What are the curly brackets in this cryptographic hash function definition?
Reading the Schnorr signature Wikipedia page, I stumbled upon the following statement:
All users agree on a cryptographic hash function $H:{0,1}^*tomathbb{Z}_q$.
What do these curly brackets mean here and how exactly is the hash function's input domain defined? Normally, you can use whatever input you want for a CHF/ PRF.
notation
asked Dec 8 at 13:45
Paul Berg
1407
add a comment |
Reading the Schnorr signature Wikipedia page, I stumbled upon the following statement:
All users agree on a cryptographic hash function $H:{0,1}^*tomathbb{Z}_q$.
What do these curly brackets mean here and how exactly is the hash function's input domain defined? Normally, you can use whatever input you want for a CHF/ PRF.
notation
asked Dec 8 at 13:45
Paul Berg
1407
2
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21
add a comment |
Reading the Schnorr signature Wikipedia page, I stumbled upon the following statement:
All users agree on a cryptographic hash function $H:{0,1}^*tomathbb{Z}_q$.
What do these curly brackets mean here and how exactly is the hash function's input domain defined? Normally, you can use whatever input you want for a CHF/ PRF.
notation
asked Dec 8 at 13:45
Paul Berg
1407
Reading the Schnorr signature Wikipedia page, I stumbled upon the following statement:
All users agree on a cryptographic hash function $H:{0,1}^*tomathbb{Z}_q$.
What do these curly brackets mean here and how exactly is the hash function's input domain defined? Normally, you can use whatever input you want for a CHF/ PRF.
notation
notation
asked Dec 8 at 13:45
Paul Berg
1407
asked Dec 8 at 13:45
Paul Berg
1407
edited Dec 8 at 16:30
asked Dec 8 at 13:45
Paul Berg
1407
asked Dec 8 at 13:45
Paul Berg
1407
asked Dec 8 at 13:45
Paul Berg
1407
1407
2
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21
add a comment |
2
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21
2
2
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21
add a comment |
1 Answer
1
active
oldest
votes
This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation.
${0,1}$ is the set consisting of $0$ and $1$, so the set of all single bits.
For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S times S$.
So strictly speaking ${0,1}^n$ refers to the set of $n$-tuples of bits, however we generally call these "bitstrings of length $n$".
Finally, ${0,1}^*$ is defined as $${0,1}^*=bigcup_{ninmathbb{N}_0}{0,1}^n.$$ I.e. it refers to the (infinite) set of all finite length bitstrings.
answered Dec 8 at 13:57
Maeher
3,49211730
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
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oldest
votes
active
oldest
votes
This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation.
${0,1}$ is the set consisting of $0$ and $1$, so the set of all single bits.
For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S times S$.
So strictly speaking ${0,1}^n$ refers to the set of $n$-tuples of bits, however we generally call these "bitstrings of length $n$".
Finally, ${0,1}^*$ is defined as $${0,1}^*=bigcup_{ninmathbb{N}_0}{0,1}^n.$$ I.e. it refers to the (infinite) set of all finite length bitstrings.
answered Dec 8 at 13:57
Maeher
3,49211730
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
add a comment |
This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation.
${0,1}$ is the set consisting of $0$ and $1$, so the set of all single bits.
For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S times S$.
So strictly speaking ${0,1}^n$ refers to the set of $n$-tuples of bits, however we generally call these "bitstrings of length $n$".
Finally, ${0,1}^*$ is defined as $${0,1}^*=bigcup_{ninmathbb{N}_0}{0,1}^n.$$ I.e. it refers to the (infinite) set of all finite length bitstrings.
answered Dec 8 at 13:57
Maeher
3,49211730
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
add a comment |
This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation.
${0,1}$ is the set consisting of $0$ and $1$, so the set of all single bits.
For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S times S$.
So strictly speaking ${0,1}^n$ refers to the set of $n$-tuples of bits, however we generally call these "bitstrings of length $n$".
Finally, ${0,1}^*$ is defined as $${0,1}^*=bigcup_{ninmathbb{N}_0}{0,1}^n.$$ I.e. it refers to the (infinite) set of all finite length bitstrings.
answered Dec 8 at 13:57
Maeher
3,49211730
This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation.
${0,1}$ is the set consisting of $0$ and $1$, so the set of all single bits.
For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S times S$.
So strictly speaking ${0,1}^n$ refers to the set of $n$-tuples of bits, however we generally call these "bitstrings of length $n$".
Finally, ${0,1}^*$ is defined as $${0,1}^*=bigcup_{ninmathbb{N}_0}{0,1}^n.$$ I.e. it refers to the (infinite) set of all finite length bitstrings.
answered Dec 8 at 13:57
Maeher
3,49211730
edited Dec 8 at 16:22
answered Dec 8 at 13:57
Maeher
3,49211730
answered Dec 8 at 13:57
Maeher
3,49211730
answered Dec 8 at 13:57
Maeher
3,49211730
3,49211730
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
add a comment |
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
1
1
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
+1 though I would add that this notation is likely borrowed from regular expressions / automata theory where '*' (aka the Kleene star operator) means "Zero or more occurrences of the preceding symbol", and is well-defined over sets in the way you describe.
– Mike Ounsworth
Dec 8 at 20:03
add a comment |
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2
This means that the domain input domain is an unlimited number of input bits that are either 0 or 1 (it's understood in context to refer to binary). Not sure this is really about crypto though. It seems this is just math.
– forest
Dec 8 at 13:55
@forest I see, should a mod move this to the math SE?
– Paul Berg
Dec 8 at 16:21