Relevant bibliographies by topics / Pollard’s Rho

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Pollard’s Rho'

Author: Grafiati
Published: 4 June 2021
Last updated: 16 February 2022

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Journal articles on the topic "Pollard’s Rho"

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Cheon, Jung Hee, Jin Hong, and Minkyu Kim. "Accelerating Pollard’s Rho Algorithm on Finite Fields." Journal of Cryptology 25, no. 2 (November 12, 2010): 195–242. http://dx.doi.org/10.1007/s00145-010-9093-7.

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Lokhande, Santosh P., Dr Indivar Gupta, and Dr Dinesh B. Kulkarni. "A Review: Solving ECDLP Problem using Pollard’s Rho Algorithm." IJARCCE 6, no. 5 (May 30, 2017): 377–80. http://dx.doi.org/10.17148/ijarcce.2017.6571.

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Candra, Ade, Mohammad Andri Budiman, and Dian Rachmawati. "On Factoring The RSA Modulus Using Tabu Search." Data Science: Journal of Computing and Applied Informatics 1, no. 1 (July 18, 2017): 30–37. http://dx.doi.org/10.32734/jocai.v1.i1-65.

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It is intuitively clear that the security of RSA cryptosystem depends on the hardness of factoring a very large integer into its two prime factors. Numerous studies about integer factorization in the field of number theory have been carried out, and as a result, lots of exact factorization algorithms, such as Fermat’s factorization algorithm, quadratic sieve method, and Pollard’s rho algorithm have been found. The factorization problem is in the class of NP (non-deterministic polynomial time). Tabu search is a metaheuristic in the field of artificial intelligence which is often used to solve NP and NP-hard problems; the result of this method is expected to be close-to-optimal (suboptimal). This study aims to factorize the RSA modulus into its two prime factors using tabu search by conducting experiments in Python programming language and to compare its time performance with an exact factorization algorithm, i.e. Pollard’s algorithm. The primality test is done with Lehmann’s algorithm.
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Budiman, M. A., and D. Rachmawati. "On factoring RSA modulus using random-restart hill-climbing algorithm and Pollard’s rho algorithm." Journal of Physics: Conference Series 943 (December 2017): 012057. http://dx.doi.org/10.1088/1742-6596/943/1/012057.

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Delaplace, Claire, and Alexander May. "Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation?" Journal of Mathematical Cryptology 14, no. 1 (August 18, 2020): 293–306. http://dx.doi.org/10.1515/jmc-2019-0025.

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AbstractWe give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard’s Rho algorithm with square root (in the field size) complexity 𝓞(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as $\begin{array}{} p^{\frac 3 4}, \end{array}$ only their computation is yet too costly.
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Bridy, Andrew, and Derek Garton. "The Cycle Structure of Unicritical Polynomials." International Mathematics Research Notices 2020, no. 23 (October 26, 2018): 9120–47. http://dx.doi.org/10.1093/imrn/rny232.

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Abstract A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $ {{\mathbb{F}}}_p$. The questions of whether and in what sense these families are random have been studied extensively, spurred in part by Pollard’s famous “rho” algorithm for integer factorization (the heuristic justification of which is the conjectural randomness of one such family). However, the cycle structure of these families cannot be random, since in any such family, the number of cycles of a fixed length in any dynamical system in that family is bounded. In this paper, we show that the cycle statistics of many of these families are as random as possible. As a corollary, we show that most members of these families have many cycles, addressing a conjecture of Mans et al.
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Yokoyama, Kazuhiro, Masaya Yasuda, Yasushi Takahashi, and Jun Kogure. "Complexity bounds on Semaev’s naive index calculus method for ECDLP." Journal of Mathematical Cryptology 14, no. 1 (October 30, 2020): 460–85. http://dx.doi.org/10.1515/jmc-2019-0029.

