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Algebraic Attacks over GF(2 k ), Application to HFE Challenge 2 and Sflash-v2 [chapter]

Nicolas T. Courtois
2004 Lecture Notes in Computer Science  
Similarly, the fastest alleged algebraic attack on AES due to Courtois, Pieprzyk, Murphy and Robshaw uses a MQ system over GF (2 8 ).  ...  For example Sflash, the fastest known signature scheme (cf. [1]), is based on MQ equations over GF (2 7 ), and Patarin's 500 $ HFE Challenge 2 is over GF (2 4 ).  ...  We showed that for systems of low degree equations over fields GF (2 k ), it is not hard to avoid additional solutions in the algebraic extension or at infinity, that would make algebraic attacks fail.  ... 
doi:10.1007/978-3-540-24632-9_15 fatcat:iolptwuuxraglhhkgztmkhzzhq

On Serre's conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

Chandrashekhar Khare, Jean-Pierre Wintenberger
2009 Annals of Mathematics  
We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations ρ of the absolute Galois group of Q.  ...  For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that a ρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function.  ...  Then there is a totally real field F that is Galois over Q of even degree, unramified above p, and even split above p if ρ| Dp is irreducible, im(ρ) = im(ρ| GF ), and ρ| GF (µp) absolutely irreducible  ... 
doi:10.4007/annals.2009.169.229 fatcat:g6n7dybrk5hklfrj3er27jcrsq

Two Alerts for Design of Certain Stream Ciphers: Trapped LFSR and Weak Resilient Function over GF(q) [chapter]

Paul Camion, Miodrag J. Mihaljević, Hideki Imai
2003 Lecture Notes in Computer Science  
It is shown that although certain keystream generators over GF(q) are resistant against correlation and linear complexity based attacks, they are vulnerable by some novel attacks.  ...  GF(q) proposed at CRYPTO'96.  ...  In a general case the combination generators can be constructed over GF(q), q > 2. This assumes employment of LFSRs over GF(q), as well as combining functions over GF(q).  ... 
doi:10.1007/3-540-36492-7_14 fatcat:xcd3ezmpyvfxxnfduj5xhijzoy

Geometry of 𝑞 and 𝑞,𝑡-analogs in combinatorial enumeration [chapter]

Mark Haiman, Alexander Woo
2007 Geometric Combinatorics  
We have a corresponding action on Z by g · (X, F • ) := (gXg −1 , gF • ), so the fibers of π over points in the same G orbit are isomorphic. Let Y µ = π −1 (P ) for some point P ∈ O µ .  ...  Now use the notion of attack defined earlier to define twt(T ) = #{(c, c )|c, c ∈ λ, T (c) > T (c ), and c attacks c }. Theorem 11.  ... 
doi:10.1090/pcms/013/05 fatcat:ogrls2wzg5ar5jvo7qwicilmh4

Secure Generators of q-valued Pseudo-Random Sequences on Arithmetic Polynomials [article]

Oleg Finko and Sergey Dichenko and Dmitry Samoylenko
2018 arXiv   pre-print
A technique for controlling errors in the functioning of nodes for the formation of q-valued pseudo-random sequences (PRS) operating under both random errors and errors generated through intentional attack  ...  by an attacker is provided, in which systems of characteristic equations are realized by arithmetic polynomials that allow the calculation process to be parallelized and, in turn, allow the use of redundant  ...  Fig. 2 shows main types of modification of PRS over the GF(q) field.  ... 
arXiv:1809.02452v1 fatcat:j3xg62vtynhmrgdric5zzzcgmq

Generic and {$q$}-rational representation theory

Edward Cline, Brian Parshall, Leonard Scott
1999 Publications of the Research Institute for Mathematical Sciences  
First, we present an explicipt Morita equivalence connecting GL n (q] with the theory of g-Schur algebras, extending a unipotent block equivalence of Takeuchi [T].  ...  Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of finite and algebraic groups, and we formalize the notion of a "  ...  -Schur algebras over the ring TL [q, q" 1 ] of Laurent polynomials.  ... 
doi:10.2977/prims/1195144189 fatcat:xoetegfnhzhavargjje7wfo3l4

Class fields of abelian extensions of Q

B. Mazur, A. Wiles
1984 Inventiones Mathematicae  
Let R be a nontrivial local Z p-algebra of finite cardinality with maximal ideal denoted m R . Then the following conditions are equivalent.  ...  The value of g(c/a) is seen to lie in (gF [Pp] " It is easy to show that g (c/a)" is in (gF and furthermore lies in WE.  ...  There is a natural isogeny of abelian varieties over k: ,q ~lgusa(p";m).~ av {Pic~ H)/x) } ~ 11 pic~ gF i Hgi~B j i=1 w 8. Examples 1. The canonical model Of Xl(ap")/zpt~,j.  ... 
doi:10.1007/bf01388599 fatcat:7tu25wekq5atnaa7jfjv2qpopm

Excursions into Algebra and Combinatorics at q=0 [article]

