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A New Provably Secure Cryptosystem Using Dedekind Domain Direct Product Approach

Amir Hassani Karbasi
2018 Ratio Mathematica  
Our proposed scheme is an improvement 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 =´´´.  ... 
doi:10.23755/rm.v34i0.404 doaj:83ab2c7b067f4eb89c721357d721aa95 fatcat:fdf6dduxabgrlcakxf4at6kmhm

Using Primitive Subgroups to Do More with Fewer Bits [chapter]

K. Rubin, A. Silverberg
2004 Lecture Notes in Computer Science  
This paper gives a survey 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.  ... 
doi:10.1007/978-3-540-24847-7_2 fatcat:rswga3wcora57kw2agp36obaie

Number Theoretic Transforms for Secure Signal Processing [article]

Alberto Pedrouzo-Ulloa, Juan Ramón Troncoso-Pastoriza, Fernando Pérez-González
2018 arXiv   pre-print
This usage 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.  ... 
arXiv:1607.05229v2 fatcat:mrlorcxrrrg7lbfr7khn7mr2xq

Two Generic Constructions of Probabilistic Cryptosystems and Their Applications [chapter]

Guilhem Castagnos
2008 Lecture Notes in Computer Science  
In this paper, we build, in a generic way, two asymmetric 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.  ... 
doi:10.1007/978-3-540-85855-3_7 fatcat:rjdnj3msd5bzneoc5mznxomrne

Cryptography: A Comparison of Public Key Systems

Tzvetalin S. Vassilev, Andrew Twizell
2013 Algorithms Research  
Among the 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.  ... 
doi:10.5923/j.algorithms.20120105.01 fatcat:67b3nojgznhdvktslc7cx6dpr4

A More Compact Representation of XTR Cryptosystem

2008 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Therefore, the proposed variant 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].  ... 
doi:10.1093/ietfec/e91-a.10.2843 fatcat:o57gxgybprcphbfr26hekydt44

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,.  ... 

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]

Thomas Pöppelmann, Tim Güneysu
2012 Lecture Notes in Computer Science  
We give instantiations 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 .  ... 
doi:10.1007/978-3-642-33481-8_8 fatcat:53fo5qtatbe4nifvlaku7nxlvm

An efficient probabilistic public-key cryptosystem over quadratic fields quotients

Guilhem Castagnos
2007 Finite Fields and Their Applications  
This system appears to be an alternative to schemes based on the RSA 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 .  ... 
doi:10.1016/j.ffa.2006.05.004 fatcat:5wweofxi2nhilg5qbebkyxuwlq

New number-theoretic cryptographic primitives

Éric Brier, Houda Ferradi, Marc Joye, David Naccache
2019 Journal of Mathematical Cryptology  
The generalized signature schemes replace the Jacobi symbol 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.  ... 
doi:10.1515/jmc-2019-0035 fatcat:4qn7el77gnbh5kdfhm3bfpakji

Spectral modular arithmetic for binary extension fields

Gokay Saldamli, Yoo-Jin Baek, Cetin Kaya Koc
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.  ... 
doi:10.1109/iccsn.2011.6014061 fatcat:6nyjlkahzjffnmckbmbfsuo6ie

On the discrete logarithm problem [article]

Cristian Cobeli
2008 arXiv   pre-print
Let p>2 be prime and g a 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  ... 
arXiv:0811.4182v1 fatcat:wawxkl6sgvdulmy6r44lhxpgpm

Non-invertible key exchange protocol

Luis Adrian Lizama-Perez
2020 SN Applied Sciences  
The 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.  ... 
doi:10.1007/s42452-020-2791-3 fatcat:u3owvhttb5du5cbmx5qzuqex6y

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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