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AbstractSince Semaev introduced summation polynomials in 2004, a number of studies have been devoted to improving the index calculus method for solving the elliptic curve discrete logarithm problem (ECDLP) with better complexity than generic methods such as Pollard’s rho method and the baby-step and giant-step method (BSGS). In this paper, we provide a deep analysis of Gröbner basis computation for solving polynomial systems appearing in the point decomposition problem (PDP) in Semaev’s naive index calculus method. Our analysis relies on linear algebra under simple statistical assumptions on summation polynomials. We show that the ideal derived from PDP has a special structure and Gröbner basis computation for the ideal is regarded as an extension of the extended Euclidean algorithm. This enables us to obtain a lower bound on the cost of Gröbner basis computation. With the lower bound, we prove that the naive index calculus method cannot be more efficient than generic methods.
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Abro, Adeel, Zhongliang Deng, and Kamran Ali Memon. "A Lightweight Elliptic-Elgamal-Based Authentication Scheme for Secure Device-to-Device Communication." Future Internet 11, no. 5 (May 7, 2019): 108. http://dx.doi.org/10.3390/fi11050108.

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Device-to-Device (D2D) is a major part of 5G that will facilitate deployments with extended coverage where devices can act as users or relays. These relays normally act as decode and forward relays (semi-intelligent devices) with limited computational and storage capabilities. However, introducing such a technology, where users can act as relays, presents a wide range of security threats, in particular, rogue relay devices or man in the middle attacks (M-I-T-M). Second, passing fewer control messages is always advisable when considering authenticity and secrecy. To mitigate M-I-T-M and to reduce communication costs, this paper presents a lightweight elliptic-ElGamal-based authentication scheme using PKI (FHEEP) in D2D communication. Pollard’s rho and Baby Step, Giant Step (BSGS) methods are used to evaluate the authenticity and secrecy of our proposed scheme. The communication cost is calculated based on the comparative analysis indicating that our proposed scheme outperforms the baseline protocol. The proposed scheme can be used for any infrastructure architecture that will enhance the security of any D2D settings with better performance.
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Peng, Yang, Tomoyuki Nagase, Shan You, and Toshiki Kanamoto. "A VOHE System for Underwater Communications." Electronics 9, no. 10 (September 23, 2020): 1557. http://dx.doi.org/10.3390/electronics9101557.

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This paper presents a new method for encrypting holographic information based on optical and acoustic signals called a Virtual Optical Holographic Encryption (VOHE) system for underwater communications that can be applicable for communications between deep submergence research vehicles. The transmission medium is composed of a combination of optical signals and acoustic signals together to form the VOHE system for transmitting system information. The optical encryption system provides essential parameters for constructing secure communications such as the propagation wavelength (λ) and focal length (f) of the Fourier lens, which are considered as keys for implementing encryption and decryption processes. An expanded RSA (ERSA) algorithm using a complex function sends system information (λ, f) as a message to a receiver. To determine accuracy of the information retrieved by the proposed technique, the minimum mean square error (MSE) was conducted to evaluate the accuracy of the received signal. The VOHE system employs virtual optical encryption system was simulated based on COMSOL Multiphysics simulation software. Finally, the National Institute of Standards and Technology (NIST) method and Pollard’s rho method were separately applied to evaluate the proposed ERSA algorithm. Obtained results showed that ERSA is able to achieve a more significant security level than RSA.
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Aminudin, Aminudin, and Eko Budi Cahyono. "A Practical Analysis of the Fermat Factorization and Pollard Rho Method for Factoring Integers." Lontar Komputer : Jurnal Ilmiah Teknologi Informasi 12, no. 1 (March 29, 2021): 33. http://dx.doi.org/10.24843/lkjiti.2021.v12.i01.p04.

