Storage SM mechanism
The Storage SM is one of zkProver’s secondary state machines, and thus receives instructions from the Main SM called Storage Actions. The Storage Actions basically refer to the verification of CRUD operations executed by the Main SM.
The Storage SM is designed as a microprocessor and it is therefore composed of three parts:
 Storage Assembly code,
 Storage executor code,
 Storage PIL code.
Storage assembly¶
The Storage Assembly is the interpreter between the Main State Machine and its own Executor.
The Storage SM receives instructions from the Main SM written in zkASM. It then generates a JSONfile containing the corresponding rules and logic, which are stored in a special ROM for the Storage SM.
The Storage SM has a primary Storage Assembly code that maps each instruction of the Main SM to the secondary Assembly code corresponding to each basic operation. These basic operations are mainly the CREATE, READ, UPDATE and DELETE, as discussed in previous sections.
Considering some special cases, there are eight secondary Storage Assembly codes allinall, each for a distinct basic operation. We list these in the table below.
Storage Actions  File Names  Code Names  Action Selectors In Primary zkASM Code 

READ  Get  Get  isGet() 
UPDATE  Set_Update  SU  isSetUpdate() 
CREATE new value at a found leaf  Set_InsertFound  SIF  isSetInsertFound() 
CREATE new value at a zero node  Set_InsertNotFound  SINF  isSetInsertNotFound() 
DELETE last nonzero node  Set_DeleteLast  SDL  isSetDeleteLast() 
DELETE leaf with nonzero sibling  Set_DeleteFound  SDF  isSetDeleteFound() 
DELETE leaf with zero sibling  Set_DeleteNotFound  SDNF  isSetDeleteNotFound() 
SET a zero node to zero  Set_ZeroToZero  SZTZ  isSetZeroToZero() 
Input and output states of the Storage SM are literally SMTs, given in the form of; the Merkle roots, the relevant siblings, as well as the keyvalue pairs.
Note that state machines use registers in the place of variables. All values needed, for carrying out the basic operations, are stored by the primary Assembly code in the following registers;
HASH_LEFT, HASH_RIGHT, OLD_ROOT, NEW_ROOT, VALUE_LOW, VALUE_HIGH, SIBLING_VALUE_HASH, RKEY, SIBLING_RKEY, RKEY_BIT, LEVEL.
The SIBLING_VALUE_HASH and SIBLING_RKEY registers are only used by the Set_InsertFound and the Set_DeleteFound secondary Assembly codes. The rest of the registers are used in all the secondary Assembly codes.
SMT Action selectors in primary assembly code¶
The Primary Assembly Code maps the Main SM instructions to the relevant Storage Actions using selectors. Like switches can either be ON or OFF, selectors can either be 1 or 0, where 1 means the action is selected for execution, while 0 means the instruction does not tally with the required action so a “jump if zero” JMPZ is applied.
The primary Assembly code uses selectors by following the sequence in which these Storage Actions are listed in the above table. That is,
 It first checks if the required action is a Get. If it is so, the storage_sm_get.zkasm code is fetched for execution.
 If not, it checks if the required action is Set_Update. If it is so, the storage_sm_set_update.zkasm code is fetched for execution.
 If not, it continues to check if the required action is Set_InsertFound. If it is so, the storage_sm_set_insert_found.zkasm code is fetched for execution.
 If not, it continues in the same way until the correct action is selected, in which case the corresponding code is fetched for execution.
That’s all the primary Storage Assembly code does and the details of how each of the SMT Actions is stipulated in the individual secondary Assembly codes.
The primary and secondary Storage Assembly files are stored as JSONfiles in the Storage ROM, ready to be fetched as function calls by the Storage Executor.
The UPDATE zkASM code¶
Take as an example the Set_UPDATE zkASM code. The primary Storage Assembly code uses the selector isSetUpdate() for Set_UPDATE.
Note that an UPDATE involves the following actions:
 Reconstructs the corresponding key, from both the remaining key found at the leaf and keybits used to navigate to the leaf.
 Ascertains that indeed the old value was included in the old root,
 Carries out the UPDATE of the old value with the new value, as well as updating all nodes along the path from the leaf to the root.
There is only one Set_UPDATE Assembly code, storage_sm_set_update.zkasm, for all the above three computations.
Key reconstruction in zkASM¶
Key Reconstruction is achieved in two steps: positioning of the bit “1” in the LEVEL register, and using the LEVEL register to climb the RKey. That is, append the path bit last used in navigation to the correct RKey part.
 Positioning the bit “1” in the LEVEL register
The Set_UPDATE zkASM code, first initialises the LEVEL register to (1,0,0,0). Then uses the GetLevelBit() function to read the two leastsignificant bits of the leaf level, which happens in two cases, each with its own two subcases:
Case 1. If the leastsignificant bit of leaf level is 0, then the GetLevelBit() function is used again to read the second leastsignificant bit of the leaf level.
 Subcase 1.1: If the second leastsignificant bit of the leaf level is 0, it means the leaf level is a multiple of 4, which is equivalent to 0 because leaf level works in modulo 4. So, the LEVEL register must remain as (1,0,0,0).
 Subcase 1.2: If the second leastsignificant bit of the leaf level is 1, it means the leaf level in its binary form ends with a 10. Hence, leaf level is a number of the form 2 + 4k, for some positive integer k. As a result, the LEVEL register must be rotated to the position, (0,0,1,0). The code therefore applies ROTATE_LEVEL twice to LEVEL = (1,0,0,0) in order to bring it to (0,0,1,0).
Case 2. If the leastsignificant bit of leaf level is 1; then, the LEVEL register is rotated three times to the left, using ROTATE_LEVEL, and bringing the LEVEL register to (0,1,0,0). Next, the GetLevelBit() function is used again to read the second leastsignificant bit of the leaf level.
 Subcase 2.1: If the second leastsignificant bit of the leaf level is 0, it means the leaf level in its binary form ends with a 01. That is, leaf level is a number of the form 1 + 4k, for some positive integer k. And thus, the LEVEL register must remain in its current position, (0,1,0,0). So it does not need to be rotated.

