​This question asks about using the Sage functionality for computing in Finite Fields.​

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erqql7766

Computer Science

ISOL 535

University of the Cumberlands

ISOL

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  1. Use Sage to create a finite field with 17 elements
    In this field calculate:
    The difference: 13 – 16
    The sum: 11 + 10
    The quotient: 1/2
    The product: 3 * 8
    The multiplicative inverse of: 5
  2. Use Sage to create a finite field with 32 elements.  Let 'a' denote the primitive element.
    In this field Calculate:
    The difference: (a^2 + a) - (a + 1)
    The multiplicative inverse of: a^4 + a + 1
    The quotient (a^2 + 1)/(a^4 + a + 1)
  3. Use Sage to create a finite field with 5^3 elements.  Let 'alpha' denote the primitive element.
    In this field Calculate:
    The sum: (3*alpha^2 + 4*alpha) - (alpha^2 + 3)
    The multiplicative inverse of: (alpha + 1)
    The product: (alpha + 2)*(alpha + 3)
  4. Use sage to create a finite field with 503,777,509 elements.
    In this field calculate:
    The quotient: 123,456,789/456,555,333
    The multiplicative inverse of : 987,654,321
    The difference: 789,123,456 - 444,333,111

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Sage includes robust support for working with finite fields and performing finite field arithmetic. To initialize a finite field with prime order use the GF command passing the order as the parameter. sage: F = GF(2) sage: F Finite Field of size 2 sage: F = GF(37) sage: F Finite Field of size 37 sage: p = 95131 sage: K = GF(p) sage: K Finite Field of size 95131 To initialize a field with a prime power order use the GF command with the following syntax (to keep track of the primitive element of the extension field.) sage: F. = GF(128) sage: F Finite Field in a of size 2^7 To do arithmetic in finite fields use the following syntax: sage: sage: sage: sage: 22 sage: 21 sage: 17 sage: 31 sage: 25 sage: 25 K a b a = = = - GF(37) K(3) K(18) b a + b a * b a/b a^-1 1/a To do arithmetic in a finite field with a prime power order, specify elements using the primitive element: sage: sage: sage: sage: a^5 + sage: a^5 + sage: a^3 + sage: a^5 + sage: a^5 + sage: a^5 + F. = GF(128) b = a^2 + 1 c = a^5 + a^3 + 1 b - c a^3 + a^2 b + c a^3 + a^2 b*c a^2 + a b/c a^3 + a^2 + a b^-1 a^3 + a 1/b a^3 + a
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QUESTIONS
Question 1
Using the GF command
A finite field with 17 elements
Sage: F = GF (17)
Sage: F
The difference: 13-16 = 40
The sum: 11+ 10= 21
The quotient 1/2= 12
The produc...


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