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

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numbers. The set G of generators of M is quite sparse, and its members seem related tonumbers of the form Φ n (2).The distribution of the values of r among the set of all odd primes p leads to ageneralization of Artin’s Conjecture concerning primitive roots modulo p. We generalizedthe conjectures of Blake, Gao and Lambert, and of Tromp, Zhang and Zhao, and provedthe generalization of the Blake, Gao and Lambert conjecture.ix
Chapter 1Introduction1.1 MotivationsLinear feedback shift register sequences have been used in a variety of important applications,such as CDMA (Code Division Multiple Access) communications, spread spectrumcommunications, CW radar, bit error rate measurements, error correcting codes, andstream ciphers. There are two main advantages for using linear feedback shift registersequences: they are extremely fast and easy to implement both in hardware and software,and they can be readily analyzed using algebraic techniques. A thorough introductionto the theory of shift register sequences is in the book by Golomb [5].A typical n-stage binary linear feedback shift register (known as a Fibonacci configuration)is illustrated in Figure 1.1. Both the feedback coefficientsc 1 , c 2 , . . . , c n1
Page 2: IRREDUCIBLE POLYNOMIALS WHICH DIVID
Page 6 and 7: Table of ContentsDedicationAcknowle
Page 8 and 9: List of Figures1.1 The n-stage bina
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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