This document explains the concept of LSFR and GCD calculation with solved examples. It covers the definition of LSFR and GCD, their equations, and how to calculate them. The document also includes a bibliography of sources for further reading.
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Surname1 Student’s Name Professor’s Name Subject Date Question 1 a) (1, 7)(0, 2) x1=1y1=0 x2=7y2=2 Calculating the linear equation (y-y1) =y2−x1 x2−x1 (x-x1) y-0 =2−0 7−1(x-1) y=2 6(x-1) y=1 3(x-1) b) (1, 6)(0, 1) x1=1y1=0 x2=6y2=1 Calculating the linear equation (y-y1) =y2−x1 x2−x1 (x-x1)
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Surname2 y-0 =1−0 6−1(x-1) y=1 5(x-1) y=1 5(x-1) Question 1 a) LSFR is defined as the element of pseudo random generator which generates a set of encryption keys (Kim 3). The general formula LSFR are easily generalized in any finite model; GF (p). LSFR is easily implemented in software and hardware thus they are widely applied in stream cipher (Gómez 4;Denning 42). The LFSR equation:xn+5= xn+ xn+3 When x=0, then X0+5= x0+ x0+3 =0+0 = 0 When x=1, then X1+5= x1+ x1+3 =1+0 = 1 When x=2, then X2+5= x2+ x2+3 =0+0 = 0 When x=3, then X3+5= x3+ x2+3 =0 = 1 When x=4, then X4+5= x4+ x4+3 =0+0 = 0 Thus the LSFR sequence becomes: 0, 1, 0, 1,0…… Therefore, the first 20 bits are: 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1 b) The first 20 bit follows a PN-sequence by investigation. The PN-sequences a period generalized using the formula2n−1. Therefore, the period of LSFR is2n−1. Question 2 Definition
Surname3 Let “a” and “b” be the integers and non-zero. The GCD of a and b, written as the gcd(a,b) , is the integer “d” with the two main properties : i)d is a common divisor of a and b i.e d|a and d|b ii)Ѱ of c, if c is a common divisor of both a and b, then c≤d. In other words, for all integers c, if c|a and c|b, then c≤d. Solution Integers a and b are called relatively prime iff their GCD is 1 (Bhandari 33). Given two numbers;4883 and 4369 We have to use the extended Euclidean algorithm to find the GCD of4883 and 4369 and express them as a linear combination. Step 1: Calculating the GCD 4883= 1(4369) +514 Thus 514 = 4883-4369 4883 4369= 1 rem 514 But 4369 514= 8 rem 257 And, 4883 514=9 rem 257 Thus 514 257= 2 rem 0 Since the remainder is 0, the greatest common divisor is the divisor, 257
Surname4 Therefore, GCD = 257 Step 2: Expressing GCD as a linear combination 257 = -8(4883) +9(4369) Therefore, the greatest common divisor of4883 and 4369 is 257 Number 4 a) Let n= 391 Factorizing 391 gives; 391= 17*23 n-1 = 390 Φ (n) = Φ (17) Φ = (23) = 16*22 = 352 2390=2352,238 29=512 thus29= 512(mod391) 238≡285 (mid391) Fermat Theorem with a=2, n=391ged (a, n) =1 So, aΦ(n)= 1(mod, n) 2Φ(n)= 1(mod, n)⇒2352≡≡1(mod391) 2391−1=2390=2352,238 Thus 2391−1≡2352=2352,238(mod391) 2390≡238(mod391)
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Surname5 2391≡238(mod391)⇒2285(mod391) b) 2j≡1 (mod, n) Applying the Fermat theorem J =Φ(n) =Φ(391) = 352 thus J = 352 Work Cited Bhandari, Sandeepak. "A New Era Of Cryptography : Quantum Cryptography".International Journal On Cryptography And Information Security, vol 6, no. 3/4, 2016, pp. 31- 37.Academy And Industry Research Collaboration Center (AIRCC), doi:10.5121/ijcis.2016.6403. D. Denning. “Cryptography and Data Security Addison-Wesley Publishing Company”. 2013. Gómez, Pardo J. L.Introduction to Cryptography with Maple. Heidelberg: Springer-Verlag, 2013. Internet resource. Kim, Kwangjo. "Cryptography: A New Open Access Journal".Cryptography, vol 1, no. 1, 2016, p. 1.MDPI AG, doi:10.3390/cryptography1010001.