How easy is collision search? Application to DES
EUROCRYPT '89 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves
Mathematics of Computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Algorithms for exponentiation in finite fields
Journal of Symbolic Computation
Improving the parallelized Pollard lambda search on anomalous binary curves
Mathematics of Computation
On random walks for Pollard's Rho method
Mathematics of Computation
Faster Attacks on Elliptic Curve Cryptosystems
SAC '98 Proceedings of the Selected Areas in Cryptography
Identity-Based Encryption from the Weil Pairing
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Speeding up the Discrete Log Computation on Curves with Automorphisms
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Supersingular Curves in Cryptography
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Speeding Up Pollard's Rho Method for Computing Discrete Logarithms
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Information Processing Letters
Short Signatures from the Weil Pairing
Journal of Cryptology
A subexponential algorithm for the discrete logarithm problem with applications to cryptography
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Speeding Up the Pollard Rho Method on Prime Fields
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in 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
Improved Pollard rho method for computing discrete logarithms over finite extension fields
Journal of Computational and Applied Mathematics
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In this paper, we propose a variant of the Pollard rho method. We use an iterating function whose image size is much smaller than its domain and hence reaches a collision faster than the original iterating function. We also explicitly show how this general method can be applied to multiplicative subgroups of finite fields with large extension degree. The construction for finite fields uses a distinctive feature of the normal basis representation, namely, that the p -th power of an element is just the cyclic shift of its normal basis representation, when the underlying field is of characteristic p . This makes our method appropriate for hardware implementations. On multiplicative subgroups of ${\mathbf{F}_{p^m}}$, our method shows time complexity advantage over the original Pollard rho method by a factor of approximately $\frac{3p-3}{4p-3}\sqrt{m}$. Through the MOV reduction, our method can be applied to pairing-based cryptosystems over binary or ternary fields. Hence our algorithm suggests that the order of subgroups, on which the pairing-based cryptosystems rely, needs to be increased by a factor of approximately m .