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The development of public-key cryptography generation using the factoring method is very important in practical cryptography applications. In cryptographic applications, the urgency of factoring is very risky because factoring can crack public and private keys, even though the strength in cryptographic algorithms is determined mainly by the key strength generated by the algorithm. However, solving the composite number to find the prime factors is still very rarely done. Therefore, this study will compare the Fermat factorization algorithm and Pollard rho by finding the key generator public key algorithm's prime factor value. Based on the series of test and analysis factoring integer algorithm using Fermat's Factorization and Pollards' Rho methods, it could be concluded that both methods could be used to factorize the public key which specifically aimed to identify the prime factors. During the public key factorizing process within 16 bytes – 64 bytes, Pollards' Rho's average duration was significantly faster than Fermat's Factorization.
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Dissertations / Theses on the topic "Pollard’s Rho"

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Bucic, Ida. "Pollard's rho method." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-85886.

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In this work we are going to investigate a factorization method that was invented by John Pollard. It makes possible to factorize medium large integers into a product of prime numbers. We will run a C++ program and test how do different parameters affect the results. There will be a connection drawn between the Pollard's rho method, the Birthday paradox and the Floyd's cycle finding algorithm. In results we will find a polynomial function that has the best effectiveness and performance for Pollard's rho method.
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Kouchaki, Barzi Behnaz. "Points of High Order on Elliptic Curves : ECDSA." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-58449.

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This master thesis is about Elliptic Curve Digital Signature Algorithm or ECDSA and two of the known attacks on this security system. The purpose of this thesis is to find points that are likely to be points of high order on an elliptic curve. If we have a point P of high order and if Q = mP, then we have a large set of possible values of m. Therefore it is hard to solve the Elliptic Curve Discrete Logarithm Problem or ECDLP. We have investigated on the time of finding the solution of ECDLP for a certain amount of elliptic curves based on the order of the point which is used to create the digital signatures by those elliptic curves. Method: Algebraic Structure of elliptic curves over finite fields and Discrete logarithms. This has been done by two types of attacks namely Baby Step, Giant Step and Pollard’s Rho and all of the programming parts has been done by means of Mathematica. Conclusion: We have come into a conclusion of having the probable good points which are the points of high order on elliptic curves through the mentioned attacks in which solving the ECDLP is harder if these points have been used in generating the digital signature. These probable good points can be estimated by means of a function we have come up with. The input of this function is the order of the point and the output is the time of finding the answer of ECDLP.
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Falk, Jenny. "On Pollard's rho method for solving the elliptic curve discrete logarithm problem." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-85516.

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Cryptosystems based on elliptic curves are in wide-spread use, they are considered secure because of the difficulty to solve the elliptic curve discrete logarithm problem. Pollard's rho method is regarded as the best method for attacking the logarithm problem to date, yet it is still not efficient enough to break an elliptic curve cryptosystem. This is because its time complexity is O(√n) and for uses in cryptography the value of n will be very large. The objective of this thesis is to see if there are ways to improve Pollard's rho method. To do this, we study some modifications of the original functions used in the method. We also investigate some different functions proposed by other researchers to see if we can find a version that will improve the performance. From the experiments conducted on these modifications and functions, we can conclude that we get an improvement in the performance for some of them.
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Podhorský, Jiří. "Integer Factorization on the GPU." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2014. http://www.nusl.cz/ntk/nusl-412908.

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This work deals with factorization, a decomposition of composite numbers on prime numbers and possibilities of its parallelization. It summarizes also the best known algorithms for factoring and most popular platforms for the implementation of these algorithms on the graphics card. The main part of the thesis deals with the design and implementation of hardware acceleration current fastest algorithm on the graphics card by using the OpenCL framework. Subsequently, the work provides a comparison of speeds accelerated algorithm implemented in this work with other versions of the best known algorithms for factoring, processed serially. In conclusion, the work discussed length of RSA key needed for safe operation without the possibility of breaking in real time interval.
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Breitenbacher, Dominik. "Paralelizace faktorizace celých čísel z pohledu lámání RSA." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2015. http://www.nusl.cz/ntk/nusl-234905.

