Irreducible Polynomials Which Divide Trinomials Over GF(2). - The ...

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List of Figures1.1 The n-stage binary linear feedback shift register . . . . . . . . . . . . . . . 2vii
AbstractShift-register sequences, also known as pseudorandom sequences, or pseudonoise sequences,have played increasingly important roles in many important applications. Thesimplest linear feedback shift registers to generate binary sequences involve only two taps,which corresponds to a trinomial over GF(2). It is therefore of interest to know whichirreducible polynomials f(x) divide trinomials over GF(2), since the output sequencescorresponding to f(x) can be obtained from a two-tap linear feedback shift register (witha suitable initial state) if and only if f(x) divides some trinomial t(x) = x m + x a + 1 overGF(2).In this thesis we develop the theory of irreducible polynomials which do, or do not,divide trinomials over GF(2). Abundant theorems and results are presented relating tothe primitivity t and the index r, which is the number of irreducible factors of the t thcyclotomic polynomial Φ t (x) over GF(2). The set of all positive (odd) integers t suchthat the irreducible polynomials of odd primitivity t > 1 divide trinomials over GF(2)form a multiplicative module M, which is closed with respect to multiplication by oddviii
Page 2: IRREDUCIBLE POLYNOMIALS WHICH DIVID
Page 6 and 7: Table of ContentsDedicationAcknowle
Page 10 and 11: numbers. The set G of generators of
Page 12 and 13: and the initial statea −1 , a −
Page 14 and 15: for each n ≥ 1.(Here φ(n) is the
Page 16 and 17: Φ t (x) over GF(2). We show how th
Page 18 and 19: Conversely, if h(α) = 0, we can di
Page 20 and 21: Hence, if f(x) divides no trinomial
Page 22 and 23: This occurs if and only if 2 is a p
Page 24 and 25: Then for some s, 1 ≤ s ≤ t −
Page 26 and 27: This computationally useful criteri
Page 28 and 29: ut3(p − 1)/2r < pfor all r > 1, w
Page 30 and 31: Corollary 1. Let p > 3 be a prime a
Page 32 and 33: Note that when r = 2 and p = 7, nei
Page 34 and 35: Thent 1 (x)t ∗ 1(x)t 3 (x)t ∗ 3
Page 36 and 37: We may therefore reduce all exponen
Page 38 and 39: Table 2.1: Frequency distribution o
Page 40 and 41: Chapter 3The Multiplicative ModuleI
Page 42 and 43: It was mentioned that among the eig
Page 44 and 45: Table 3.1: Prime generators of the
Page 46 and 47: 60, 787 = 89 · 683 =Φ 11 (2)Φ 22
Page 48 and 49: Table 3.4: Factors of Φ n (2) vers
Page 50 and 51: A remarkable quantitative version (
Page 52 and 53: and⎧⎪⎨F 7 (r) ∼ =⎪⎩3 if
Page 54 and 55: Table 4.1: Numerical results of F 2
Page 56 and 57: Table 4.2: Numerical results of F 3
Page 58 and 59:
Table 4.3: Collections of possible
Page 60 and 61:
4.2 Generalized TZZ and BGL Conject
Page 62 and 63:
In fact, the following is known [2]
Page 64 and 65:
Since GCD(2 n − 1, v) = 1, η j
Page 66 and 67:
conjectures of Blake, Gao and Lambe
Page 68:
[13] N. Zierler and J. Brillhart, O
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Irreducible Polynomials Which Divide Trinomials Over GF(2). - The ...

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