In GF(2 8), find the multiplicative inverse of
(x6 +x5+x2+1) modulo (x
8 + x 6 + x 5 + x 2 +
1). Use Euclidean table to show the intermediate steps
Answers
The multiplicative inverse of (x^6 + x^5 + x^2 + 1) modulo (x^8 + x^6 + x^5 + x^2 + 1) in GF(2^8) is (x^7 + x^4 - x - 1).
We are given the polynomial (x^6 + x^5 + x^2 + 1) and we want to find its multiplicative inverse modulo (x^8 + x^6 + x^5 + x^2 + 1) in GF(2^8).
Perform polynomial division
We divide the modulo polynomial by the given polynomial:
(x^8 + x^6 + x^5 + x^2 + 1) = (x^6 + x^5 + x^2 + 1)(x^2 + x) + (x^5 + x^2 + 1)
We have obtained the remainder (x^5 + x^2 + 1) and updated the polynomials.
Continue polynomial division
We divide the previous divisor (x^6 + x^5 + x^2 + 1) by the remainder:
(x^6 + x^5 + x^2 + 1) = (x^5 + x^2 + 1)(x + 1) + (x^4 + x^3 + 1)
Again, we have obtained the remainder (x^4 + x^3 + 1) and updated the polynomials.
Repeat division
We continue dividing the previous divisor by the remainder:
(x^5 + x^2 + 1) = (x^4 + x^3 + 1)(x + 1) + (x^3 + x + 1)
Once again, we have obtained the remainder (x^3 + x + 1) and updated the polynomials.
Final division
We continue dividing the previous divisor by the remainder:
(x^4 + x^3 + 1) = (x^3 + x + 1)(x + 1) + 0
At this point, the remainder is zero, and we have reached the end of the Euclidean algorithm.
Finding the inverse
Now, we need to find the Bezout coefficients to determine the inverse. We can work our way up to the given equation, replacing the remainders with the previous polynomials, as follows:
(x^4 + x^3 + 1) = (x^5 + x^2 + 1) - (x^3 + x + 1)(x + 1)
(x^3 + x + 1) = (x^6 + x^5 + x^2 + 1) - [(x^8 + x^6 + x^5 + x^2 + 1) - (x^6 + x^5 + x^2 + 1)(x^2 + x) - (x^5 + x^2 + 1)](x + 1) - (x^5 + x^2 + 1)
(x^3 + x + 1) = (x^6 + x^5 + x^2 + 1) - (x^8 + x^6 + x^5 + x^2 + 1)(x + 1) + (x^6 + x^5 + x^2 + 1)(x^2 + x)(x + 1) + (x^5 + x^2 + 1)(x + 1) - (x^5 + x^2 + 1)
Simplifying the above equation, we obtain:
(x^3 + x + 1) = x^6 + x^7 + x^4 + x^3 + 1
0 = x^7 + x^4 - x - 1
Therefore, the multiplicative inverse of (x^6 + x^5 + x^2 + 1) modulo (x^8 + x^6 + x^5 + x^2 + 1) in GF(2^8) is (x^7 + x^4 - x - 1).
The intermediate steps of the Euclidean algorithm are shown to illustrate how we arrived at the inverse.
To learn more about multiplicative inverse visit : https://brainly.com/question/1682347
#SPJ11
Related Questions
Determine the equation of the graph
Answers
I need help with this so I can pass my Algebra 1 class
Answers
Solutions for he inequalities are as follows;
9. n ≤ -6 10. a ≥ 15 11. p ≥ -17 12. v < 4
13. The slope is 1 14. The slope is 0.83
15. The slope is 0.22 16. the slope is 0.32
How do you solve the following inequalities?The solution to the following inequalities is as follows;
9. -26 ≥ 5n + 4 10. -116 ≥ 4 - 8a
5n + 4 ≤ -26 4 - 8a ≤ -116
5n ≤ -26 - 4 - 8a ≤ -116 - 4 ⇒ - 8a ≤ -120
5n ≤ -30 multiply both side by -1
Divide both sides by 5 8a ≥ 120 ⇒ a ≥ 15
n ≤ -6
11. 61 ≥ -3p + 10 12. -18 < -7v + 10
-3p + 10 ≤ 61 -7v + 10 > -18
-3p ≤ 61 - 10 -7v > -18 -10
-3p ≤ 51 -7v > -28
Divide both sides by -3 v < 4
p ≥ -17
13 To find the slope between two points (x1, y1) and (x2, y2), we use the formula: Slope = (y2 - y1) / (x2 - x1)
some of the coordinates for graph 13 are (-1, 0) (0, 1) (-2, -1) (1, 2)
When we input the values into the coordinates, its becomes
Slope between (-1, 0) and (0, 1):
Slope = (1 - 0) / (0 - (-1)) = 1 / 1 = 1
Slope between (0, 1) and (-2, -1):
Slope = (-1 - 1) / (-2 - 0) = -2 / -2 = 1
the slope for all pairs of coordinates is 1.
