Mixing
Notation: What does $bf{Z}^*_R$ mean in the context of groups?
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I recently stumbled upon the following definition for a hash function:
To get this we define $hat{h}=hat{h}_{r}: {{0,1}}^* rightarrow {bf Z}_R$
as follows:
$$ hat{h}_r(m) = ({sum_{i=1}^{k} m_i cdot r^{i-1}}) bmod R $$
where $R$ is a 161-bit prime, $r$ (the key of $hat{h}$) is a uniformly distributed
element in ${bf Z}_{R}^*, [...]$
Source: http://www.wisdom.weizmann.ac.il/~naor/p_r_func/pr/pr.html
As far as I understand, $textbf{Z}_R$ is the additive group of integers $mod R$. Is this the case?
If so, then what is $textbf{Z}^*_R$?
group-theory notation
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
$endgroup$
add a comment |
$begingroup$
I recently stumbled upon the following definition for a hash function:
To get this we define $hat{h}=hat{h}_{r}: {{0,1}}^* rightarrow {bf Z}_R$
as follows:
$$ hat{h}_r(m) = ({sum_{i=1}^{k} m_i cdot r^{i-1}}) bmod R $$
where $R$ is a 161-bit prime, $r$ (the key of $hat{h}$) is a uniformly distributed
element in ${bf Z}_{R}^*, [...]$
Source: http://www.wisdom.weizmann.ac.il/~naor/p_r_func/pr/pr.html
As far as I understand, $textbf{Z}_R$ is the additive group of integers $mod R$. Is this the case?
If so, then what is $textbf{Z}^*_R$?
group-theory notation
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
$endgroup$
$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
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– freakish
Jan 24 at 16:50
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
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It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19
add a comment |
$begingroup$
I recently stumbled upon the following definition for a hash function:
To get this we define $hat{h}=hat{h}_{r}: {{0,1}}^* rightarrow {bf Z}_R$
as follows:
$$ hat{h}_r(m) = ({sum_{i=1}^{k} m_i cdot r^{i-1}}) bmod R $$
where $R$ is a 161-bit prime, $r$ (the key of $hat{h}$) is a uniformly distributed
element in ${bf Z}_{R}^*, [...]$
Source: http://www.wisdom.weizmann.ac.il/~naor/p_r_func/pr/pr.html
As far as I understand, $textbf{Z}_R$ is the additive group of integers $mod R$. Is this the case?
If so, then what is $textbf{Z}^*_R$?
group-theory notation
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
$endgroup$
I recently stumbled upon the following definition for a hash function:
To get this we define $hat{h}=hat{h}_{r}: {{0,1}}^* rightarrow {bf Z}_R$
as follows:
$$ hat{h}_r(m) = ({sum_{i=1}^{k} m_i cdot r^{i-1}}) bmod R $$
where $R$ is a 161-bit prime, $r$ (the key of $hat{h}$) is a uniformly distributed
element in ${bf Z}_{R}^*, [...]$
Source: http://www.wisdom.weizmann.ac.il/~naor/p_r_func/pr/pr.html
As far as I understand, $textbf{Z}_R$ is the additive group of integers $mod R$. Is this the case?
If so, then what is $textbf{Z}^*_R$?
group-theory notation
group-theory notation
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
asked Jan 24 at 16:46
CryptoFanCryptoFan
1
1
$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
$endgroup$
– freakish
Jan 24 at 16:50
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
$begingroup$
It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19
add a comment |
$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
$endgroup$
– freakish
Jan 24 at 16:50
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
$begingroup$
It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19
$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
$endgroup$
– freakish
Jan 24 at 16:50
$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
$endgroup$
– freakish
Jan 24 at 16:50
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
$begingroup$
It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19
$begingroup$
It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19
add a comment |
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$begingroup$
It's the group of invertibles of $mathbb{Z}_R$: en.wikipedia.org/wiki/Unit_(ring_theory) In case when $R$ is prime it is simply $mathbb{Z}_Rbackslash{0}$.
$endgroup$
– freakish
Jan 24 at 16:50
$begingroup$
Thank you, that answered the question :)
$endgroup$
– CryptoFan
Jan 24 at 16:52
$begingroup$
It's a theorem that $mathbf{Z}_R^*$ is cyclic if $R$ is prime. (This is often exploited by simple random number generators - see Knuth, The Art of Computer Programming, Vol.2?.)
$endgroup$
– user1729
Jan 24 at 17:19