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< ol class = "chapter" > < li class = "chapter-item expanded affix " > < a href = "../index.html" > Orchard< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts.html" > < strong aria-hidden = "true" > 1.< / strong > Concepts< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../concepts/preliminaries.html" > < strong aria-hidden = "true" > 1.1.< / strong > Preliminaries< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../user.html" > < strong aria-hidden = "true" > 2.< / strong > User Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../user/keys.html" > < strong aria-hidden = "true" > 2.1.< / strong > Creating keys and addresses< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/creating-notes.html" > < strong aria-hidden = "true" > 2.2.< / strong > Creating notes< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/spending-notes.html" > < strong aria-hidden = "true" > 2.3.< / strong > Spending notes< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/integration.html" > < strong aria-hidden = "true" > 2.4.< / strong > Integration into an existing chain< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design.html" > < strong aria-hidden = "true" > 3.< / strong > Design< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/keys.html" > < strong aria-hidden = "true" > 3.1.< / strong > Keys and addresses< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/actions.html" > < strong aria-hidden = "true" > 3.2.< / strong > Actions< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/commitments.html" class = "active" > < strong aria-hidden = "true" > 3.3.< / strong > Commitments< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/commitment-tree.html" > < strong aria-hidden = "true" > 3.4.< / strong > Commitment tree< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/nullifiers.html" > < strong aria-hidden = "true" > 3.5.< / strong > Nullifiers< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/signatures.html" > < strong aria-hidden = "true" > 3.6.< / strong > Signatures< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/circuit.html" > < strong aria-hidden = "true" > 3.7.< / strong > Circuit< / a > < / li > < / ol > < / li > < / ol >
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< h1 > < a class = "header" href = "#commitments" id = "commitments" > Commitments< / a > < / h1 >
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< p > As in Sapling, we require two kinds of commitment schemes in Orchard:< / p >
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< ul >
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< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord" > < span class = "mord mathit" > H< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > p< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > c< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > is a linearly homomorphic commitment scheme with perfect hiding,
and strong binding reducible to DL.< / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68333em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathit" > S< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > t< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > are commitment schemes with perfect hiding, and
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strong binding reducible to DL.< / li >
< / ul >
< p > By " strong binding" we mean that the scheme is collision resistant on the input and
randomness.< / p >
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< p > We instantiate < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord" > < span class = "mord mathit" > H< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > p< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > c< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > with a Pedersen commitment, and use it for
value commitments:< / p >
< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.44444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathsf" > c< / span > < span class = "mord mathsf" style = "margin-right:0.01389em;" > v< / span > < / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.008448em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathit" > H< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > p< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > c< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.758448em;" > < span style = "top:-2.4530000000000003em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > r< / span > < span class = "mord mathsf mtight" > c< / span > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > v< / span > < / span > < / span > < / span > < / span > < span style = "top:-3.1473400000000002em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" > c< / span > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > v< / span > < / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.247em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > v< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / span > < / p >
< p > We instantiate < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68333em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathit" > S< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > t< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > with Sinsemilla, and use them
for all other commitments:< / p >
< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathsf" > i< / span > < span class = "mord mathsf" style = "margin-right:0.01389em;" > v< / span > < span class = "mord mathsf" > k< / span > < / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.1834479999999998em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathit" > S< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > t< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.9334479999999998em;" > < span style = "top:-2.4530000000000003em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > r< / span > < span class = "mord mathsf mtight" > i< / span > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > v< / span > < span class = "mord mathsf mtight" > k< / span > < / span > < / span > < / span > < / span > < span style = "top:-3.1473400000000002em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" > i< / span > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > v< / span > < span class = "mord mathsf mtight" > k< / span > < / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.247em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathsf" > a< / span > < span class = "mord mathsf" > k< / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathsf" > n< / span > < span class = "mord mathsf" > k< / span > < / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / span >
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.44444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathsf" > c< / span > < span class = "mord mathsf" > m< / span > < / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.7473380000000001em;" > < span style = "top:-2.4530000000000003em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" style = "margin-right:0.01389em;" > r< / span > < span class = "mord mathsf mtight" > c< / span > < span class = "mord mathsf mtight" > m< / span > < / span > < / span > < / span > < / span > < span style = "top:-3.1362300000000003em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord mathsf mtight" > c< / span > < span class = "mord mathsf mtight" > m< / span > < / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.247em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord text" > < span class = "mord" > rest of note< / span > < / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / span > < / p >
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< p > This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient
2021-01-21 04:15:54 -08:00
Sinsemilla instead of Bowe--Hopwood Pedersen hashes.< / p >
< p > Note that we also deviate from Sapling by using < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathit" > S< / span > < span class = "mord mathit" > h< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > r< / span > < span class = "mord mathit" > t< / span > < span class = "mord mathit" > C< / span > < span class = "mord mathit" > o< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > m< / span > < span class = "mord mathit" > i< / span > < span class = "mord mathit" > t< / span > < / span > < / span > < / span > < / span > to deriving < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathsf" > i< / span > < span class = "mord mathsf" style = "margin-right:0.01389em;" > v< / span > < span class = "mord mathsf" > k< / span > < / span > < / span > < / span > < / span >
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instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit,
at the cost of requiring < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathsf" style = "margin-right:0.01389em;" > r< / span > < span class = "mord mathsf" > i< / span > < span class = "mord mathsf" style = "margin-right:0.01389em;" > v< / span > < span class = "mord mathsf" > k< / span > < / span > < / span > < / span > < / span > to be part of the full viewing key.< / p >
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