14. (-2, 0) (-3, -1), (3, 4)
Slope between (0, 1.5) and (-3, -1):
Slope = (-1 - 1.5) / (-3 - 0) = -2.5 / -3 ≈ 0.83
Slope between (-3, -1) and (3, 4):
Slope = (4 - (-1)) / (3 - (-3)) = 5 / 6 ≈ 0.83
15. Find the slope of the line through each pair of points
(12, -9) (-15, -15)
Using the formula Slope = (y2 - y1) / (x2 - x1), we input the values to be
Slope = (-15 - (-9)) / (-15 - 12)
= 0.22
16. (-11, -15) (11, -8)
Slope = (y2 - y1) / (x2 - x1)
For the given coordinates:
Slope between (-11, -15) and (11, -8):
Slope = (-8 - (-15)) / (11 - (-11)) = 0.32
Find more exercises on solving inequalities;
https://brainly.com/question/20383699
#SPJ1
What is the slope of the following line?
Answers
Find the are of the polygon 16m 10m 20m 8m
Answers
Answer:
Step-by-step explanation:
AABC is dilated by a factor of 1/2 to produce AA'B'C',
What is BC, the length of BC after the dilation? What is the measure of ZA?
A. B'C' = 32, m2A = 28°
B. B'C' = 17. m2A=62
C. B'C' = 8. m ZA=28
D. B'C' = 8. m ZA' = 14
Answers
Answer:
your answer will be C. B´C=8. m ZA=28
Step-by-step explanation:
hope it helps have a nice day
After dilation of ΔABC, the length of B'C' = 8m and ∠A' = 28°.
What is Dilation?Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor.
When the triangle undergoes dilation, the length of its sides gets multiplied by 1/2 (dilation factor) but the angles remain the same ( since, the shape has to remain same).
Learn more about dilation here,
https://brainly.com/question/13176891
#SPJ2
Number graph ranging from negative eight to eight on the x axis and negative eight to eight on the y axis. Two lines that are perpendicular to each other intersect at (three, zero). The line with a positive slope is labeled a and the line with a negative slope is labeled b.
Use the graph to state the solution for the system.
x – y = 3 (line a)
x + y = 3 (line b)
Responses
(0, 3)
(0, 3)
(3, 0)
(3, 0)
(–3, 0)
(–3, 0)
(0, –3)
(0, –3)
Answers
The solution for the system is (0,3) as line a meets the y-axis at (0, -3) and the Intersection point of a and b is (3,0). This makes it obvious that the solution for x will be 0 when the solution for y is 3
Given that the number graph ranges from negative eight to eight on the x-axis and negative eight to eight on the y-axis.
Two lines that are perpendicular to each other intersect at (three, zero). The line with a positive slope is labeled a, and the line with a negative slope is labeled b.
The system is given byx – y = 3 (line a)x + y = 3 (line b)We know that both the lines a and b pass through (3,0).
Let's first find the points of intersection of the line a with the x-axis and the y-axis.
To find the point of intersection of the line a with the x-axis, put y = 0 in the equation x - y = 3x - 0 = 3x = 3So, the point of intersection of line a with the x-axis is (3,0).
To find the point of intersection of the line a with the y-axis, put x = 0 in the equation x - y = 3 0 - y = 3y = -3So, the point of intersection of line a with the y-axis is (0,-3).