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This paper follows up the factorization of integers. Factorization is the most popular and used method for RSA cryptoanalysis. The SIQS was chosen as a factorization method that will be used in this paper. Although SIQS is the fastest method (up to 100 digits), it can't be effectively computed at polynomial time, so it's needed to look up for options, how to speed up the method as much as possible. One of the possible ways is paralelization. In this case OpenMP was used. Other possible way is optimalization. The goal of this paper is also to show, how easily is possible to use paralelizion and thanks to detailed analyzation the source codes one can reach relatively large speed up. Used method of iterative optimalization showed itself as a very effective tool. Using this method the implementation of SIQS achieved almost 100 multiplied speed up and at some parts of the code even more.
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Judge, Lyndon Virginia. "Design Methods for Cryptanalysis." Thesis, Virginia Tech, 2012. http://hdl.handle.net/10919/35980.

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Security of cryptographic algorithms relies on the computational difficulty of deriving the secret key using public information. Cryptanalysis, including logical and implementation attacks, plays an important role in allowing the security community to estimate their cost, based on the computational resources of an attacker. Practical implementations of cryptanalytic systems require complex designs that integrate multiple functional components with many parameters. In this thesis, methodologies are proposed to improve the design process of cryptanalytic systems and reduce the cost of design space exploration required for optimization. First, Bluespec, a rule-based HDL, is used to increase the abstraction level of hardware design and support efficient design space exploration. Bluespec is applied to implement a hardware-accelerated logical attack on ECC with optimized modular arithmetic components. The language features of Bluespec support exploration and this is demonstrated by applying Bluespec to investigate the speed area tradeoff resulting from various design parameters and demonstrating performance that is competitive with prior work. This work also proposes a testing environment for use in verifying the implementation attack resistance of secure systems. A modular design approach is used to provide separation between the device being tested and the test script, as well as portability, and openness. This yields an open-source solution that supports implementation attack testing independent of the system platform, implementation details, and type of attack under evaluation. The suitability of the proposed test environment for implementation attack vulnerability analysis is demonstrated by applying the environment to perform an implementation attack on AES. The design of complex cryptanalytic hardware can greatly benefit from better design methodologies and the results presented in this thesis advocate the importance of this aspect.
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Hitchcock, Yvonne Roslyn. "Elliptic Curve Cryptography for Lightweight Applications." Queensland University of Technology, 2003. http://eprints.qut.edu.au/15838/.

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Elliptic curves were first proposed as a basis for public key cryptography in the mid 1980's. They provide public key cryptosystems based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP) , which is so called because of its similarity to the discrete logarithm problem (DLP) over the integers modulo a large prime. One benefit of elliptic curve cryptosystems (ECCs) is that they can use a much shorter key length than other public key cryptosystems to provide an equivalent level of security. For example, 160 bit ECCs are believed to provide about the same level of security as 1024 bit RSA. Also, the level of security provided by an ECC increases faster with key size than for integer based discrete logarithm (dl) or RSA cryptosystems. ECCs can also provide a faster implementation than RSA or dl systems, and use less bandwidth and power. These issues can be crucial in lightweight applications such as smart cards. In the last few years, ECCs have been included or proposed for inclusion in internationally recognized standards. Thus elliptic curve cryptography is set to become an integral part of lightweight applications in the immediate future. This thesis presents an analysis of several important issues for ECCs on lightweight devices. It begins with an introduction to elliptic curves and the algorithms required to implement an ECC. It then gives an analysis of the speed, code size and memory usage of various possible implementation options. Enough details are presented to enable an implementer to choose for implementation those algorithms which give the greatest speed whilst conforming to the code size and ram restrictions of a particular lightweight device. Recommendations are made for new functions to be included on coprocessors for lightweight devices to support ECC implementations Another issue of concern for implementers is the side-channel attacks that have recently been proposed. They obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time and the information is then used to break implementations that have not incorporated appropriate defences. A new method of defence to protect an implementation from the simple power analysis (spa) method of attack is presented in this thesis. It requires 44% fewer additions and 11% more doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm. The algorithm forms a contribution to the current range of possible spa defences which has a good speed but low memory usage. Another topic of paramount importance to ECCs for lightweight applications is whether the security of fixed curves is equivalent to that of random curves. Because of the inability of lightweight devices to generate secure random curves, fixed curves are used in such devices. These curves provide the additional advantage of requiring less bandwidth, code size and processing time. However, it is intuitively obvious that a large precomputation to aid in the breaking of the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve which would be unavailable for a random curve. Therefore, it would appear that fixed curves are less secure than random curves, but quantifying the loss of security is much more difficult. The thesis performs an examination of fixed curve security taking this observation into account, and includes a definition of equivalent security and an analysis of a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. A lower bound on the expected time to solve such ECDLPs using this method is presented, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. It is concluded that adding a total of 11 bits to the size of a fixed curve provides an equivalent level of security compared to random curves. The final part of the thesis deals with proofs of security of key exchange protocols in the Canetti-Krawczyk proof model. This model has been used since it offers the advantage of a modular proof with reusable components. Firstly a password-based authentication mechanism and its security proof are discussed, followed by an analysis of the use of the authentication mechanism in key exchange protocols. The Canetti-Krawczyk model is then used to examine secure tripartite (three party) key exchange protocols. Tripartite key exchange protocols are particularly suited to ECCs because of the availability of bilinear mappings on elliptic curves, which allow more efficient tripartite key exchange protocols.
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Yoshida, Kayo. "Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem." Thesis, 2009. http://hdl.handle.net/10012/4219.

