How to solve xor equations with multiple variable? [closed]
$begingroup$
I have 9 variables $$a_1,a_2,a_3 ... a_9 $$ and want to extract their values using the following equations.
$$
a_1 oplus a_4 = D1
$$
$$
a_2 oplus a_5 = D2
$$
$$
a_3 oplus a_6 = D3
$$
$$
a_4 oplus a_7 = D4
$$
$$
a_5 oplus a_8 = D5
$$
$$
a_6 oplus a_9 = D6
$$
$$
a_1 oplus a_7 = D7
$$
$$
a_2 oplus a_8 = D8
$$
$$
a_3 oplus a_9 = D9
$$
$$D1, D2, D3, ... D9 $$ are given interger values.
How to solve this kind of question or specifically this question?
discrete-mathematics logic
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
$endgroup$
closed as off-topic by RRL, John Bentin, user334732, amWhy, mrtaurho Dec 17 '18 at 19:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, John Bentin, user334732, amWhy, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I have 9 variables $$a_1,a_2,a_3 ... a_9 $$ and want to extract their values using the following equations.
$$
a_1 oplus a_4 = D1
$$
$$
a_2 oplus a_5 = D2
$$
$$
a_3 oplus a_6 = D3
$$
$$
a_4 oplus a_7 = D4
$$
$$
a_5 oplus a_8 = D5
$$
$$
a_6 oplus a_9 = D6
$$
$$
a_1 oplus a_7 = D7
$$
$$
a_2 oplus a_8 = D8
$$
$$
a_3 oplus a_9 = D9
$$
$$D1, D2, D3, ... D9 $$ are given interger values.
How to solve this kind of question or specifically this question?
discrete-mathematics logic
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
$endgroup$
closed as off-topic by RRL, John Bentin, user334732, amWhy, mrtaurho Dec 17 '18 at 19:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, John Bentin, user334732, amWhy, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18
add a comment |
$begingroup$
I have 9 variables $$a_1,a_2,a_3 ... a_9 $$ and want to extract their values using the following equations.
$$
a_1 oplus a_4 = D1
$$
$$
a_2 oplus a_5 = D2
$$
$$
a_3 oplus a_6 = D3
$$
$$
a_4 oplus a_7 = D4
$$
$$
a_5 oplus a_8 = D5
$$
$$
a_6 oplus a_9 = D6
$$
$$
a_1 oplus a_7 = D7
$$
$$
a_2 oplus a_8 = D8
$$
$$
a_3 oplus a_9 = D9
$$
$$D1, D2, D3, ... D9 $$ are given interger values.
How to solve this kind of question or specifically this question?
discrete-mathematics logic
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
$endgroup$
I have 9 variables $$a_1,a_2,a_3 ... a_9 $$ and want to extract their values using the following equations.
$$
a_1 oplus a_4 = D1
$$
$$
a_2 oplus a_5 = D2
$$
$$
a_3 oplus a_6 = D3
$$
$$
a_4 oplus a_7 = D4
$$
$$
a_5 oplus a_8 = D5
$$
$$
a_6 oplus a_9 = D6
$$
$$
a_1 oplus a_7 = D7
$$
$$
a_2 oplus a_8 = D8
$$
$$
a_3 oplus a_9 = D9
$$
$$D1, D2, D3, ... D9 $$ are given interger values.
How to solve this kind of question or specifically this question?
discrete-mathematics logic
discrete-mathematics logic
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
asked Dec 17 '18 at 15:11
vikas2ccvikas2cc
15
15
closed as off-topic by RRL, John Bentin, user334732, amWhy, mrtaurho Dec 17 '18 at 19:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, John Bentin, user334732, amWhy, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by RRL, John Bentin, user334732, amWhy, mrtaurho Dec 17 '18 at 19:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, John Bentin, user334732, amWhy, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18
add a comment |
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The system need not be consistent.