Similarly, let's find the points of intersection of the line b with the x-axis and the y-axis.
To find the point of intersection of the line b with the x-axis, put y = 0 in the equation x + y = 3x + 0 = 3x = -3So, the point of intersection of line b with the x-axis is (-3,0).
To find the point of intersection of the line b with the y-axis, put x = 0 in the equation x + y = 3 0 + y = 3y = 3So, the point of intersection of line b with the y-axis is (0,3).
Now, we can plot the graph of the two lines a and b and then find the points of intersection as follows:
Therefore, the solution for the system is (0,3) as line a meets the y-axis at (0, -3) and the intersection point of a and b is (3,0). This makes it obvious that the solution for x will be 0 when the solution for y is 3.
Hence, the answer is (0, 3).Option B. (0, 3)
For more questions on Intersection .
https://brainly.com/question/30429663
#SPJ8
ou need to buy some chicken for dinner tonight. you found an ad showing that the store across town has it on sale for $3.09 a pound, which is cheaper than your usual neighborhood store, which sells it for $3.29 a pound. is it worth the extra drive? first, determine what information you need to answer this question, then click here to display that info (along with other info). how much chicken will you be buying? 3 pounds how much far are the two stores? my neighborhood store is 2.3 miles away, and takes about 7 minutes. the store across town is 8.4 miles away, and takes about 22 minutes. how kind of mileage does your car get? it averages about 23 miles per gallon in the city. how many gallons does your car hold? about 14 gallons how much is gas? about $3.87/gallon right now. going to the close store is cheaper going to the further store is cheaper the cheaper option saves you $ give your answer to the nearest cent.
Answers
1) $9.27 (far store), $9.87 (near store)
2) $0.169 per mile
3) 4.6 miles , 16.8 miles
4) $0.78, $2.84
5) $10.05 to the far store, $12.71 to the near store
What is Cost?
A mathematical formula known as the cost function is used to graph how production costs will change depending on the output level. To put it another way, it calculates the overall cost of production based on the quantity that is produced.
1) Let's compare chicken costs:
$3.09 * 3 = $9.27 (far store)
$3.29 * 3 = $9.87 (near store)
2) Now let's look at how much it costs to travel in your car:
\(\frac{3.87}{23} = 0.169\ per \ mile\)
= $0.169 per mile.
3) Next, let's look how much it costs to travel to each store from the gas used. Multiply by 2 because you not only have to drive there, but also back home for the round trip:
2.3 miles * 2 = 4.6 miles
8.4 miles * 2 = 16.8 miles
4) Now to find the cost of gas for the respective trips:
$0.169 * 4.6 = $0.78
$0.169 * 16.8 = $2.84
5) We can just eyeball it from here that it's not worth it to travel, but for completion's sake, let's add the gas and chicken costs together:
$9.27 + $0.78 = $10.05 to the far store
$9.87 + $2.84 = $12.71 to the near store
To know more about Cost visit,
https://brainly.com/question/2292799
#SPJ4
Mixed in a drawer are 8 blue socks, 8 white socks, and 2 gray socks. You pull out two socks, one at a time, without looking. Find the probability of getting 2 socks of the same color The probability of getting 2 socks of the same color is
Answers
The probability of getting 2 socks of the same color isP(two socks of same color) = P(2 blue socks) + P(2 white socks) + P(2 gray socks)= 28/153 + 28/153 + 1/153= 57/153 = 19/51. Hence, the required probability is 19/51.
The probability of getting 2 socks of the same color is given by P(two socks of same color) = P(2 blue socks) + P(2 white socks) + P(2 gray socks).Given, There are 8 blue socks. There are 8 white socks. There are 2 gray socks. Now, we will find the probability of getting 2 blue socks: Probability of selecting one blue sock from 8 blue socks is 8/18.
Probability of selecting another blue sock from 7 blue socks is 7/17.Probability of getting 2 blue socks is P(2 blue socks) = 8/18 * 7/17 = 28/153. Now, we will find the probability of getting 2 white socks: Probability of selecting one white sock from 8 white socks is 8/18.Probability of selecting another white sock from 7 white socks is 7/17.Probability of getting 2 white socks is P(2 white socks) = 8/18 * 7/17 = 28/153.
