reorganised chapter and corrected definition of VRF transcripts

docs/sect-block-production.md

@@ -120,6 +120,46 @@ The numbers should be treated as 64-bit rational numbers.
 :::
 ### -sec-num- Primary Block Production Lottery {#id-primary-block-production-lottery}
 
+###### Definition -def-num- BABE Slot VRF transcript, output, and proof  {#defn-babe-vrf-transcript}
+:::definition
+
+The BABE block production lottery requires a specific transcript structure ([Definition -def-num-ref-](id-cryptography-encoding#defn-vrf-transcript)). That structure is used by both primary slots ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery)) and secondary slots ([Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
+
+$$
+{t}_{{1}}\leftarrow\text{Transcript}{\left(\text{'BABE'}\right)}
+$$
+$$
+{t}_{{2}}\leftarrow\text{append}{\left({t}_{{1}},\text{'slot number'},{s}\right)}
+$$
+$$
+{t}_{{3}}\leftarrow\text{append}{\left({t}_{{2}},\text{'current epoch'},{e}_{{i}}\right)}
+$$
+$$
+{t}_{{4}}\leftarrow\text{append}{\left({t}_{{3}},\text{'chain randomness'},{r}\right)}
+$$
+$$
+{t}_{{5}}\leftarrow\text{append}{\left({t}_{{4}},\text{'vrf-nm-pk'},{p}_{{k}}\right)}
+$$
+$$
+{t}_{{6}}\leftarrow\text{meta-ad}{\left({t}_{{5}},\text{'VRFHash'},\text{False}\right)}
+$$
+$$
+{t}_{{7}}\leftarrow\text{meta-ad}{\left({t}_{{6}},{64}_{\text{le}},\text{True}\right)}
+$$
+$$
+{h}\leftarrow\text{prf}{\left({t}_{{7}},\text{False}\right)}
+$$
+$$
+{d}={s}_{{k}}\cdot{h}
+$$
+$$
+{\pi}\leftarrow\text{dleq\_prove}{\left({t}_{{7}},{h}\right)}
+$$
+
+The operators are defined in [Definition -def-num-ref-](id-cryptography-encoding#defn-strobe-operations), $\text{dleq\_prove}$ in [Definition -def-num-ref-](id-cryptography-encoding#defn-vrf-dleq-prove). The computed outputs, ${d}$ and ${\pi}$, are included in the block Pre-Digest ([Definition -def-num-ref-](sect-block-production#defn-babe-header)).
+
+:::
+
 A block producer aiming to produce a block during ${\mathcal{{E}}}_{{n}}$ should run the $\text{Block-Production-Lottery}$ algorithm to identify the slots it is awarded. These are the slots during which the block producer is allowed to build a block. The session secret key, ${s}{k}$, is the block producer lottery secret key, and ${n}$ is the index of the epoch for whose slots the block producer is running the lottery.
 
 ###### Algorithm -algo-num- Block Production Lottery {#algo-block-production-lottery}
@@ -136,7 +176,6 @@ where $\text{Epoch-Randomness}$ is defined in ([Definition -def-num-ref-](sect-b
 :::info
 The secondary slots ([Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)) are running alongside the primary block production lottery and mainly serve as a fallback to in case no authority was selected in the primary lottery.
 :::
-
 ###### Definition -def-num- Secondary Slots {#defn-babe-secondary-slots}
 :::definition
 
@@ -160,46 +199,7 @@ $$
 
 - ${A}_{{l}}$ is the lengths of the authority list ([Definition -def-num-ref-](chap-sync#defn-authority-list)).
 
