In Mathematics is there a discrete logarithm function?
$begingroup$
I find it difficult to understand this part in this book.
Because, as far as I know, there is no unique function or formula for discrete logarithms. I cann't understand what this formula does. Is this formula to direct calculate the discrete logarithm?
Or is there a direct unique formula or funtion that calculates special discrete logarithms?
discrete-mathematics special-functions math-history cryptography discrete-logarithms
asked Jan 23 at 5:14
NewuserNewuser
314213
$endgroup$
add a comment |
$begingroup$
I find it difficult to understand this part in this book.
Because, as far as I know, there is no unique function or formula for discrete logarithms. I cann't understand what this formula does. Is this formula to direct calculate the discrete logarithm?
Or is there a direct unique formula or funtion that calculates special discrete logarithms?
discrete-mathematics special-functions math-history cryptography discrete-logarithms
asked Jan 23 at 5:14
NewuserNewuser
314213
$endgroup$
$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
2
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03
add a comment |
$begingroup$
I find it difficult to understand this part in this book.
Because, as far as I know, there is no unique function or formula for discrete logarithms. I cann't understand what this formula does. Is this formula to direct calculate the discrete logarithm?
Or is there a direct unique formula or funtion that calculates special discrete logarithms?
discrete-mathematics special-functions math-history cryptography discrete-logarithms
asked Jan 23 at 5:14
NewuserNewuser
314213
$endgroup$
I find it difficult to understand this part in this book.
Because, as far as I know, there is no unique function or formula for discrete logarithms. I cann't understand what this formula does. Is this formula to direct calculate the discrete logarithm?
Or is there a direct unique formula or funtion that calculates special discrete logarithms?
discrete-mathematics special-functions math-history cryptography discrete-logarithms
discrete-mathematics special-functions math-history cryptography discrete-logarithms
asked Jan 23 at 5:14
NewuserNewuser
314213
asked Jan 23 at 5:14
NewuserNewuser
314213
edited Jan 23 at 5:37
Newuser
asked Jan 23 at 5:14
NewuserNewuser
314213
asked Jan 23 at 5:14
NewuserNewuser
314213
asked Jan 23 at 5:14
NewuserNewuser
314213
314213
$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
2
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03
add a comment |
$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
2
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03
$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
2
2
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The definition of the discrete logarithm used there: if $g$ is a primitive root mod $p$ - that is, it generates the multiplicative group mod $p$ - and $unotequiv 0mod p$ then $log_g u$ is $min{L: g^L equiv umod n, Lge 0}$. This is always an integer in $[0,p-2]$.
Now, what's with the formula? An integer in $[0,p-1]$ can be interpreted mod $p$, and the possible values are all distinct. Throw in an arbitrary value at zero, and this can be interpreted as a function from $mathbb{Z}/p$ to itself. It shouldn't be, but it can. Every such function can be written as a polynomial function of degree at most $p-1$, and someone went to the trouble of figuring out what that is in this case. The arbitrary value assigned to zero in this case, by the way, is $0$ if $p=2$ and $-1$ if $p$ is odd.
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
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add a comment |
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1 Answer
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$begingroup$
The definition of the discrete logarithm used there: if $g$ is a primitive root mod $p$ - that is, it generates the multiplicative group mod $p$ - and $unotequiv 0mod p$ then $log_g u$ is $min{L: g^L equiv umod n, Lge 0}$. This is always an integer in $[0,p-2]$.
Now, what's with the formula? An integer in $[0,p-1]$ can be interpreted mod $p$, and the possible values are all distinct. Throw in an arbitrary value at zero, and this can be interpreted as a function from $mathbb{Z}/p$ to itself. It shouldn't be, but it can. Every such function can be written as a polynomial function of degree at most $p-1$, and someone went to the trouble of figuring out what that is in this case. The arbitrary value assigned to zero in this case, by the way, is $0$ if $p=2$ and $-1$ if $p$ is odd.
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
$endgroup$
add a comment |
$begingroup$
The definition of the discrete logarithm used there: if $g$ is a primitive root mod $p$ - that is, it generates the multiplicative group mod $p$ - and $unotequiv 0mod p$ then $log_g u$ is $min{L: g^L equiv umod n, Lge 0}$. This is always an integer in $[0,p-2]$.
Now, what's with the formula? An integer in $[0,p-1]$ can be interpreted mod $p$, and the possible values are all distinct. Throw in an arbitrary value at zero, and this can be interpreted as a function from $mathbb{Z}/p$ to itself. It shouldn't be, but it can. Every such function can be written as a polynomial function of degree at most $p-1$, and someone went to the trouble of figuring out what that is in this case. The arbitrary value assigned to zero in this case, by the way, is $0$ if $p=2$ and $-1$ if $p$ is odd.
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
$endgroup$
add a comment |
$begingroup$
The definition of the discrete logarithm used there: if $g$ is a primitive root mod $p$ - that is, it generates the multiplicative group mod $p$ - and $unotequiv 0mod p$ then $log_g u$ is $min{L: g^L equiv umod n, Lge 0}$. This is always an integer in $[0,p-2]$.
Now, what's with the formula? An integer in $[0,p-1]$ can be interpreted mod $p$, and the possible values are all distinct. Throw in an arbitrary value at zero, and this can be interpreted as a function from $mathbb{Z}/p$ to itself. It shouldn't be, but it can. Every such function can be written as a polynomial function of degree at most $p-1$, and someone went to the trouble of figuring out what that is in this case. The arbitrary value assigned to zero in this case, by the way, is $0$ if $p=2$ and $-1$ if $p$ is odd.
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
$endgroup$
The definition of the discrete logarithm used there: if $g$ is a primitive root mod $p$ - that is, it generates the multiplicative group mod $p$ - and $unotequiv 0mod p$ then $log_g u$ is $min{L: g^L equiv umod n, Lge 0}$. This is always an integer in $[0,p-2]$.
Now, what's with the formula? An integer in $[0,p-1]$ can be interpreted mod $p$, and the possible values are all distinct. Throw in an arbitrary value at zero, and this can be interpreted as a function from $mathbb{Z}/p$ to itself. It shouldn't be, but it can. Every such function can be written as a polynomial function of degree at most $p-1$, and someone went to the trouble of figuring out what that is in this case. The arbitrary value assigned to zero in this case, by the way, is $0$ if $p=2$ and $-1$ if $p$ is odd.
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
answered Jan 23 at 5:53
jmerryjmerry
11.6k1526
11.6k1526
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$begingroup$
You need to unknow what you know, because it is wrong. There very mush is a function, and behold, even a formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:47
$begingroup$
@IvanNeretin unique formula?
$endgroup$
– Newuser
Jan 23 at 5:52
$begingroup$
There is no such thing as unique formula.
$endgroup$
– Ivan Neretin
Jan 23 at 5:53
2
$begingroup$
Once you have fixed a generator $g$, every non-zero element is of the form $g^k$ for some integer $k$. If you restrict $1leq k leq p-1$, there there is exactly only one $k$. So the function must be unique: if $h = g^r$ then the discrete log function $f$ must give you exactly $f(h) = r$. However since we can define multiple formulas for $f$, there is no unique formula. In particular the one you listed, by Wells, is one such formula (I guess your question would extend to proving this formula).
$endgroup$
– Yong Hao Ng
Jan 23 at 6:03