A public key cryptosystem and a signature scheme based on discrete logarithms
Proceedings of CRYPTO 84 on Advances in cryptology
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Efficient Publicly Verifiable Secret Sharing Schemes with Fast or Delayed Recovery
ICICS '99 Proceedings of the Second International Conference on Information and Communication Security
An Efficient Verifiable Encryption Scheme for Encryption of Discrete Logarithms
CARDIS '98 Proceedings of the The International Conference on Smart Card Research and Applications
Gradual and Verifiable Release of a Secret
CRYPTO '87 A Conference on the Theory and Applications of Cryptographic Techniques on Advances in Cryptology
Wallet Databases with Observers
CRYPTO '92 Proceedings of the 12th Annual International Cryptology Conference on Advances in Cryptology
Statistical Zero Knowledge Protocols to Prove Modular Polynomial Relations
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
A Statistically-Hiding Integer Commitment Scheme Based on Groups with Hidden Order
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Guaranteed Correct Sharing of Integer Factorization with Off-Line Shareholders
PKC '98 Proceedings of the First International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
Security proofs for signature schemes
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Proving in zero-knowledge that a number is the product of two safe primes
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
Efficient proofs that a committed number lies in an interval
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
A Verifiable Secret Sharing Scheme Based on the Chinese Remainder Theorem
INDOCRYPT '08 Proceedings of the 9th International Conference on Cryptology in India: Progress in Cryptology
An efficient rational secret sharing scheme based on the Chinese remainder theorem
ACISP'11 Proceedings of the 16th Australasian conference on Information security and privacy
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Checking whether a committed integer lies in a specific interval has many cryptographic applications. In Eurocrypt'98, Chan et al. proposed an instantiation (CFT Proof). Based on CFT, Boudot presented a popular range-bounded commitment scheme in Eurocrypt'2000. Both CFT Proof and Boudot Proof are based on the encryption E(x, r) = gxhr mod n, where n is an RSA modulus whose factorization is unknown by the prover. They did not use a single base as usual. Thus an increase in cost occurs. In this paper, we show that it suffices to adopt a single base. The cost of the modified Boudot Proof is about half of that of the original scheme. Moreover, the key restriction in the original scheme, i.e., both the discrete logarithm of g in base h and the discrete logarithm of h in base g are unknown by the prover, which is a potential menace to the Boudot Proof, is definitely removed.