Decrypting Multi-Prime RSA with e, N, and factors of N givenWhat makes RSA secure by using prime numbers?Knowing N and a relationship between the 2 prime numbers(p and q) find p and q generated with RSACan multi-prime RSA be used to create an abuse-resistant lawful interception mechanism?What's wrong with RSA and OpenSSL?Is it possible to check if a number is the product of two primes without factorizing it?
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Decrypting Multi-Prime RSA with e, N, and factors of N given
What makes RSA secure by using prime numbers?Knowing N and a relationship between the 2 prime numbers(p and q) find p and q generated with RSACan multi-prime RSA be used to create an abuse-resistant lawful interception mechanism?What's wrong with RSA and OpenSSL?Is it possible to check if a number is the product of two primes without factorizing it?
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$begingroup$
I was wondering if there was any way to compute the private key $d$ when knowing only $e$ and $N$, and being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$. I've been searching for days and I can't find any way.
rsa multi-prime-rsa
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment
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$begingroup$
I was wondering if there was any way to compute the private key $d$ when knowing only $e$ and $N$, and being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$. I've been searching for days and I can't find any way.
rsa multi-prime-rsa
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
3
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago
add a comment
|
$begingroup$
I was wondering if there was any way to compute the private key $d$ when knowing only $e$ and $N$, and being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$. I've been searching for days and I can't find any way.
rsa multi-prime-rsa
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I was wondering if there was any way to compute the private key $d$ when knowing only $e$ and $N$, and being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$. I've been searching for days and I can't find any way.
rsa multi-prime-rsa
rsa multi-prime-rsa
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
11.2k3 gold badges29 silver badges55 bronze badges
asked 8 hours ago
DominicDominic
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
DominicDominic
61 bronze badge
asked 8 hours ago
DominicDominic
61 bronze badge
61 bronze badge
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Dominic is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
3
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago
add a comment
|
$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
3
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago
$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
3
3
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago
add a comment
|
1 Answer
1
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oldest
votes
$begingroup$
In multi-prime RSA, with the factorization of $N$ into primes $p$, $q$, $r$, $s$ known, computing $d$ can be done as usual, e.g.$$dgets e^-1bmodoperatornamelcm(p-1,q-1,r-1,s-1)$$or$$dgets e^-1bmod((p-1)(q-1)(r-1)(s-1))$$
Note: the rationale of using multi-prime RSA is to obtain speedups that require not using $d$; but using $d$ will work anyway, only slower.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
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$begingroup$
In multi-prime RSA, with the factorization of $N$ into primes $p$, $q$, $r$, $s$ known, computing $d$ can be done as usual, e.g.$$dgets e^-1bmodoperatornamelcm(p-1,q-1,r-1,s-1)$$or$$dgets e^-1bmod((p-1)(q-1)(r-1)(s-1))$$
Note: the rationale of using multi-prime RSA is to obtain speedups that require not using $d$; but using $d$ will work anyway, only slower.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
$endgroup$
add a comment
|
$begingroup$
In multi-prime RSA, with the factorization of $N$ into primes $p$, $q$, $r$, $s$ known, computing $d$ can be done as usual, e.g.$$dgets e^-1bmodoperatornamelcm(p-1,q-1,r-1,s-1)$$or$$dgets e^-1bmod((p-1)(q-1)(r-1)(s-1))$$
Note: the rationale of using multi-prime RSA is to obtain speedups that require not using $d$; but using $d$ will work anyway, only slower.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
$endgroup$
add a comment
|
$begingroup$
In multi-prime RSA, with the factorization of $N$ into primes $p$, $q$, $r$, $s$ known, computing $d$ can be done as usual, e.g.$$dgets e^-1bmodoperatornamelcm(p-1,q-1,r-1,s-1)$$or$$dgets e^-1bmod((p-1)(q-1)(r-1)(s-1))$$
Note: the rationale of using multi-prime RSA is to obtain speedups that require not using $d$; but using $d$ will work anyway, only slower.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
$endgroup$
In multi-prime RSA, with the factorization of $N$ into primes $p$, $q$, $r$, $s$ known, computing $d$ can be done as usual, e.g.$$dgets e^-1bmodoperatornamelcm(p-1,q-1,r-1,s-1)$$or$$dgets e^-1bmod((p-1)(q-1)(r-1)(s-1))$$
Note: the rationale of using multi-prime RSA is to obtain speedups that require not using $d$; but using $d$ will work anyway, only slower.
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
edited 5 hours ago
kelalaka
11.2k3 gold badges29 silver badges55 bronze badges
11.2k3 gold badges29 silver badges55 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
answered 7 hours ago
fgrieufgrieu
86.4k7 gold badges192 silver badges378 bronze badges
answered 7 hours ago
fgrieufgrieu
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add a comment
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Dominic is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
"being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$" - does this mean that you know/can find the values of $p, q, r, s$, or merely that you know that $N$ has 4 factors?
$endgroup$
– Ella Rose♦
8 hours ago
$begingroup$
I have found p, q, r and s. But I have no clue what way to go next. I've tried calculating the totient but I don't think that's the good way because I didn't get the answer I wanted.
$endgroup$
– Dominic
8 hours ago
$begingroup$
Thank you!! I will try it
$endgroup$
– Dominic
8 hours ago
3
$begingroup$
@fgrieu: that sounds like an answer; why don't you post it as one...
$endgroup$
– poncho
7 hours ago