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The Boneh-Boyen signature scheme is a short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman (SDH) assumption. The primary objective of this thesis is to examine the relationship between the Boneh-Boyen signature scheme and SDH. The secondary objective is to survey surrounding topics such as the generic group model, related signature schemes, intractability assumptions, and the relationship to identity-based encryption (IBE) schemes. Along these lines, we analyze the plausibility of the SDH assumption using the generic bilinear group model. We present the security proofs for the Boneh-Boyen signature scheme, with the addition of a small improvement in one of the probability bounds. Our main contribution is to give the reduction in the reverse direction; that is, to show that if the SDH problem can be solved then the Boneh-Boyen signature scheme can be forged. This contribution represents the first known proof of equivalence between the SDH problem and Boneh-Boyen signatures. We also discuss the algorithm of Cheon for solving the SDH problem. We analyze the implications of Cheon's algorithm for the security of the Boneh-Boyen signature scheme, accompanied by a brief discussion on how to counter the attack.
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Book chapters on the topic "Pollard’s Rho"

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Kuhn, Fabian, and René Struik. "Random Walks Revisited: Extensions of Pollard’s Rho Algorithm for Computing Multiple Discrete Logarithms." In Selected Areas in Cryptography, 212–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45537-x_17.

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Miller, Stephen D., and Ramarathnam Venkatesan. "Spectral Analysis of Pollard Rho Collisions." In Lecture Notes in Computer Science, 573–81. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11792086_40.

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Teske, Edlyn. "Speeding up Pollard's rho method for computing discrete logarithms." In Lecture Notes in Computer Science, 541–54. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0054891.

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Cheon, Jung Hee, Jin Hong, and Minkyu Kim. "Speeding Up the Pollard Rho Method on Prime Fields." In Advances in Cryptology - ASIACRYPT 2008, 471–88. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-89255-7_29.

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Bos, Joppe W., Thorsten Kleinjung, and Arjen K. Lenstra. "On the Use of the Negation Map in the Pollard Rho Method." In Lecture Notes in Computer Science, 66–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14518-6_9.

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Bernstein, Daniel J., Tanja Lange, and Peter Schwabe. "On the Correct Use of the Negation Map in the Pollard rho Method." In Public Key Cryptography – PKC 2011, 128–46. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-19379-8_8.

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Kim, Minkyu, Jung Hee Cheon, and Jin Hong. "Subset-Restricted Random Walks for Pollard rho Method on ${\mathbf{F}_{p^m}}$." In Public Key Cryptography – PKC 2009, 54–67. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00468-1_4.

