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A New Provably Secure Cryptosystem Using Dedekind Domain Direct Product Approach

2018
*
Ratio Mathematica
*

Our proposed scheme is an improvement

doi:10.23755/rm.v34i0.404
doaj:83ab2c7b067f4eb89c721357d721aa95
fatcat:fdf6dduxabgrlcakxf4at6kmhm
*of*ETRU*cryptosystem*. ETRU is an NTRU-Like*public*-*key**cryptosystem*based on the Eisenstein integers Z [f_3 ] where f_3 is a*primitive*cube*root**of**unity*. ... In this paper, we present a new NTRU-Like*public*-*key**cryptosystem**with*security provably based on the worst-case hardness*of*the approximate lattice problems (NP-hard problems) in some structured lattices ... n+1-th*root**of**unity*modulo (q1, q2, q3, q4) component-wise. q R R q R R q R R q R =´´´. ...##
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Using Primitive Subgroups to Do More with Fewer Bits
[chapter]

2004
*
Lecture Notes in Computer Science
*

This paper gives a survey

doi:10.1007/978-3-540-24847-7_2
fatcat:rswga3wcora57kw2agp36obaie
*of*some ways to improve the efficiency*of*discrete log-based cryptography by using the restriction*of*scalars and the geometry and arithmetic*of*algebraic tori and abelian varieties ... (The polynomial Φ n (x) has integer coefficients, and its (complex)*roots*are the*primitive*n-th*roots**of**unity*.) ... These conditions ensure that F q (ζ r ) contains F q 6 , where ζ r is a*primitive*r-th*root**of**unity*. (Note that if the order*of*q (mod r) is 6, then Φ 6 (q) is divisible by 6, so is not prime. ...##
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Number Theoretic Transforms for Secure Signal Processing
[article]

2018
*
arXiv
*
pre-print

This usage

arXiv:1607.05229v2
fatcat:mrlorcxrrrg7lbfr7khn7mr2xq
*of*NTTs paired*with*appropriate signal pre- and post-coding enables a whole range*of*easily composable signal processing operations comprising, among others, filtering, generalized convolutions ... This work presents a novel and comprehensive set*of*approaches and*primitives*to efficiently process signals in an encrypted form, by using Number Theoretic Transforms (NTTs) in innovative ways. ...*Of*course, when the different*powers**of*the N -th*root**of**unity*are known, φ(N ) is disclosed. ...##
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Two Generic Constructions of Probabilistic Cryptosystems and Their Applications
[chapter]

2008
*
Lecture Notes in Computer Science
*

In this paper, we build, in a generic way, two asymmetric

doi:10.1007/978-3-540-85855-3_7
fatcat:rjdnj3msd5bzneoc5mznxomrne
*cryptosystems**with*a careful study*of*their security. ... By implementing these constructions*with*quotients*of*Z, elliptic curves and quadratic fields quotients we get some*cryptosystems*yet described in the past few years and provide variants that achieve higher ... The decomposition*of*an element*of*G in a product*of*a k th*root**of**unity*by a k th*power*is less obvious, unless one knows the values*of*λ and k. ...##
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Cryptography: A Comparison of Public Key Systems

2013
*
Algorithms Research
*

Among the

doi:10.5923/j.algorithms.20120105.01
fatcat:67b3nojgznhdvktslc7cx6dpr4
*cryptosystems*currently in use, the*public**key**cryptosystems*have a special place. ... Here we provide an overview*of*different*cryptosystems*, including but not limited to an examination and comparison*of*five different influential*public**key**cryptosystems*. ... Andrew Twizell would like to acknowledge the help and support*of*Dr. Vassilev during the work on this paper. ...##
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A More Compact Representation of XTR Cryptosystem

2008
*
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
*

Therefore, the proposed variant

doi:10.1093/ietfec/e91-a.10.2843
fatcat:o57gxgybprcphbfr26hekydt44
*of*XTR*cryptosystem*is one*of*the fastest*public*-*key**cryptosystems**with*the smallest compression ratio. ... XTR is one*of*the most efficient*public*-*key**cryptosystems*that allow us to compress the communication bandwidth*of*their ciphertext. ... Acknowledgements The work reported in this paper was supported by the IT R&D program*of*MIC/IITA. [2005-S088-04, Development*of*Security technology for Secure RFID/USN Service]. ...##
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Page 6664 of Mathematical Reviews Vol. , Issue 99j
[page]

1999
*
Mathematical Reviews
*

The paper concerns the explicit computation

*of*some Jacobi sums that may be useful for*public**key**cryptosystems*. ... If r = 2n + 1 is an odd integer and # is a*primitive*rth*root**of**unity*over F, we denote the Gauss period B+B-' EF o> by a,. ...##
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Page 4542 of Mathematical Reviews Vol. , Issue 2000f
[page]

