One-way functions and P=NP
$begingroup$
This site contains various discussions of one-way functions and their relation to P versus NP.
Some of these discussions use a language
$L={(x',y) ~mid~ x'le x text{ and } f(x)=y }$, where $f:Sigma^*toSigma^*$ is the one-way function and $x'le x$ is the prefix relation.
Now one central claim is that this language $L$ is contained in NP, since the word $x$ is a YES-certificate for $(x',y)in L$.
I do not see why this claim is justified.
Why is the length of the certificate $x$ polynomially bounded in the length of $(x',y)$?
Couldn't it be possible that $x$ is exponentially long in $y$ and $x'$, but $f(x)$ is short and quickly computable from $x$?
one-way-function
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
$endgroup$
add a comment |
$begingroup$
This site contains various discussions of one-way functions and their relation to P versus NP.
Some of these discussions use a language
$L={(x',y) ~mid~ x'le x text{ and } f(x)=y }$, where $f:Sigma^*toSigma^*$ is the one-way function and $x'le x$ is the prefix relation.
Now one central claim is that this language $L$ is contained in NP, since the word $x$ is a YES-certificate for $(x',y)in L$.
I do not see why this claim is justified.
Why is the length of the certificate $x$ polynomially bounded in the length of $(x',y)$?
Couldn't it be possible that $x$ is exponentially long in $y$ and $x'$, but $f(x)$ is short and quickly computable from $x$?
one-way-function
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
$endgroup$
$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56
add a comment |
$begingroup$
This site contains various discussions of one-way functions and their relation to P versus NP.
Some of these discussions use a language
$L={(x',y) ~mid~ x'le x text{ and } f(x)=y }$, where $f:Sigma^*toSigma^*$ is the one-way function and $x'le x$ is the prefix relation.
Now one central claim is that this language $L$ is contained in NP, since the word $x$ is a YES-certificate for $(x',y)in L$.
I do not see why this claim is justified.
Why is the length of the certificate $x$ polynomially bounded in the length of $(x',y)$?
Couldn't it be possible that $x$ is exponentially long in $y$ and $x'$, but $f(x)$ is short and quickly computable from $x$?
one-way-function
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
$endgroup$
This site contains various discussions of one-way functions and their relation to P versus NP.
Some of these discussions use a language
$L={(x',y) ~mid~ x'le x text{ and } f(x)=y }$, where $f:Sigma^*toSigma^*$ is the one-way function and $x'le x$ is the prefix relation.
Now one central claim is that this language $L$ is contained in NP, since the word $x$ is a YES-certificate for $(x',y)in L$.
I do not see why this claim is justified.
Why is the length of the certificate $x$ polynomially bounded in the length of $(x',y)$?
Couldn't it be possible that $x$ is exponentially long in $y$ and $x'$, but $f(x)$ is short and quickly computable from $x$?
one-way-function
one-way-function
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
asked Dec 17 '18 at 14:33
AlexisAlexis
1284
1284
$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56
add a comment |
$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56
$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56
$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, it could be that in the language you give, $x$ is exponentially long in $(y,x')$, and $f$ is an efficiently computable one-way function (note that it only has to run in time polynomial in its input length, so $f(x)$ needs not be computable in time polynomial in $(y,x')$).
However, this is really a minor issue: the answers to this question that you read are simply a bit informal, and only give an intuition of the proof that OWF implies $P neq NP$. Intuitively, to fix this, modify your language as follows:
$L={(1^n, x',y) ~mid~ exists x, |x| = n, x'le x, text{ and } f(x)=y }$,
where $1^n$ means a sequence of $n$ consecutive one, which exactly allows to fix the issue you point out (note that here $x'le x$ means $x'$ is a prefix of $x$).
Note: the second answer to the question you link to does provide a link to an exercise sheet which contains the more formal solution.
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
$endgroup$
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
add a comment |
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$begingroup$
Yes, it could be that in the language you give, $x$ is exponentially long in $(y,x')$, and $f$ is an efficiently computable one-way function (note that it only has to run in time polynomial in its input length, so $f(x)$ needs not be computable in time polynomial in $(y,x')$).
