Strong primes are easy to find
Proc. of the EUROCRYPT 84 workshop on Advances in cryptology: theory and application of cryptographic techniques
The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms
SIAM Journal on Computing
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Diffie-Hillman is as Strong as Discrete Log for Certain Primes
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Speeding Up Pollard's Rho Method for Computing Discrete Logarithms
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Short Signatures from the Weil Pairing
Journal of Cryptology
Lower bounds for discrete logarithms and related problems
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
A verifiable random function with short proofs and keys
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Collusion resistant broadcast encryption with short ciphertexts and private keys
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
Hierarchical identity based encryption with constant size ciphertext
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Pairing-Based cryptography at high security levels
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
A tapestry of identity-based encryption: practical frameworks compared
International Journal of Applied Cryptography
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
General Ad Hoc Encryption from Exponent Inversion IBE
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
Multi-Identity Single-Key Decryption without Random Oracles
Information Security and Cryptology
One-Round ID-Based Blind Signature Scheme without ROS Assumption
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
Efficient Protocols for Set Membership and Range Proofs
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Efficient Identity-Based Encryption with Tight Security Reduction
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Efficient Fair Exchange from Identity-Based Signature
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
Towards Black-Box Accountable Authority IBE with Short Ciphertexts and Private Keys
Irvine Proceedings of the 12th International Conference on Practice and Theory in Public Key Cryptography: PKC '09
Verifiable Random Functions from Identity-Based Key Encapsulation
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Efficient Non-interactive Range Proof
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem
Pairing '09 Proceedings of the 3rd International Conference Palo Alto on Pairing-Based Cryptography
How to Prove Security of a Signature with a Tighter Security Reduction
ProvSec '09 Proceedings of the 3rd International Conference on Provable Security
Is the Notion of Divisible On-Line/Off-Line Signatures Stronger than On-Line/Off-Line Signatures?
ProvSec '09 Proceedings of the 3rd International Conference on Provable Security
Certificate-Free Attribute Authentication
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Pairing-friendly elliptic curves with small security loss by Cheon's algorithm
ICISC'07 Proceedings of the 10th international conference on Information security and cryptology
Blind identity-based encryption and simulatable oblivious transfer
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
Zero-knowledge sets with short proofs
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
CCA2 secure IBE: standard model efficiency through authenticated symmetric encryption
CT-RSA'08 Proceedings of the 2008 The Cryptopgraphers' Track at the RSA conference on Topics in cryptology
Choosing key sizes for cryptography
Information Security Tech. Report
Additive combinatorics and discrete logarithm based range protocols
ACISP'10 Proceedings of the 15th Australasian conference on Information security and privacy
Expressive key-policy attribute-based encryption with constant-size ciphertexts
PKC'11 Proceedings of the 14th international conference on Practice and theory in public key cryptography conference on Public key cryptography
Group-oriented fair exchange of signatures
Information Sciences: an International Journal
Controversy corner: Improving security of q-SDH based digital signatures
Journal of Systems and Software
Solving DLP with auxiliary input over an elliptic curve used in TinyTate library
WISTP'11 Proceedings of the 5th IFIP WG 11.2 international conference on Information security theory and practice: security and privacy of mobile devices in wireless communication
Verifiable delegation of computation over large datasets
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Forward-Secure and searchable broadcast encryption with short ciphertexts and private keys
ASIACRYPT'06 Proceedings of the 12th international conference on Theory and Application of Cryptology and Information Security
On pairing-based cryptosystems
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
A new signature scheme without random oracles from bilinear pairings
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
Concise mercurial vector commitments and independent zero-knowledge sets with short proofs
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Attribute-based encryption schemes with constant-size ciphertexts
Theoretical Computer Science
Using equivalence classes to accelerate solving the discrete logarithm problem in a short interval
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Universal designated verifier signatures without random oracles or non-black box assumptions
SCN'06 Proceedings of the 5th international conference on Security and Cryptography for Networks
Efficient identity-based encryption with tight security reduction
CANS'06 Proceedings of the 5th international conference on Cryptology and Network Security
Revocation of direct anonymous attestation
INTRUST'10 Proceedings of the Second international conference on Trusted Systems
Solving a DLP with auxiliary input with the ρ-algorithm
WISA'11 Proceedings of the 12th international conference on Information Security Applications
Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments
TCC'12 Proceedings of the 9th international conference on Theory of Cryptography
Solving a discrete logarithm problem with auxiliary input on a 160-bit elliptic curve
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Restricted identification scheme and diffie-hellman linking problem
INTRUST'11 Proceedings of the Third international conference on Trusted Systems
Remarks on Cheon's algorithms for pairing-related problems
Pairing'07 Proceedings of the First international conference on Pairing-Based Cryptography
Identity-based encryption: how to decrypt multiple ciphertexts using a single decryption key
Pairing'07 Proceedings of the First international conference on Pairing-Based Cryptography
Identity-based proxy signature from pairings
ATC'07 Proceedings of the 4th international conference on Autonomic and Trusted Computing
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Let g be an element of prime order p in an abelian group and $\alpha\in {{\mathbb Z}}_p$. We show that if g, gα, and $g^{\alpha^d}$ are given for a positive divisor d of p–1, we can compute the secret α in $O(\log p \cdot (\sqrt{p/d}+\sqrt d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. If $g^{\alpha^i}$ (i=0,1,2,..., d) are provided for a positive divisor d of p+1, α can be computed in $O(\log p \cdot (\sqrt{p/d}+d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. This implies that the strong Diffie-Hellman problem and its related problems have computational complexity reduced by $O(\sqrt d)$ from that of the discrete logarithm problem for such primes. Further we apply this algorithm to the schemes based on the Diffie-Hellman problem on an abelian group of prime order p. As a result, we reduce the complexity of recovering the secret key from $O(\sqrt p)$ to $O(\sqrt{p/d})$ for Boldyreva's blind signature and the original ElGamal scheme when p–1 (resp. p+1) has a divisor d ≤p1/2 (resp. d ≤p1/3) and d signature or decryption queries are allowed.