MACM 401/MATH 801/CMPT 891

Assignment 5

数学算法代写 Question 1 Factorization in Zp[x] (20 marks) (a) Factor the following polynomials over Z11 using the Cantor-Zassenhaus algorithm.

Question 1

Factorization in Zp[x] (20 marks)

(a) Factor the following polynomials over Z11 using the Cantor-Zassenhaus algorithm.

a₁= x⁴ + 8x² + 6x + 8,

a₂ = x⁶ + 3x⁵ − x⁴ + 2x³ − 3x + 3,

a₃ = x⁸ + x⁷ + x⁶ + 2x⁴ + 5x³ + 2x² + 8.

Use Maple to do all polynomial arithmetic, that is, you can use the Gcd(…) mod p and Powmod(…) mod p commands etc., but not Factor(…) mod p.

(b) Compute the square-roots of the integers a = 3, 5, 7 in the integers modulo p, if they exist, for p = 10²⁰ + 129 = 100000000000000000129 by factoring the polynomial x² − a in Zp[x] using the Cantor-Zassenhaus algorithm. Show your working. You will have to use Powmod here.

Question 2  数学算法代写

Factorization in Z[x] (25 marks)

Factor the following polynomials in Z[x].

a₁ = x¹⁰− 6x⁴ + 3x² + 13

a₂ = 8x⁷ + 12x⁶ + 22x⁵ + 25x⁴ + 84x³ + 110x² + 54x + 9

a₃ = 9x⁷ + 6x⁶ − 12x⁵ + 14x⁴ + 15x³ + 2x² − 3x + 14

a₄ = x¹¹ + 2x ¹⁰ + 3x⁹ − 10x⁸ − x⁷ − 2x⁶ + 16x⁴ + 26x³+ 4x² + 51x − 170

For each polynomial, fifirst compute its square free factorization. You may use the Maple command gcd(…) to do this. Now factor each non-linear square-free factor as follows. Use the Maple command Factor(…) mod p to factor the square-free factors over Z_p modulo the primes p = 13, 17, 19, 23. From this information, determine whether each polynomial is irreducible over Z or not. If not irreducible, try to discover what the irreducible factors are by considering combinations of the modular factors and Chinese remaindering (if necessary) and trial division over Z.

Using Chinese remaindering here is not effiffifficient in general. Why?

Thus for the polynomial a₄, use Hensel lifting instead; using a prime of your choice from 13, 17, 19, 23, Hensel lift each factor mod p, then determine the irreducible factorization of a₄ over Z.

Question 3  

The linear x-adic Newton iteration (15 marks)

XadicSqrt := proc(a,x::name,p::prime)
# Input a(x) in Zp[x]
# Output sqrt(a) if it is in Zp[x] else FAIL
local a0,u0,u,i,k,d,m,e,uk;
if a=0 then return 0; fi;
d := degree(a,x);
if irem(d,2) <> 0 then return FAIL fi;
a0 := coeff(a,x,0);
if a0 = 0 then error “not implemented”; fi;
u0 := numtheory[msqrt](a0,p);
if u0 = FAIL then return FAIL; fi;
i := modp(1/2/u0,p);
u := u0;
e := Expand( a-u^2 ) mod p;
for k from 1 to d/2 do
uk := coeff(e,x,k)*i mod p;
u := u + uk*x^k;
e := Expand( a-u^2 ) mod p;
od;
if e = 0 then u else FAIL fi;
end:

The algorithm is correct but not effiffifficient. Let us fifind out why and then fifix it.  数学算法代写 Let deg a = d and let T(d) be the number of arithmetic operations algorithm XadicLift does in Zp. Assuming the polynomial multiplication u2 uses a classical quadratic algorithm, show that T(d) is cubic in d. You are to modify the computation of the error so that that T(d) ∈ O(d² ). You must show that T(d) ∈ O(d² ). Implement your modifified algorithm in Maple – just modify my code appropriately. Finally make your Maple code work for inputs where a0 = 0. Test your code on the following inputs for p = 11.

a = (9x³ + 3x² + 5x + 6)² , a = x³ + (9x³ + 3x²  + 5x + 6)2 , a = (9x³  + 3x²  + 5x)² .

Hint: The error e_k+1 = a − (u^（k+1）) ²  = a − (u^（k） + u_kx^k)² .

Use the error e_k = a − u^（k）2 from the previous iteration to calculate e_k+1 faster.

Question 4  数学算法代写

Integration and Difffferentiation in Maple (10 marks)

(a) You were probably taught that the derivative of tan x is sec² x. Difffferentiate tan x in Maple. Now use Maple to show that Maple’s answer equals sec² x.

Question 5  数学算法代写

Symbolic Integration (15 marks)

Implement a Maple procedure INT (you may use Int if you prefer) that evaluates antiderivatives ∫f(x)dx. For a constant c and positive integer n your Maple procedure should apply

You may ignore the constant of integration. NOTE: e^x in Maple is  exp(x), i.e. it’s a function not a power. HINT: use the diff command for difffferentiation to determine if a Maple expression is a constant wrt x. Test your program on the following.

> INT( 2*f(x) + 3*y*x/2 + 3*ln(2), x );
> INT( x^2*exp(x) + 2*x*exp(x), x );
> INT( 4*x^3*ln(x), x );
> INT( 2*exp(-x) + ln(2*x+1), x );