Tom Denton
2011 arXiv   pre-print
We explore combinatorics associated with the degenerate Hecke algebra at q=0, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results.  ...  Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of -trivial monoids.  ...  Then the quantum Kac-Moody algebra U q associated with Cartan datum (A, P ∨ , P, Π ∨ , Π) is the associative algebra over Q(q) with the unit 1 generated by the symbols e i , f i (i ∈ I) and q h (h ∈ P  ... 
arXiv:1108.4379v1 fatcat:j4he4lbcdzdd7bw7vetpv3b7sa

Maximal subgroups of PSp6n(q) stabilizing spreads of totally isotropic planes

R.H Dye
1986 Journal of Algebra  
The projective symplectic group PSp,,(q) fixes a symplectic polarity A in (6n -1 )-dimensional projective space PG(6n -1, q) over the field GF(q) with q elements.  ...  Let K = GF(q) and L = GF(q3), and take o to be a primitive root of L.  ... 
doi:10.1016/0021-8693(86)90063-3 fatcat:cy2q56t5nbgb3fyzfa7753peju

q-Legendre transformation: partition functions and quantization of the Boltzmann constant

Artur E Ruuge, Freddy van Oystaeyen
2010 Journal of Physics A: Mathematical and Theoretical  
The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics.  ...  The latter is an infinite dimensional noncommutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of "deformations of deformations" of the Weyl algebra  ...  Define q g,f , for any monomials f, g ∈ A q , by f g = q g,f gf .  ... 
doi:10.1088/1751-8113/43/34/345203 fatcat:k7prpaum75akdk3czhqwlrgwle

Dependable and Secure Sensor Data Storage with Dynamic Integrity Assurance

Q. Wang, K. Ren, W. Lou, Y. Zhang
2009 IEEE INFOCOM 2009 - The 28th Conference on Computer Communications  
To further dynamically ensure the integrity of the distributed data shares, we then propose an efficient data integrity verification scheme exploiting the technique of algebraic signatures.  ...  Extensive security and performance analysis shows that the proposed schemes have strong resistance against various attacks and are practical for WSNs.  ...  The symbols x i can be one-byte or 2-byte words as the elements of the Galois Field GF (2 q )(e.g., q = 8, 16).  ... 
doi:10.1109/infcom.2009.5062006 dblp:conf/infocom/WangRLZ09 fatcat:xd2s65jil5bung555bcp4xmxeq

On the Complexity of the Rank Syndrome Decoding Problem

Philippe Gaborit, Olivier Ruatta, Julien Schrek
2016 IEEE Transactions on Information Theory  
For C a [n, k] code over GF (q m ) that one wants to decode for an error of rank r, the 1996 attack by Chabaud and Stern [13] has an exponential term in q (m−r)(r−1) and the 2003 attack by Ourivski and  ...  Let C be a random [n, k] rank code over GF (q m ) and let y = x + e be a received word such that x ∈ C and rank(e) = r.  ...  . , β r ), and E a matrix of size r × n over GF (q). Equivalently, writing each coordinate of β in a basis B of GF (q m ) over GF (q), the vector β can also be seen as a m × r matrix over GF (q).  ... 
doi:10.1109/tit.2015.2511786 fatcat:quynxesm5vfsvpwkfgdw73bfyi

On the complexity of the Rank Syndrome Decoding problem [article]

Philippe Gaborit and Olivier Ruatta and Julien Schrek
2013 arXiv   pre-print
In this paper we propose two new generic attacks on the Rank Syndrome Decoding (RSD) problem Let C be a random [n,k] rank code over GF(q^m) and let y=x+e be a received word such that x ∈ C and the Rank  ...  The second attack is based on a algebraic attacks: based on the theory of q-polynomials introduced by Ore we propose a new algebraic setting for the RSD problem that permits to consider equations and  ...  over the finite field GF(q m ).  ... 
arXiv:1301.1026v1 fatcat:gw4zl7fytragplpv5gmmyzcbmm

Simplifying algebraic attacks with univariate analysis

Tor Helleseth, Sondre Ronjom
2011 2011 Information Theory and Applications Workshop  
A usual way of presenting such ciphers is in terms of multivariate equations over GF(2). Another way is in terms of the trace-representation of the sequences, but still with respect to GF(2).  ...  Such a view simplifies theory on algebraic attacks on such ciphers and provides an alternative view of the Rønjom-Helleseth attack.  ...  In algebraic attacks (in univariate form) we want to determine a univariate Boolean function g(x) ∈ GF(q)[x] such that f (x)g(x) = 0 for any x ∈ GF(q) and g(x) ∈ GF(2) for any x ∈ GF(q).  ... 
doi:10.1109/ita.2011.5743578 dblp:conf/ita/HellesethR11 fatcat:37elv3fpdzg4tgdbn5hm2tmv7e

New Construction for Balanced Boolean Functions with Very High Nonlinearity

2007 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Our construction also have other desirable properties like high algebraic degree and large linear span.  ...  Proposition 1 (Sarkar and Maitra [20, Proposition 2 and 3]) Let f : GF(2) N → GF(2) be a balanced On Algebraic Attack and Our Construction Recently, algebraic attack has been applied successfully to  ...  It was found that the algebraic immunity is less than the optimal value n/2. In the same paper, they also measured the algebraic immunity of the trace of popular power functions over GF(2 n ).  ... 
doi:10.1093/ietfec/e90-a.1.29 fatcat:deebndzngzgx5cbw5jnrrpx4v4
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