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

2004
*
Lecture Notes in Computer Science
*

Similarly, the fastest alleged

doi:10.1007/978-3-540-24632-9_15
fatcat:iolptwuuxraglhhkgztmkhzzhq
*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. ...##
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On Serre's conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

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

doi:10.4007/annals.2009.169.229
fatcat:g6n7dybrk5hklfrj3er27jcrsq
*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 ...##
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Two Alerts for Design of Certain Stream Ciphers: Trapped LFSR and Weak Resilient Function over GF(q)
[chapter]

2003
*
Lecture Notes in Computer Science
*

It is shown that although certain keystream generators

doi:10.1007/3-540-36492-7_14
fatcat:xcd3ezmpyvfxxnfduj5xhijzoy
*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*). ...##
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Geometry of 𝑞 and 𝑞,𝑡-analogs in combinatorial enumeration
[chapter]

2007
*
Geometric Combinatorics
*

We have a corresponding action on Z by g · (X, F • ) := (gXg −1 ,

doi:10.1090/pcms/013/05
fatcat:ogrls2wzg5ar5jvo7qwicilmh4
*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. ...##
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Secure Generators of q-valued Pseudo-Random Sequences on Arithmetic Polynomials
[article]

2018
*
arXiv
*
pre-print

A technique for controlling errors in the functioning of nodes for the formation of

arXiv:1809.02452v1
fatcat:j3xg62vtynhmrgdric5zzzcgmq
*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. ...##
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Generic and {$q$}-rational representation theory

1999
*
Publications of the Research Institute for Mathematical Sciences
*

First, we present an explicipt Morita equivalence connecting GL n (

doi:10.2977/prims/1195144189
fatcat:xoetegfnhzhavargjje7wfo3l4
*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. ...##
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Class fields of abelian extensions of Q

1984
*
Inventiones Mathematicae
*

Let R be a nontrivial local Z p-

doi:10.1007/bf01388599
fatcat:7tu25wekq5atnaa7jfjv2qpopm
*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. ...##
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Excursions into Algebra and Combinatorics at q=0
[article]

2011
*
arXiv
*
pre-print

We explore combinatorics associated with the degenerate Hecke

arXiv:1108.4379v1
fatcat:j4he4lbcdzdd7bw7vetpv3b7sa
*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 ...##
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Maximal subgroups of PSp6n(q) stabilizing spreads of totally isotropic planes

1986
*
Journal of Algebra
*

The projective symplectic group PSp,,(

doi:10.1016/0021-8693(86)90063-3
fatcat:cy2q56t5nbgb3fyzfa7753peju
*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. ...##
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q-Legendre transformation: partition functions and quantization of the Boltzmann constant

2010
*
Journal of Physics A: Mathematical and Theoretical
*

The basic idea is to go from the

doi:10.1088/1751-8113/43/34/345203
fatcat:k7prpaum75akdk3czhqwlrgwle
*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*. ...##
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Dependable and Secure Sensor Data Storage with Dynamic Integrity Assurance

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

doi:10.1109/infcom.2009.5062006
dblp:conf/infocom/WangRLZ09
fatcat:xd2s65jil5bung555bcp4xmxeq
*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). ...##
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On the Complexity of the Rank Syndrome Decoding Problem

2016
*
IEEE Transactions on Information Theory
*

For C a [n, k] code

doi:10.1109/tit.2015.2511786
fatcat:quynxesm5vfsvpwkfgdw73bfyi
*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*). ...##
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On the complexity of the Rank Syndrome Decoding problem
[article]

2013
*
arXiv
*
pre-print

In this paper we propose two new generic

arXiv:1301.1026v1
fatcat:gw4zl7fytragplpv5gmmyzcbmm
*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 ). ...##
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Simplifying algebraic attacks with univariate analysis

2011
*
2011 Information Theory and Applications Workshop
*

A usual way of presenting such ciphers is in terms of multivariate equations

doi:10.1109/ita.2011.5743578
dblp:conf/ita/HellesethR11
fatcat:37elv3fpdzg4tgdbn5hm2tmv7e
*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*). ...##
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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

doi:10.1093/ietfec/e90-a.1.29
fatcat:deebndzngzgx5cbw5jnrrpx4v4
*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 ). ...
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