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Somsuk, Kritsanapong. "Decreasing Size of Parameter for Computing Greatest Common Divisor to Speed up New Factorization Algorithm Based on Pollard Rho." In Lecture Notes in Electrical Engineering, 399–405. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-46578-3_47.

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Kim, Jeong Han, Ravi Montenegro, Yuval Peres, and Prasad Tetali. "A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm." In Lecture Notes in Computer Science, 402–15. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-79456-1_27.

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Viswanath, M. K., and K. P. Vidya. "POLLARD'S RHO SPLIT KNOWLEDGE SCHEME." In Series in Machine Perception and Artificial Intelligence, 379–90. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812773036_0026.

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Conference papers on the topic "Pollard’s Rho"

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Ikuta, Ken, Sho Joichi, Kazuya Kobayashi, Md Al-Amin Khandaker, Takuya Kusaka, and Yasuyuki Nogami. "A Study on the Parameter of the Distinguished Point Method in Pollard’s Rho Method for ECDLP." In 2018 International Symposium on Information Theory and Its Applications (ISITA). IEEE, 2018. http://dx.doi.org/10.23919/isita.2018.8664405.

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Miura, Hiromasa, Rikuya Matsumura, Takuya Kusaka, and Yasuyuki Nogami. "Analysis of a method to eliminate fruitless cycles for Pollard’s rho method with skew Frobenius mapping over a Barreto-Naehrig curve." In 2020 Eighth International Symposium on Computing and Networking (CANDAR). IEEE, 2020. http://dx.doi.org/10.1109/candar51075.2020.00029.

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Harish, G., G. Punith Kumar, Anjan K. Koundinya, Y. V. Pramod, N. K. Srinath, G. E. Raghavendra Kumar, R. Sandeep, Archit Shukla, and Madan Acharya. "Parallelization of Pollard's Rho Integer factorization algorithm." In the Second International Conference. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2393216.2393224.

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Ezzouak, Siham, Mohammed Elamrani, and Abdelmalek Azizi. "Improving Pollard's Rho attack on elliptic curve cryptosystems." In 2012 International Conference on Multimedia Computing and Systems (ICMCS). IEEE, 2012. http://dx.doi.org/10.1109/icmcs.2012.6320288.

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Sadkhan, Sattar B. "A Proposed Developments of Pollards Rho Method for Attacking the ECDLP." In 2021 7th International Engineering Conference “Research & Innovation amid Global Pandemic" (IEC). IEEE, 2021. http://dx.doi.org/10.1109/iec52205.2021.9476119.

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Kim, Jeong Han, Ravi Montenegro, and Prasad Tetali. "Near Optimal Bounds for Collision in Pollard Rho for Discrete Log." In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07). IEEE, 2007. http://dx.doi.org/10.1109/focs.2007.38.

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Kim, Jeong Han, Ravi Montenegro, and Prasad Tetali. "Near Optimal Bounds for Collision in Pollard Rho for Discrete Log." In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07). IEEE, 2007. http://dx.doi.org/10.1109/focs.2007.4389494.

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Wienardo, Fajar Yuliawan, Intan Muchtadi-Alamsyah, and Budi Rahardjo. "Implementation of Pollard Rho attack on elliptic curve cryptography over binary fields." In THE 5TH INTERNATIONAL CONFERENCE ON MATHEMATICS AND NATURAL SCIENCES. AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4930641.

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Wang, Ping, and Fangguo Zhang. "Improving the Parallelized Pollard Rho Method for Computing Elliptic Curve Discrete Logarithms." In 2013 Fourth International Conference on Emerging Intelligent Data and Web Technologies (EIDWT). IEEE, 2013. http://dx.doi.org/10.1109/eidwt.2013.55.

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Miura, Hiromasa, Ken Ikuta, Sho Joichi, Takuya Kusaka, and Yasuyuki Nogami. "Analysis of the fruitless cycle of Pollard's rho method based attack for solving ECDLP over Barreto-Naehrig curves." In 2019 34th International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC). IEEE, 2019. http://dx.doi.org/10.1109/itc-cscc.2019.8793449.

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