2000
*
Mathematical Reviews
*

For two linear shift-register sequences (a;);cn, and (b;);en, over GF(p) and the

*primitive*complex pth*root**of**unity*¢ := e*/?) ... Here we present a*public*-*key**cryptosystem*based on similar ideas, but*with*much less data expansion.” {For the entire collection see MR 2000e:94041.} ...##
###
Towards Efficient Arithmetic for Lattice-Based Cryptography on Reconfigurable Hardware
[chapter]

2012
*
Lecture Notes in Computer Science
*

We give instantiations

doi:10.1007/978-3-642-33481-8_8
fatcat:53fo5qtatbe4nifvlaku7nxlvm
*of*recently proposed parameter sets for homomorphic and*public*-*key*encryption. ... In practice, ideal lattices defined in the polynomial ring Zp[x]/ x n + 1 allow the reduction*of*the generally very large*key*sizes*of*lattice constructions. ... Let ω be a*primitive*2n-th*root**of**unity*in Z p . ...##
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An efficient probabilistic public-key cryptosystem over quadratic fields quotients

2007
*
Finite Fields and Their Applications
*

This system appears to be an alternative to schemes based on the RSA

doi:10.1016/j.ffa.2006.05.004
fatcat:5wweofxi2nhilg5qbebkyxuwlq
*primitive*and has a full computational cost smaller than the El Gamal EC*cryptosystem*. ... We present a new probabilistic*cryptosystem*working in quadratic fields quotients. Computation in such objects can be done efficiently*with*Lucas sequences which help to design a fast system. ... We summarize the computational costs and*keys*sizes*of*these*cryptosystems**with*these*key*length values in Table 2 . ...##
###
New number-theoretic cryptographic primitives

2019
*
Journal of Mathematical Cryptology
*

The generalized signature schemes replace the Jacobi symbol

doi:10.1515/jmc-2019-0035
fatcat:4qn7el77gnbh5kdfhm3bfpakji
*with*higher-*power*residue symbols. ... Given*of*their very unique design, the proposed signature schemes seem to be overlooked "missing species" in the corpus*of*known signature algorithms. ... If ζ r is an r-th*primitive**root**of**unity*, the ring ℤ[ζ r ] is not necessarily norm-Euclidean. ...##
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Spectral modular arithmetic for binary extension fields

2011
*
2011 IEEE 3rd International Conference on Communication Software and Networks
*

*With*some careful parameter selection, the method yields highly parallel architectures for operations involving several field multiplications such as the scalar multiplication calculation

*of*elliptic curve ... We describe a method

*of*carrying multiplication in the binary extension fields. ... Notation 1: Assume that ω is a principal d-th

*root*

*of*

*unity*; we let γ(t) be the spectral polynomial

*with*coefficients consists

*of*the negative

*powers*

*of*ω i.e. ...

##
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On the discrete logarithm problem
[article]

2008
*
arXiv
*
pre-print

Let p>2 be prime and g a

arXiv:0811.4182v1
fatcat:wawxkl6sgvdulmy6r44lhxpgpm
*primitive**root*modulo p. ... We present an argument for the fact that discrete logarithms*of*the numbers in any arithmetic progression are uniformly distributed in [1,p] and raise some questions on the subject. ... Then,*with*the development*of*cryptography, their importance raised considerably, especially after Diffie and Hellman [1] created the*key*exchange algorithm, the first practical*public**key**cryptosystem*...##
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Non-invertible key exchange protocol

2020
*
SN Applied Sciences
*

The

doi:10.1007/s42452-020-2791-3
fatcat:u3owvhttb5du5cbmx5qzuqex6y
*key*exchange protocol introduced by Diffie and Hellman [8]. The asymmetric*key*encryption algorithm for*public*-*key*cryptography*of*Taher ElGamal [9]. ... The protocol demonstrates minimum required time for encryption/ decryption processes when is compared*with*the main*public**key*algorithms as Diffie-Hellman, ElGamal or RSA. ... This is so because the prime integer r is chosen such that 2 is a*primitive**root**of*r. Fortunately, the condition p = q = 2 produces acceptable lengths*of*the*public*and private*keys*. ...##
###
Page 1664 of Mathematical Reviews Vol. , Issue 88c
[page]

1988
*
Mathematical Reviews
*

The Zetterberg codes have length n = 2” + 1, m even, and a generator polynomial g(x) which is the minimal polynomial

*of*a*primitive**root**of**unity*in GF(2?"). ... These results bear on the reliability*of*the Solovay-Strassen fast Monte Carlo test for primality and*of**public**key**cryptosystems*. 94B Algebraic theory*of*error-correcting codes See also 94018. 88c:94021 ...
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