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Hardness of Computing Individual Bits for One-Way Functions on Elliptic Curves [chapter]

Alexandre Duc, Dimitar Jetchev
2012 Lecture Notes in Computer Science  
We prove that if one can predict any of the bits of the input to an elliptic curve based one-way function over a finite field, then we can invert the function.  ...  Our result implies that all the bits of the functions defined above are hard-to-compute assuming these functions are one-way.  ...  We are grateful to Dan Boneh, David Freeman, Rosario Gennaro, Florian Hess, Eike Kiltz, Arjen Lenstra, Amin Shokrollahi, Igor Shparlinski, Martijn Stam, Serge Vaudenay and Ramarathnam Venkatesan for helpful  ... 
doi:10.1007/978-3-642-32009-5_48 fatcat:paujmgl2bvcwflnfs6gqz5wch4

Hard-Core Predicates for a Diffie-Hellman Problem over Finite Fields [chapter]

Nelly Fazio, Rosario Gennaro, Irippuge Milinda Perera, William E. Skeith
2013 Lecture Notes in Computer Science  
prove that every bit (and every segment predicate) of the elliptic curve Diffie-Hellman problem is hard-core; • We define the notion of partial one-way function over finite fields F p 2 and prove that  ...  every bit (and every segment predicate) of one of the input coordinates for these functions is hard-core.  ...  Ministry of Defence or the U.K. Government. The U.S. and U.K. Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.  ... 
doi:10.1007/978-3-642-40084-1_9 fatcat:giygm7mzx5ekxhqyi5p7ypjin4

How to Keep Your Secrets in a Post-Quantum World

Kristin Lauter
2020 Notices of the American Mathematical Society  
Cryptographic systems are often built on the premise that certain math problems are very hard to solve, in the sense that known solutions require enormous computational time and resources.  ...  It is now a flourishing branch of mathematics that, in addition to encryption, also provides other tools to protect security and privacy of individuals, enterprises, data, systems, and transactions.  ...  Elliptic curve cryptosystems are based on the hardness of solving ECDLP, the discrete logarithm problem in the abelian group of points on an elliptic curve over a finite field.  ... 
doi:10.1090/noti2004 fatcat:jq44tmpfdnb2pl6rpa5scvl3qi

An efficient discrete log pseudo random generator [chapter]

Sarvar Patel, Ganapathy S. Sundaram
1998 Lecture Notes in Computer Science  
The exponentiation function in a finite field of order p (a prime number) is believed to be a one-way function. It is well known that O(log tog p) bits are simultaneously hazd for this function.  ...  We consider a special case of this problem, the discrete logarithm with short exponents, which is also believed to be hard to compute.  ...  Informally, a hard bit B(.) for a one way function f(.) is a bit which is as hard to compute as it is to invert f.  ... 
doi:10.1007/bfb0055737 fatcat:lyrdtbkcovcvfdut3sb2eocfti

Resource Guide for Teaching Post-Quantum Cryptography [article]

Joshua Holden
2022 arXiv   pre-print
This article gives a tour of the main types of systems under consideration and the teaching resources available for instructors who want to teach them.  ...  Unfortunately, many experts expect quantum computers to make common forms of public-key cryptography obsolete in the near future.  ...  A big advantage of this type of cipher is that a lot of current cryptography already uses elliptic curves, although in a way that's not secure against quantum computers.  ... 
arXiv:2207.00558v1 fatcat:nq7q7tx3tncszok2wlvm5hzo3a

Fog Computing: Comprehensive Approach for Security Data Theft Attack Using Elliptic Curve Cryptography and Decoy Technology

Mai Trung Dong, Xianwei Zhou
2016 OALib  
Elliptic Curve Cryptography (ECC) and decoy technology in such a constrained environment along with the other two aspects of ECC, namely its security and efficiency.  ...  Fog computing extends cloud computing, provides the services like data, compute, storage and application to end user. It improves the quality of service and also reduces latency.  ...  Diffie-Hellman key exchange We use the same one-way function of Elliptic Curve to consider a Diffie-Hellman key exchange protocol.  ... 
doi:10.4236/oalib.1102802 fatcat:hntozyqqnzgmbiiq6jwqw5lbea

Pairing for Greenhorn: Survey and Future Perspective [article]

Mahender Kumar, Satish Chand
2021 arXiv   pre-print
Pairing is the most powerful tool in cryptography that maps two points on the elliptic curve to the group over the finite field.  ...  We also analyze the bandwidth and computational efficiency of pairing and submitting those pairing suitable for implementing a cryptographic protocol for lightweight devices.  ...  That means, has zeros and poles on . One way to keep track of the zeroes and poles of a function is to compute the divide of f, where poles are related to projective coordinates.  ... 
arXiv:2108.12392v1 fatcat:lpwnif5f5vaxzcapdtqda76aca

Comparative Evaluation of Elliptic Curve Cryptography Based Homomorphic Encryption Schemes for a Novel Secure Multiparty Computation

