Unknown Exponent In Modular Equation
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If $(9^4)^x equiv 12 pmod23$, then how do I find $x$?
I have tried solving this, but I was thinking if there is a step by step formula. I know that any number from the group order may suffice.
Will I still be able to find $x$ when the numbers involved are huge, such as, a large prime instead of $23$ and a large number instead of $4$?
modular-arithmetic
edited Aug 19 at 3:55
Math Lover
12.6k21232
asked Aug 19 at 2:36
DENN
62
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up vote
1
down vote
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If $(9^4)^x equiv 12 pmod23$, then how do I find $x$?
I have tried solving this, but I was thinking if there is a step by step formula. I know that any number from the group order may suffice.
Will I still be able to find $x$ when the numbers involved are huge, such as, a large prime instead of $23$ and a large number instead of $4$?
modular-arithmetic
edited Aug 19 at 3:55
Math Lover
12.6k21232
asked Aug 19 at 2:36
DENN
62
This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $(9^4)^x equiv 12 pmod23$, then how do I find $x$?
I have tried solving this, but I was thinking if there is a step by step formula. I know that any number from the group order may suffice.
Will I still be able to find $x$ when the numbers involved are huge, such as, a large prime instead of $23$ and a large number instead of $4$?
modular-arithmetic
edited Aug 19 at 3:55
Math Lover
12.6k21232
asked Aug 19 at 2:36
DENN
62
If $(9^4)^x equiv 12 pmod23$, then how do I find $x$?
I have tried solving this, but I was thinking if there is a step by step formula. I know that any number from the group order may suffice.
Will I still be able to find $x$ when the numbers involved are huge, such as, a large prime instead of $23$ and a large number instead of $4$?
modular-arithmetic
edited Aug 19 at 3:55
Math Lover
12.6k21232
asked Aug 19 at 2:36
DENN
62
edited Aug 19 at 3:55
Math Lover
12.6k21232
edited Aug 19 at 3:55
Math Lover
12.6k21232
edited Aug 19 at 3:55
Math Lover
12.6k21232
12.6k21232
asked Aug 19 at 2:36
DENN
62
asked Aug 19 at 2:36
DENN
62
asked Aug 19 at 2:36
DENN
62
62
This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49
add a comment |Â
This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49
This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49
This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49
add a comment |Â
1 Answer
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$$3^8x-1equiv4pmod23equiv3^3$$
$$iff3^8x-4equiv1$$
$3^2notequiv1,3^3equiv4,3^11equiv9(4^3)equiv1$
$$implies8x-4equiv0pmod11$$
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$$3^8x-1equiv4pmod23equiv3^3$$
$$iff3^8x-4equiv1$$
$3^2notequiv1,3^3equiv4,3^11equiv9(4^3)equiv1$
$$implies8x-4equiv0pmod11$$
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
add a comment |Â
up vote
0
down vote
$$3^8x-1equiv4pmod23equiv3^3$$
$$iff3^8x-4equiv1$$
$3^2notequiv1,3^3equiv4,3^11equiv9(4^3)equiv1$
$$implies8x-4equiv0pmod11$$
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
add a comment |Â
up vote
0
down vote
up vote
0
down vote
$$3^8x-1equiv4pmod23equiv3^3$$
$$iff3^8x-4equiv1$$
$3^2notequiv1,3^3equiv4,3^11equiv9(4^3)equiv1$
$$implies8x-4equiv0pmod11$$
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
$$3^8x-1equiv4pmod23equiv3^3$$
$$iff3^8x-4equiv1$$
$3^2notequiv1,3^3equiv4,3^11equiv9(4^3)equiv1$
$$implies8x-4equiv0pmod11$$
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
answered Aug 19 at 2:44
lab bhattacharjee
215k14152264
215k14152264
add a comment |Â
add a comment |Â
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This is a difficult problem in general. Look up discrete logarithm. A lot of cryptography is predicated on this problem continuing to be difficult.
– Callus
Aug 19 at 2:49