exactly match known algorithms. Our generic lower bounds also extend to decisional variants of the decoding problem, and to groups in which the decisional Diffie-Hellman (DDH) problem is easy. This suggests that hardness of decoding in the exponent is a qualitatively new assumption that lies “between” the DDH and CDH assumptions.