To be consistent, we require $$D_1 oplus D_4 = D_7$$
$$D_2 oplus D_5 = D_8$$
$$D_3 oplus D_6 = D_9$$
Suppose those conditions are satisfied, the whole system of equations is equivalent to the first $6$ equations.
We have $6$ independent equations and $9$ variables, we have $3$ degree of freedom. Fix $a_7, a_8, a_9$, use backward substitution to solve for the other variables.
Remark: Notice that this is a linear system of equations, we can use Gaussian elimination.
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
|
show 3 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The system need not be consistent.
To be consistent, we require $$D_1 oplus D_4 = D_7$$
$$D_2 oplus D_5 = D_8$$
$$D_3 oplus D_6 = D_9$$
Suppose those conditions are satisfied, the whole system of equations is equivalent to the first $6$ equations.
We have $6$ independent equations and $9$ variables, we have $3$ degree of freedom. Fix $a_7, a_8, a_9$, use backward substitution to solve for the other variables.
Remark: Notice that this is a linear system of equations, we can use Gaussian elimination.
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
|
show 3 more comments
$begingroup$
The system need not be consistent.
To be consistent, we require $$D_1 oplus D_4 = D_7$$
$$D_2 oplus D_5 = D_8$$
$$D_3 oplus D_6 = D_9$$
Suppose those conditions are satisfied, the whole system of equations is equivalent to the first $6$ equations.
We have $6$ independent equations and $9$ variables, we have $3$ degree of freedom. Fix $a_7, a_8, a_9$, use backward substitution to solve for the other variables.
Remark: Notice that this is a linear system of equations, we can use Gaussian elimination.
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
|
show 3 more comments
$begingroup$
The system need not be consistent.
To be consistent, we require $$D_1 oplus D_4 = D_7$$
$$D_2 oplus D_5 = D_8$$
$$D_3 oplus D_6 = D_9$$
Suppose those conditions are satisfied, the whole system of equations is equivalent to the first $6$ equations.
We have $6$ independent equations and $9$ variables, we have $3$ degree of freedom. Fix $a_7, a_8, a_9$, use backward substitution to solve for the other variables.
Remark: Notice that this is a linear system of equations, we can use Gaussian elimination.
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
The system need not be consistent.
To be consistent, we require $$D_1 oplus D_4 = D_7$$
$$D_2 oplus D_5 = D_8$$
$$D_3 oplus D_6 = D_9$$
Suppose those conditions are satisfied, the whole system of equations is equivalent to the first $6$ equations.
We have $6$ independent equations and $9$ variables, we have $3$ degree of freedom. Fix $a_7, a_8, a_9$, use backward substitution to solve for the other variables.
Remark: Notice that this is a linear system of equations, we can use Gaussian elimination.
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 17 '18 at 15:17
Siong Thye GohSiong Thye Goh
101k1466118
101k1466118
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
|
show 3 more comments
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
How to fix a7,a8,a9? can you please explain?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:20
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
there are some degree of freedom (meaning the solution is not unique) let them be whatever that you want and you can express the rest in terms of them, for example $a_6 = D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:21
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
and how to apply backward substitution while having xor in equations?
$endgroup$
– vikas2cc
Dec 17 '18 at 15:22
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
for example, after I express $a_6 = D_6 oplus a_9$, I also know that $a_3 = D_3 oplus a_6 = D_3 oplus D_6 oplus a_9$.
$endgroup$
– Siong Thye Goh
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
$begingroup$
ok i understand the equation $$ a_6 = D6 oplus a_9$$.
$endgroup$
– vikas2cc
Dec 17 '18 at 15:23
|
show 3 more comments
$begingroup$
Hi Vikas and welcome to MSE. You're more likely to receive help if you demonstrate some effort on your own part.
$endgroup$
– user334732
Dec 17 '18 at 19:15
$begingroup$
$$a_1 oplus a_4 = D1$$ is equivalent to $$a_1 + a_4 = D1 pmod 2$$. So you can solve this like any system of linear equations, just do all the arithmetic mod 2.
$endgroup$
– DanielV
Dec 17 '18 at 19:18