Now, we will find the probability of getting 2 gray socks: Probability of selecting one gray sock from 2 gray socks is 2/18.Probability of selecting another gray sock from 1 gray sock is 1/17.Probability of getting 2 gray socks is P(2 gray socks) = 2/18 * 1/17 = 1/153.
Therefore,The probability of getting 2 socks of the same color isP(two socks of same color) = P(2 blue socks) + P(2 white socks) + P(2 gray socks)= 28/153 + 28/153 + 1/153= 57/153 = 19/51. Hence, the required probability is 19/51.
Learn more about Probability
brainly.com/question/30034780
#SPJ11
what is the length of leg s of the triangle below?
Answers
Answer:
8
Step-by-step explanation:
the length of the hypotenuse is s times the square root of 2
The length of leg s in the right angled triangle given is 8.
What is the length of s?
A triangle is a three-sided polygon with three edges and three vertices. the sum of angles in a triangle is 180 degrees. A right angled triangle is a triangle in which of its angle measure 90 degrees.
Length of s : sin 45 = opposite / hypotenuse
1/√2 = opposite / 8√2
Opposite = 8√2 x 1/2= 8
To learn more about triangles, please check: https://brainly.com/question/22949981
#SPJ2
in how many positive four-digit integers that are not multiples of $1111$ do the digits form an arithmetic sequence? (the digits must form an arithmetic sequence, in order. for example, the number $5137$ does not count.)
Answers
$8\times9\times8\times9 = 5184$
There are a total of $8\times9\times8\times9 = 5184$ positive four-digit integers that are not multiples of 1111 and form an arithmetic sequence. To explain further, the first digit can be any integer from 1 to 8 (not 0 as the number must be four-digits long). Once the first digit is chosen, the second digit can be any integer from 1 to 9. Once both the first and second digits are chosen, the third digit can be any integer from 1 to 8, and finally the fourth digit can be any integer from 1 to 9. Therefore, the total number of four-digit integers that are not multiples of 1111 and form an arithmetic sequence is $8\times9\times8\times9 = 5184$.
Learn more about arithmetic sequence.
brainly.com/question/10396151
#SPJ11
Work out the equation of the line which passes throught the point (-1,2) and is parallel to the line y=x+4
Answers
Answer:
y = x + 3
Step-by-step explanation:
In point slope form, the equation of line is,
\(y - b = m(x - a)\)
where a and b correspond to the x and y coordinates of the given point and m is the slope
Since the line is parallel to y = x+4, it has the same slope so m = 1 since the slope of y = x+4 is 1
and putting the values of the point (-1,2), we get,
y - 2 = x - (-1)
y-2 = x + 1
y = x + 3
i kinda need help on this
Answers
Answer:
16 units²
Step-by-step explanation:
Formula for the area of a parallelogram is given as, A = b*h
Where,
Base (b) = 4
Height (h) = √(5² - 3²) = 4 (Pythagorean theorem)
Plug in the values
Area of parallelogram, A = 4*4
= 16 units²
Hertford Juniors is a basketball team.
At the end of 10 games, their mean score is 35 points per game.
At the end of 11 games, their mean score has gone down to 33 points per game.
How many points did the team score in the 11th game?
Answers
The team scored 13 points in their 11th game.
MeanThe mean is the arithmetic average of a set of given numbers.
Let x represent the 11 score and a represent the sum of the first ten game points. Hence:
a/10 = 35
a = 350
After the 11th game the mean was 33, hence:
(a + x)/11 = 33
a + x = 363
x = 363 - a = 363 - 350
x = 13
The team scored 13 points in their 11th game.
Find out more on Mean at: https://brainly.com/question/1136789
4+(x+y)=(4+ )+y please answer I need it now
Answers
x=0
4+(x+y)=(4+)+y
4 + x + y = 4 + y
Subtract y from both sides
4 + x + y - y = 4 + y - y
Simplify
4 + x = 4
Subtract 4 from both sides
4 + x - 4 = 4 - 4
Simplify
x = 0
What does it mean to be closed under addition and subtraction?
