scholarly journals The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

2010 ◽  
Vol 3 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Finite Fields ◽  
Correspondence Principle ◽  
Gowers Norm

scholarly journals True complexity of polynomial progressions in finite fields

2021 ◽  
pp. 1-53
Author(s):  
Borys Kuca
Keyword(s):  
Finite Fields ◽  
Linear Forms ◽  
Gowers Norm ◽  
Counting Lemma

Abstract The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls the counting operator of the progression over finite fields of large characteristic. We give a conjecture that relates true complexity to algebraic relations between the terms of the progression, and we prove it for a number of progressions, including $x, x+y, x+y^{2}, x+y+y^{2}$ and $x, x+y, x+2y, x+y^{2}$ . As a corollary, we prove an asymptotic for the count of certain progressions of complexity 1 in subsets of finite fields. In the process, we obtain an equidistribution result for certain polynomial progressions, analogous to the counting lemma for systems of linear forms proved by Green and Tao.

scholarly journals The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

2011 ◽  
Vol 16 (1) ◽  
pp. 121-188 ◽  
Author(s):  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Finite Fields ◽  
Gowers Norm

scholarly journals Ergodic theorem involving additive and multiplicative groups of a field and patterns

2015 ◽  
Vol 37 (3) ◽  
pp. 673-692 ◽  
Author(s):  
VITALY BERGELSON ◽  
JOEL MOREIRA
Keyword(s):  
Finite Field ◽  
Ergodic Theorem ◽  
Finite Fields ◽  
Correspondence Principle ◽  
Combinatorial Problems ◽  
Alternative Proof ◽  
Large Set ◽  
Sidon Sets ◽  
New Variant ◽  
Multiplicative Groups

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Notes on three-pass protocol

2020 ◽  
Vol 25 (4) ◽  
pp. 4-9
Author(s):  
Yerzhan R. Baissalov ◽  
Ulan Dauyl
Keyword(s):  
Finite Fields ◽  
Matrix Algebras ◽  
Associative Structures

The article discusses primitive, linear three-pass protocols, as well as three-pass protocols on associative structures. The linear three-pass protocols over finite fields and the three-pass protocols based on matrix algebras are shown to be cryptographically weak.

About the Complexity of Square Root Extraction Algorithms in Finite Fields and Residue Rings

Vestnik MEI ◽  
2018 ◽  
Vol 5 (5) ◽  
pp. 79-88
Author(s):  
Sergey B. Gashkov ◽  
◽  
Aleksandr B. Frolov ◽  
Elizaveta Р. Popova ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Square Root ◽  
Root Extraction
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