However, this is really a minor issue: the answers to this question that you read are simply a bit informal, and only give an intuition of the proof that OWF implies $P neq NP$. Intuitively, to fix this, modify your language as follows:
$L={(1^n, x',y) ~mid~ exists x, |x| = n, x'le x, text{ and } f(x)=y }$,
where $1^n$ means a sequence of $n$ consecutive one, which exactly allows to fix the issue you point out (note that here $x'le x$ means $x'$ is a prefix of $x$).
Note: the second answer to the question you link to does provide a link to an exercise sheet which contains the more formal solution.
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
$endgroup$
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
add a comment |
$begingroup$
Yes, it could be that in the language you give, $x$ is exponentially long in $(y,x')$, and $f$ is an efficiently computable one-way function (note that it only has to run in time polynomial in its input length, so $f(x)$ needs not be computable in time polynomial in $(y,x')$).
However, this is really a minor issue: the answers to this question that you read are simply a bit informal, and only give an intuition of the proof that OWF implies $P neq NP$. Intuitively, to fix this, modify your language as follows:
$L={(1^n, x',y) ~mid~ exists x, |x| = n, x'le x, text{ and } f(x)=y }$,
where $1^n$ means a sequence of $n$ consecutive one, which exactly allows to fix the issue you point out (note that here $x'le x$ means $x'$ is a prefix of $x$).
Note: the second answer to the question you link to does provide a link to an exercise sheet which contains the more formal solution.
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
$endgroup$
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
add a comment |
$begingroup$
Yes, it could be that in the language you give, $x$ is exponentially long in $(y,x')$, and $f$ is an efficiently computable one-way function (note that it only has to run in time polynomial in its input length, so $f(x)$ needs not be computable in time polynomial in $(y,x')$).
However, this is really a minor issue: the answers to this question that you read are simply a bit informal, and only give an intuition of the proof that OWF implies $P neq NP$. Intuitively, to fix this, modify your language as follows:
$L={(1^n, x',y) ~mid~ exists x, |x| = n, x'le x, text{ and } f(x)=y }$,
where $1^n$ means a sequence of $n$ consecutive one, which exactly allows to fix the issue you point out (note that here $x'le x$ means $x'$ is a prefix of $x$).
Note: the second answer to the question you link to does provide a link to an exercise sheet which contains the more formal solution.
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
$endgroup$
Yes, it could be that in the language you give, $x$ is exponentially long in $(y,x')$, and $f$ is an efficiently computable one-way function (note that it only has to run in time polynomial in its input length, so $f(x)$ needs not be computable in time polynomial in $(y,x')$).
However, this is really a minor issue: the answers to this question that you read are simply a bit informal, and only give an intuition of the proof that OWF implies $P neq NP$. Intuitively, to fix this, modify your language as follows:
$L={(1^n, x',y) ~mid~ exists x, |x| = n, x'le x, text{ and } f(x)=y }$,
where $1^n$ means a sequence of $n$ consecutive one, which exactly allows to fix the issue you point out (note that here $x'le x$ means $x'$ is a prefix of $x$).
Note: the second answer to the question you link to does provide a link to an exercise sheet which contains the more formal solution.
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
edited Dec 17 '18 at 14:51
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
answered Dec 17 '18 at 14:47
Geoffroy CouteauGeoffroy Couteau
8,49011733
8,49011733
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
add a comment |
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Sorry, but with your modification it is not clear anymore how to invert function $f$ in polynomial time, in case $L$ is in $P$. For exploting $L$, it seems that now I need some a priori bound on $n$, but this might be exponentially large in the length of $y$.
$endgroup$
– Alexis
Dec 17 '18 at 15:04
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
Inverting $f$ needs not be polynomial in the output size $y$, but still in the input size $x$, which is the case if $L$ is in $P$. You should check the detailed solution given on page 2-3 of the exercise sheet I link to (courses.cs.ut.ee/all/MTAT.07.004/2016_fall/uploads/solution/…).
$endgroup$
– Geoffroy Couteau
Dec 17 '18 at 15:29
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
$begingroup$
With polynomial runtime for f you can not generate an output f(x) with superpolynonial size - so actually any limitation on x also translates to y implicitly.
$endgroup$
– tylo
Dec 18 '18 at 11:56
add a comment |
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$begingroup$
It is likely that a proof that P=NP is not an effective proof. Crypto works about the same if the effort to crack n bit keys is 2^n or n^65536.
$endgroup$
– Joshua
Dec 17 '18 at 19:56