Answers
The set of rational numbers is closed under addition, subtraction, multiplication, and division (division by zero is not defined) because if you complete any of these operations on rational numbers, the solution is always a rational number. -1 and 1.
What is a closed set ?
A Closed Set Math has a way of explaining a lot of things, and one of those explanations is called a closed set.
Closure property holds for addition and multiplication of whole numbers. The sum of any two whole numbers will always be a whole number, i.e. if a and b are any two whole numbers, a + b will be a whole number. The difference between any two whole numbers may or may not be a whole number. Hence, the whole numbers are not closed under subtraction.
According to the given question:
A set is closed under addition if you can add any two numbers in the set and still have a number in the set as a result. A set is closed under (scalar) multiplication if you can multiply any two elements, and the result is still a number in the set.
Learn more about closed under addition and subtraction here ;
https://brainly.com/question/25292125
#SPJ4
Use the integral test to determine whether the series is convergent or divergent.
Answers
We need to find the function f(n) whose terms are the same as the series in question. We can then integrate this function from n=1 to infinity and determine if the integral is convergent or divergent. If it is convergent, then the series is convergent. If it is divergent, then the series is also divergent.
To determine whether a series is convergent or divergent using the integral test, we need to first check if the series satisfies three conditions:
1) The terms of the series are positive.
2) The terms of the series are decreasing.
3) The series has an infinite number of terms.
Assuming these conditions are satisfied, we can use the integral test which states that if the integral of the function f(x) from n=1 to infinity is convergent, then the series with terms a_n = f(n) is also convergent. Conversely, if the integral is divergent, then the series is also divergent.
Learn more about integral here:
brainly.com/question/18125359
#SPJ11
is the ecosystem going down bc of pollution
Answers
Answer: yes. but luckily we have vsvo girls who are going to save the world. still waiting for them tho.. where did they go? are they hibernating or sum?
Step-by-step explanation:
Pls help! I’ll give brainliest!
A = ___ ??
B = ___ ??
Answers
Answer:
A:80 B:30
Step-by-step explanation:
b²+4 when b =-4..........
Answers
Answer:
20
Step-by-step explanation:
-4 x -4 = 16
16 + 4 = 20
Hope this helps. Pls give brainliest.
Answer:
You need to explain moreFrank joined a fitness club. He paid a monthly $50 registration fee, and dues are $12 per month.
Answers
Answer:
the answer is 50+12x. is this iready
At the beginning of the school year, Jamie had
$500 in her savings account. She wants to have at
least $200 left in the account at the end of the
school year. Each week, she withdraws $30 for
spending money. To determine how many weeks
she can spend $30, Jamie wrote and solved this
inequality
Answers
Answer:
Hmmm let me solve then I will answer
A powerful computer is purchased for $2000, but loses 20% of its value each year.
Is this growth or decay?
Answers
Answer:
Decay
Step-by-step explanation:
SCOOBY+DOOO=BUSTED;
FIND THE VALUE OF B+U+S+T+E+D ?
Answers
The value of B+U+S+T+E+D is 20. B represents the number 2, U represents the number 3, S represents the number 19, T represents the number 20, E represents the number 5, and D represents the number 4. When these numbers are added together, they equal 20.
BUSTED is an acronym for 'Believing Uncovering Solving Troubles Everywhere Dynamicly'. It is a popular phrase used by the characters in the Scooby-Doo cartoons. It is a phrase used to express the moment when the mystery is solved and the bad guy is caught.
BUSTED is an important part of the Scooby-Doo cartoon series. It expresses the moment when the mystery is solved and the bad guy is caught. It is also a great way to encourage children to believe in themselves and to use their own detective skills to solve a mystery. The value of B+U+S+T+E+D is 20, and it is a fun way to teach kids math and problem solving skills.
know more about acronym here
https://brainly.com/question/2696106#
#SPJ11
Plz help!!!!!!!!!!!!!!!!!!!
Answers
Answer:
1.
\(7x\)
2.
\(x - 5\)
3.
\(x \div 2\)
4.