-If ${i}_{{d}}$ points to the authority, that authority must claim the secondary slot by creating a BABE VRF transcript ([Definition -def-num-ref-](sect-block-production#defn-babe-vrf-transcript)). The resulting values ${o}$ and ${p}$ are then used in the Pre-Digest item ([Definition -def-num-ref-](sect-block-production#defn-babe-header)). In the case of secondary slots with plain outputs, respectively the Pre-Digest being of value *2*, the transcript respectively the VRF is skipped.
-
-:::
-###### Definition -def-num- BABE Slot VRF transcript {#defn-babe-vrf-transcript}
-:::definition
-
-The BABE block production lottery requires a specific transcript structure ([Definition -def-num-ref-](id-cryptography-encoding#defn-vrf-transcript)). That structure is used by both primary slots ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery)) and secondary slots ([Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
-
-$$
-{t}_{{1}}\leftarrow\text{Transcript}{\left(\text{'BABE'}\right)}
-$$
-$$
-{t}_{{2}}\leftarrow\text{append}{\left({t}_{{1}},\text{'slot number'},{s}\right)}
-$$
-$$
-{t}_{{3}}\leftarrow\text{append}{\left({t}_{{2}},\text{'current epoch'},{e}_{{i}}\right)}
-$$
-$$
-{t}_{{4}}\leftarrow\text{append}{\left({t}_{{3}},\text{'chain randomness'},{r}\right)}
-$$
-$$
-{t}_{{5}}\leftarrow\text{append}{\left({t}_{{4}},\text{'vrf-nm-pk'},{p}_{{k}}\right)}
-$$
-$$
-{t}_{{6}}\leftarrow\text{meta-ad}{\left({t}_{{5}},\text{'VRFHash'},\text{False}\right)}
-$$
-$$
-{t}_{{7}}\leftarrow\text{meta-ad}{\left({t}_{{6}},{64}_{\text{le}},\text{True}\right)}
-$$
-$$
-{h}\leftarrow\text{prf}{\left({t}_{{7}},\text{False}\right)}
-$$
-$$
-{o}={s}_{{k}}\cdot{h}
-$$
-$$
-{p}\leftarrow\text{dleq\_prove}{\left({t}_{{7}},{h}\right)}
-$$
-
-The operators are defined in [Definition -def-num-ref-](id-cryptography-encoding#defn-strobe-operations), $\text{dleq\_prove}$ in [Definition -def-num-ref-](id-cryptography-encoding#defn-vrf-dleq-prove). The computed outputs, ${o}$ and ${p}$, are included in the block Pre-Digest ([Definition -def-num-ref-](sect-block-production#defn-babe-header)).
+If ${i}_{{d}}$ points to the authority, that authority must claim the secondary slot by creating a BABE VRF transcript ([Definition -def-num-ref-](sect-block-production#defn-babe-vrf-transcript)). The resulting values ${d}$ and ${\pi}$ are then used in the Pre-Digest item ([Definition -def-num-ref-](sect-block-production#defn-babe-header)). In the case of secondary slots with plain outputs, respectively the Pre-Digest being of value *2*, the transcript respectively the VRF is skipped.
 
 :::
 ## -sec-num- Slot Number Calculation {#sect-slot-number-calculation}
@@ -299,7 +299,7 @@ Throughout each epoch, each block producer should run [Invoke-Block-Authoring](s
 The **Pre-Digest**, or BABE header, ${P}$, is a varying datatype of the following format:
 
 $$
-{P}={\left\lbrace\begin{matrix}{1}&\rightarrow&{\left({a}_{\text{id}},{s},{o},{p}\right)}\\{2}&\rightarrow&{\left({a}_{\text{id}},{s}\right)}\\{3}&\rightarrow&{\left({a}_{\text{id}},{s},{o},{p}\right)}\end{matrix}\right.}
+{P}={\left\lbrace\begin{matrix}{1}&\rightarrow&{\left({a}_{\text{id}},{s},{d},{\pi}\right)}\\{2}&\rightarrow&{\left({a}_{\text{id}},{s}\right)}\\{3}&\rightarrow&{\left({a}_{\text{id}},{s},{d},{\pi}\right)}\end{matrix}\right.}
 $$
 
 **where**  
@@ -309,9 +309,9 @@ $$
 
 - ${s}$ is the slot number ([Definition -def-num-ref-](sect-block-production#defn-epoch-slot)).
 
-- ${o}$ is VRF output ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery) respectively [Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
+- ${d}$ is VRF output ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery) respectively [Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
 
-- ${p}$ is VRF proof ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery) respectively [Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
+- ${\pi}$ is VRF proof ([Block-Production-Lottery](sect-block-production#algo-block-production-lottery) respectively [Definition -def-num-ref-](sect-block-production#defn-babe-secondary-slots)).
 
 The Pre-Digest must be included as a digest item of Pre-Runtime type in the header digest ([Definition -def-num-ref-](chap-state#defn-digest)) ${H}_{{d}}{\left({B}\right)}$.
 :::