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Random small Hamming weight products with applications to cryptography

Jeffrey Hoffstein, Joseph H. Silverman
2003 Discrete Applied Mathematics  
The underlying idea is to form a random exponent or multiplier as a product of factors, each of which has low Hamming weight when expanded as a sum of powers of some fast operation. ?  ...  We describe a fast method to compute random powers and multiples in certain important situations including powers in the Galois ÿeld F2n , multiples on Koblitz elliptic curves, and multiples in NTRU convolution  ...  Acknowledgements The authors would like to thank Don Coppersmith, Dan Gordon, Victor Miller, and an anonymous referee for their suggestions and help with the existing literature.  ... 
doi:10.1016/s0166-218x(02)00588-7 fatcat:7susqbaavjgkbmvdf4rmc5ebae

On the distribution of Low Hamming Weight products

Jianghua Li, Qiao Li
2020 Journal of Inequalities and Applications  
Math. 130:37-49, 2003) introduced the Low Hamming Weight products (LHWP) X = x 1 x 2 x 3 as random exponent of elements in a group or a ring to improve the operational efficiency, where each x i has Hamming  ...  The main purpose of this paper is using the analytic method and the properties of the character sums to prove the distribution of the Hamming weight products, which is related to their pseudorandomness  ...  Acknowledgements The authors would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. Funding  ... 
doi:10.1186/s13660-020-02321-z fatcat:dpl3bn6xijak7ffg6svpnqx5em

Efficient Encryption from Random Quasi-Cyclic Codes [article]

Carlos Aguilar, Olivier Blazy, Jean-Christophe Deneuville, Philippe Gaborit, Gilles Zémor
2016 arXiv   pre-print
Our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits.  ...  The framework is in the spirit of the schemes first proposed by Alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight.  ...  Our analysis allows us to give small parameters for code-based encryption in Hamming and Rank metrics.  ... 
arXiv:1612.05572v1 fatcat:up7iubangna3db2gd77tfi3ywe

Page 5268 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
This is done from the point of view of the cryptographer with a par- ticular eye on applications in quantum cryptography.  ...  Such sequences with almost perfect linear complexity profile, i.e. d-perfect sequences for small inte- gers d, are important in cryptography, particularly stream cipher systems.  ... 

Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic [chapter]

E. Öztürk, B. Sunar, E. Savaş
2004 Lecture Notes in Computer Science  
Our inversion algorithm exhibits superior performance to the Euclidean algorithm and lends itself to efficient hardware implementation due to its simplicity.  ...  The resulting architecture successfully utilizes redundant representation of elements in GF (p) and provides a low-power, high speed, and small footprint specialized elliptic curve implementation.  ...  Although the idea of using a low-Hamming weight integer modulus is not new [3] , its application to Elliptic Curve Cryptography was limited to only elliptic curves defined over Optimal Extension Fields  ... 
doi:10.1007/978-3-540-28632-5_7 fatcat:m434epm46bgxlmdp7it4mogxdy

The Hardness of Code Equivalence over $\mathbb{F}_q$ and Its Application to Code-Based Cryptography [chapter]

Nicolas Sendrier, Dimitris E. Simos
2013 Lecture Notes in Computer Science  
The code equivalence problem is to decide whether two linear codes over Fq are identical up to a linear isometry of the Hamming space.  ...  In this paper, we review the hardness of code equivalence over Fq due to some recent negative results and argue on the possible implications in code-based cryptography.  ...  Improved Version of Girault Identification Scheme Key Generation: Random [n, k] linear code with an (n − k) × n parity-check matrix H over F q -Private key: A word e ∈ F n q of small weight w -Public key  ... 
doi:10.1007/978-3-642-38616-9_14 fatcat:eufpaexm7nhcdbh7xxbee7iiqi

Post-Quantum and Code-Based Cryptography—Some Prospective Research Directions

Chithralekha Balamurugan, Kalpana Singh, Ganeshvani Ganesan, Muttukrishnan Rajarajan
2021 Cryptography  
Thus, cryptography research has also been evolving from the classical Caesar cipher to the modern cryptosystems, based on modular arithmetic to the contemporary cryptosystems based on quantum computing  ...  This threat triggered post-quantum cryptography research to design and develop post-quantum algorithms that can withstand quantum computing attacks.  ...  In the same year of 2018, Persichetti [96] conformed the Lyubashevsky scheme to random quasi-cyclic Hamming metric codes.  ... 
doi:10.3390/cryptography5040038 fatcat:wkb43zunwjagbnljc6brfou4ty

Product Subset Problem : Applications to number theory and cryptography [article]

K.A.Draziotis, V. Martidis, S. Tiganourias
2020 arXiv   pre-print
We consider applications of Subset Product Problem (SPP) in number theory and cryptography.  ...  In fact we provide an application to the problem of finding Carmichael numbers and an attack to Naccache-Stern knapsack cryptosystem, where we update previous results.  ...  The authors are grateful to Scientific Computing Center (SCC) of the Aristotle's University of Thessaloniki (Greece), for providing access to their computing facilities and their technical support.  ... 
arXiv:2002.07095v2 fatcat:kq56krfdcfgmziis5kmcct4oam

