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

2012
*
Lecture Notes in Computer Science
*

We prove that if

doi:10.1007/978-3-642-32009-5_48
fatcat:paujmgl2bvcwflnfs6gqz5wch4
*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 ...##
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Hard-Core Predicates for a Diffie-Hellman Problem over Finite Fields
[chapter]

2013
*
Lecture Notes in Computer Science
*

prove that every

doi:10.1007/978-3-642-40084-1_9
fatcat:giygm7mzx5ekxhqyi5p7ypjin4
*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. ...##
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How to Keep Your Secrets in a Post-Quantum World

2020
*
Notices of the American Mathematical Society
*

Cryptographic systems are often built

doi:10.1090/noti2004
fatcat:jq44tmpfdnb2pl6rpa5scvl3qi
*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. ...##
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An efficient discrete log pseudo random generator
[chapter]

1998
*
Lecture Notes in Computer Science
*

The exponentiation

doi:10.1007/bfb0055737
fatcat:lyrdtbkcovcvfdut3sb2eocfti
*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. ...##
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Resource Guide for Teaching Post-Quantum Cryptography
[article]

2022
*
arXiv
*
pre-print

This article gives a tour

arXiv:2207.00558v1
fatcat:nq7q7tx3tncszok2wlvm5hzo3a
*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*. ...##
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Fog Computing: Comprehensive Approach for Security Data Theft Attack Using Elliptic Curve Cryptography and Decoy Technology

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. ...

##
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Pairing for Greenhorn: Survey and Future Perspective
[article]

2021
*
arXiv
*
pre-print

Pairing is the most powerful tool in cryptography that maps two points

arXiv:2108.12392v1
fatcat:lpwnif5f5vaxzcapdtqda76aca
*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. ...##
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Comparative Evaluation of Elliptic Curve Cryptography Based Homomorphic Encryption Schemes for a Novel Secure Multiparty Computation

2014
*
Journal of Information Security
*

In this paper, we focus

doi:10.4236/jis.2014.51002
fatcat:wvansmtuy5c65bnnkjbow2ptvu
*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). ...##
###
An ECC-Based Blind Signcryption Scheme for Multiple Digital Documents

2017
*
Security and Communication Networks
*

Further, most

doi:10.1155/2017/8981606
fatcat:f2fztffbxffedgfg7gn2qhqfdi
*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*. ...##
###
Post Quantum Cryptography: Techniques, Challenges, Standardization, and Directions for Future Research
[article]

2022
*
arXiv
*
pre-print

The security

arXiv:2202.02826v1
fatcat:ghq4lqiu4rf5vpuv6gbk7ws4jm
*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. ...##
###
Hidden Pairings and Trapdoor DDH Groups
[chapter]

2006
*
Lecture Notes in Computer Science
*

Some possible cryptographic applications

doi:10.1007/11792086_31
fatcat:mydusgowcvf5nnkmuqvec6adm4
*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. ...##
###
The advantages of elliptic curve cryptography for wireless security

2004
*
IEEE wireless communications
*

However, using an

doi:10.1109/mwc.2004.1269719
fatcat:rdei23ylqnbypl2yw3572erbhi
*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. ...##
###
The Multivariate Hidden Number Problem
[chapter]

2015
*
Lecture Notes in Computer Science
*

The second considers a model that allows changing representations, where we show

doi:10.1007/978-3-319-17470-9_15
fatcat:z2ranc2yrjf7pdproy2slygh3y
*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. ...##
###
FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

2008
*
Journal of systems architecture
*

In this thesis, I propose the FPGA implementations

doi:10.1016/j.sysarc.2008.04.012
fatcat:3tlovro4lbfajgk3v2ffvheydu
*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). ...##
###
A Survey of Public-Key Cryptosystems

2004
*
SIAM Review
*

We give an overview

doi:10.1137/s0036144503439190
fatcat:wix7yrhdnzaslooaugb5ttrt7y
*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. ...
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