\(x + 4\)
Hope it helped
PLS mark BRAINLIEST
a paint can is 10 cm tall and holds approximately 535 cubic centimeters of paint. what is the approximate area of the base of the can? responses 5350 square centimeters 5350 square centimeters 53.5 square centimeters 53 point 5 square centimeters 100 square centimeters 100 square centimeters 5.35 square centimeters
Answers
the approximate area of the base of the paint can is 53.5 square centimeters.by using Volume formula is Base Area × Height
To find the approximate area of the base of the paint can, we can use the formula for the volume of a cylinder:
Volume = Base Area × Height
We are given the volume (535 cubic centimeters) and the height (10 cm). We need to solve for the Base Area. Rearranging the formula to solve for Base Area, we get:
Base Area = Volume ÷ Height
Now, substitute the given values:
Base Area = 535 cubic centimeters ÷ 10 cm
Base Area ≈ 53.5 square centimeters
So, the approximate area of the base of the paint can is 53.5 square centimeters.
To know more about area Visit:
https://brainly.com/question/20898622
#SPJ11
The approximate area of the base of the can is 53.5 square centimeters
How to determine the area of the base of the can?From the question, we have the following parameters that can be used in our computation:
Volume = 535 cubic centimeters of paint
Height of container = 10 cm
The area of the base of the can is calculated as
Base area = Volume/Height
Substitute the known values in the above equation, so, we have the following representation
Base area = 535/10
Evaluate
Base area = 53.5
Hence, the base area is 53.5 square centimeters
Read more about volume at
https://brainly.com/question/463363
#SPJ4
the slopes of a quadrilateral are -1/2, 3/2, -1/2, and 3/2. What conclusion about this shape cam be made.
Answers
The conclusion that can be made given the slopes of the quadrilateral is that: since the slopes of the opposite sides are equal, the opposite sides are parallel, therefore the quadrilateral is a parallelogram. (Option A is correct).
Recall:
The slopes of the opposite sides of a parallelogram are always parallel.When two lines have a slope that is equal, the lines are said to be parallel to each other.The slopes given shows are:
-1/2, 3/2, -1/2, and 3/2
This implies that the quadrilateral has two pairs of equal sides that are parallel to each other.
Therefore, the conclusion that can be made given the slopes of the quadrilateral is that: since the slopes of the opposite sides are equal, the opposite sides are parallel, therefore the quadrilateral is a parallelogram. (Option A is correct).Learn more about parallelogram on:
https://brainly.com/question/3050890
For the rectangle shown which equation can be used to find the value of c
A=5
B=x
C=12
Answers
Answer: B = 1.9
Step-by-step explanation:
A=5
C=12
B=x
Pythagorean Theorem: 5^2 + x^2 = 12^2
25+x^2=144
x^2=119
\(\sqrt{19}\)≈10.9
Is it true that all while numbers are integers
Answers
Answer:
n o
Step-by-step explanation:
Answer:
Yes.
Step-by-step explanation:
A whole number is a number greater than or equal to zero that does not have any fractions and/or cannot be written with a fractional component. It can also be defined as an integer, though it should be noted that integers include negative numbers. 3, 5, 14, 2, and 0 are all whole numbers and are also all integers. 1.5, 16.32, and -4.598 are not whole numbers and are thus not integers.
Compare 1.96312.... and the square root of -5 show your work
Answers
If it’s 1.96312 - 5 the answer is -3.03688 if it’s something else then i’m just gonna sound slow
What are the domain and range of the function below?
A. Domain: (-4,0)
Range: [-2,∞)
B. Domain: (-3,0)
Range: [-2,∞)
C. Domain: (-∞,∞)
Range: [-2,∞)
D. Domain:
Range: (-∞,∞)
PS don't mind that dot on the graph, didn't mean to put it there.
Answers
The domain is (-∞,∞) and the range is [-2,∞)
How to determine the domain and the range?From the graph, we can see that the x values extend at both sides of the graph.
This means that the domain is the set of real values i.e. (-∞,∞)
However, the minimum y value is -2 and it increases from their
This means that the range is [-2,∞)
Hence, the domain is (-∞,∞) and the range is [-2,∞)
Read more about domain and range at:
https://brainly.com/question/10197594
#SPJ1