Primal-Dual Distance Bounds of Linear Codes With Application to Cryptography

R. Matsumoto, K. Kurosawa, T. Itoh, T. Konno, T. Uyematsu
2006 IEEE Transactions on Information Theory  
Let N(d,d^) denote the minimum length n of a linear code C with d and d^, where d is the minimum Hamming distance of C and d^ is the minimum Hamming distance of C^.  ...  Further, for small values of d and d^, we determine N(d,d^) and give a generator matrix of the optimum linear code.  ...  ACKNOWLEDGMENT The authors would like to thank the reviewers' critical comments that improved this correspondence; in particular, Theorem 12 was improved by the reviewer's comment.  ... 
doi:10.1109/tit.2006.880050 fatcat:acrhpvkvinc75ju26ec7nnuhrm

Modified pqsigRM: RM Code-based Signature Scheme

Yongwoo Lee, Wijik Lee, Young-Sik Kim, Jong-Seon No
2020 IEEE Access  
The proposed decoder samples from coset elements with small Hamming weight for any given syndrome and efficiently finds such an element.  ...  The proposed scheme has the advantage of the pqsigRM decoder and uses public codes that are more difficult to distinguish from random codes.  ...  To find a signature with small Hamming weight, the scheme in [7] uses a sparse coset element added to a codeword with small Hamming weight.  ... 
doi:10.1109/access.2020.3026989 fatcat:ozlmqazvojeznh5zh3mwcny2aq

Selected RNS Bases for Modular Multiplication

J. C. Bajard, M. Kaihara, T. Plantard
2009 2009 19th IEEE Symposium on Computer Arithmetic  
This work proposes specific sets of optimal RNS moduli with elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight.  ...  elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight.  ...  System, can be performed without any multiplication, thanks to the property of small Hamming weights of their inverses.  ... 
doi:10.1109/arith.2009.20 dblp:conf/arith/BajardKP09 fatcat:qvzcgr24fnhh5pce4ylsvnbxhm

RankSign: an efficient signature algorithm based on the rank metric [article]

Philippe Gaborit, Olivier Ruatta, Julien Schrek, Gilles Zémor
2017 arXiv   pre-print
We explain how it is possible (depending on choices of parameters) to obtain a full decoding algorithm which is able to find a preimage of reasonable rank weight for any random syndrome with a very strong  ...  Moreover the scheme can be very fast for small base fields.  ...  For the Hamming metric, this approach leads to very flat dual matrices, ie, codes with high rate and very low Hamming distance.  ... 
arXiv:1606.00629v2 fatcat:4r3pcnwlqfczdpx3y7bad7oiue

Efficient Implementation of Arithmetic Operations in ECC over Binary Fields

G.N. Purohit, Asmita Singh Rawat
2010 International Journal of Computer Applications  
We present two techniques: (i)reducing Hamming weight of scalars in binary representation and (ii) using sliding window , for obtatining scalar multiplication in a faster manner.  ...  In this paper the implementation of arithmetic operations in ECC is described.Elliptic curve cryptography is very useful in the field of the network security because of its small key size and its high  ...  Reducing Hamming Weight of Scalars The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used.  ... 
doi:10.5120/1056-1376 fatcat:5sjiddyz7vdufjoo4imwyevb4m

RankSign: An Efficient Signature Algorithm Based on the Rank Metric [chapter]

Philippe Gaborit, Olivier Ruatta, Julien Schrek, Gilles Zémor
2014 Lecture Notes in Computer Science  
We explain how it is possible (depending on choices of parameters) to obtain a full decoding algorithm which is able to find a preimage of reasonable rank weight for any random syndrome with a very strong  ...  Moreover the scheme can be very fast for small base fields.  ...  Overall the main interest of rank-metric based cryptography is that the complexity of the best known attack grows very quickly with the size of parameters: Contrary to (Hamming) code-based or to lattice-based  ... 
doi:10.1007/978-3-319-11659-4_6 fatcat:db2gbgb37nhafgbdkelp2xpuqm

Evolutionary Approach for Finding Correlation Immune Boolean Functions of Order t with Minimal Hamming Weight [chapter]

Stjepan Picek, Sylvain Guilley, Claude Carlet, Domagoj Jakobovic, Julian F. Miller
2015 Lecture Notes in Computer Science  
Our results show that such algorithms are a viable choice when evolving Boolean functions with minimal Hamming weight and certain order of correlation immunity.  ...  Therefore, various methods to construct Boolean functions with desired properties are of direct interest.  ...  Conclusion and Future Work In this paper we investigated the evolution of Boolean functions with minimal Hamming weight and various orders of the correlation immunity property.  ... 
doi:10.1007/978-3-319-26841-5_6 fatcat:mjin66zkzng4xenn6owpv5fpeu
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