Sankita J. Patel, Ankit Chouhan, Devesh C. Jinwala
2014 Journal of Information Security  
In this paper, we focus on Elliptic Curve Cryptography based approach for Secure Multiparty Computation (SMC) problem.  ...  We propose Elliptic Curve Cryptography (ECC) based approach for SMC that is scalable in terms of computational and communication cost and avoids TTP.  ...  One way to compute the desired functionality is to use Trusted Third Party (TTP).  ... 
doi:10.4236/jis.2014.51002 fatcat:wvansmtuy5c65bnnkjbow2ptvu

An ECC-Based Blind Signcryption Scheme for Multiple Digital Documents

Chien-Hua Tsai, Pin-Chang Su
2017 Security and Communication Networks  
Further, most of the topics only deal with signing rather than encryption. Thus, we propose a new blind signature scheme for multiple digital documents based on elliptic curve cryptography (ECC).  ...  The popularity of the Internet has comprehensively altered the traditional way of communication and interaction patterns, such as e-contract negotiations, e-payment services, or digital credential processes  ...  For these parameters, the adversary or the dishonest signer then has to encounter the hardness of solving the ECDLP and the difficulty of inverting the one-way hash function.  ... 
doi:10.1155/2017/8981606 fatcat:f2fztffbxffedgfg7gn2qhqfdi

Post Quantum Cryptography: Techniques, Challenges, Standardization, and Directions for Future Research [article]

Ritik Bavdekar, Eashan Jayant Chopde, Ashutosh Bhatia, Kamlesh Tiwari, Sandeep Joshua Daniel, Atul
2022 arXiv   pre-print
The security of asymmetric algorithms like RSA, Diffie Hellman, and ECC is based on the mathematical hardness of prime factorization and discrete logarithm.  ...  The development of large quantum computers will have dire consequences for cryptography. Most of the symmetric and asymmetric cryptographic algorithms are vulnerable to quantum algorithms.  ...  ECC hardness is based on the hardness of solving the discrete logarithm problem in elliptic curve groups.  ... 
arXiv:2202.02826v1 fatcat:ghq4lqiu4rf5vpuv6gbk7ws4jm

Hidden Pairings and Trapdoor DDH Groups [chapter]

Alexander W. Dent, Steven D. Galbraith
2006 Lecture Notes in Computer Science  
Some possible cryptographic applications of this idea are given. Both of our instantiations are based on elliptic curves. The first relies on the factoring assumption for hiding the pairing.  ...  The second relies on the hardness of solving a system of multivariate equations.  ...  Acknowledgements We thank Kenny Paterson for his comments.  ... 
doi:10.1007/11792086_31 fatcat:mydusgowcvf5nnkmuqvec6adm4

The advantages of elliptic curve cryptography for wireless security

K. Lauter
2004 IEEE wireless communications  
However, using an elliptic curve cryptosystem to perform steps 2, 3, 5, and 6 can significantly decrease the computational burden on the individual parties.  ...  WEIL PAIRING ON ELLIPTIC CURVES In [21] , Boneh and Franklin proposed using the group of points on an elliptic curve for the group G 1 , the Weil pairing on the elliptic curve as the bilinear map, and  ...  Since 1999 she has been a researcher in the Cryptography and Anti-Piracy group at Microsoft Research, working on elliptic curve cryptography and related areas.  ... 
doi:10.1109/mwc.2004.1269719 fatcat:rdei23ylqnbypl2yw3572erbhi

The Multivariate Hidden Number Problem [chapter]

Steven D. Galbraith, Barak Shani
2015 Lecture Notes in Computer Science  
The second considers a model that allows changing representations, where we show hardness of individual bits for elliptic curve and pairing based functions for elliptic curves over extension fields, as  ...  Keywords: hidden number problem, bit security, hardcore bits to prove hardness of bits for Diffie-Hellman key exchange.  ...  Acknowledgements We thank the anonymous referees for their helpful comments.  ... 
doi:10.1007/978-3-319-17470-9_15 fatcat:z2ranc2yrjf7pdproy2slygh3y

FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

Hao Li, Jian Huang, Philip Sweany, Dijiang Huang
2008 Journal of systems architecture  
In this thesis, I propose the FPGA implementations of the elliptic curve point multiplication in GF (2 283 ) as well as Tate pairing computation on supersingular elliptic curve in GF (2 283 ).  ...  Elliptic curve cryptography (ECC) is an alternative to traditional techniques for public key cryptography. It offers smaller key size without sacrificing security level.  ...  The security of ECC is based on the hardness of elliptic curve discrete logarithm problem (ECDLP).  ... 
doi:10.1016/j.sysarc.2008.04.012 fatcat:3tlovro4lbfajgk3v2ffvheydu

A Survey of Public-Key Cryptosystems

Neal Koblitz, Alfred J. Menezes
2004 SIAM Review  
We give an overview of the most important public-key cryptosystems and discuss the difficult task of evaluating the merit of such systems.  ...  Speaking informally, a one-to-one function f : X → Y is "one-way" if it is easy to compute f (x) for any x ∈ X but hard to compute f −1 (y) for most randomly selected y in the range of f . 1 In [98]  ...  is the idea of using a one-way function for encryption.  ... 
doi:10.1137/s0036144503439190 fatcat:wix7yrhdnzaslooaugb5ttrt7y
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