Subcase 2.2: Otherwise, the second leastsignificant bit of the leaf level is 1, which means the leaf level in its binary form ends with a 11. Hence, leaf level is a number of the form 3 + 4k, for some positive integer k. Consequently, the LEVEL register needs to be rotated from the current position (0,1,0,0) to the position (0,0,0,1).

Using LEVEL to “climb the RKey”
The Remaining Key is fetched using the GetRKey() function and stored in the RKEY register.
When climbing the tree, there are two functions that are used in the code: the CLIMB_RKEY and the ROTATE_LEVEL.
 First, the LEVEL register is used to pinpoint the correct part of the Remaining Key to which the pathbit last used in the navigation must be appended.
 Second, the ROTATE_LEVEL is used to rotate the LEVEL register once.
 The CLIMB_RKEY is used. Firstly, to shift the value of the pinpointed RKey part one position to the left. Secondly, to insert the last used path bit to the leastsignificant position of the shiftedvalue of the pinpointed RKey part.
The above two steps are repeated until all the path bits used in navigation have been appended. Later, equality between the reconstructed key and the original key is checked.
Checking inclusion of old value in old root¶
The above key reconstruction, together with checking inclusion of the old value in the old root and updating the old value to the new value, are carried out simultaneously.
Since checking inclusion of the old value in the old root follows the same steps as the update of the old value to the new value, the corresponding lines in the Assembly code are similar. It suffices therefore to explain only one of these two computations.
Next is the discussion of the update of the old value to the new value.
Update part of Set_UPDATE¶
All values, \(\text{V}_{0123}=\big(\text{V}_{0},\text{V}_{1},\text{V}_{2},\text{V}_{3},\text{V}_{4},\text{V}_{5},\text{V}_{6},\text{V}_{7}\big)\) are 256bit long and expressed as lower half and higher half as, VALUE_LOW \(=\big(\text{V}_{0},\text{V}_{1},\text{V}_{2},\text{V}_{3}\big)\) and VALUE_HIGH \(=\big(\text{V}_{4},\text{V}_{5},\text{V}_{6},\text{V}_{7} \big)\).

Computing the new leaf value

The functions GetValueLow() and GetValueHigh() are used to fetch VALUE_LOW \(=\big(\text{V}_{0},\text{V}_{1},\text{V}_{2},\text{V}_{3}\big)\) and VALUE_HIGH \(=\big(\text{V}_{4},\text{V}_{5},\text{V}_{6},\text{V}_{7}\big)\), respectively.

The VALUE_LOW \(= \big(\text{V}_{0},\text{V}_{1},\text{V}_{2},\text{V}_{3}\big)\) is stored in a register called HASH_LEFT, whilst VALUE_HIGH \(=\big(\text{V}_{4},\text{V}_{5},\text{V}_{6},\text{V}_{7}\big)\) is stored in another register called HASH_RIGHT.

The hashed value of \(\text{V}_{0123}\) is computed using HASH0 as, \(\text{HASH0}\big(\text{HASH}\_ {\text{LEFT}}\\text{HASH}\_ {\text{RIGHT}}\big)\). Note that this is in fact, \(\text{POSEIDON}\big(0\0\0\0\\text{VALUE}\_ {\text{LOW}}\\text{VALUE}\_ {\text{HIGH}}\big)\). The hashed value is then stored in HASH_RIGHT.
!!!info This means the HASH_RIGHT and the HASH_LOW are ‘makeshift’ registers. Whenever a value is stored in it, the old value that was previously stored therein is simply pushed out. They hold values only for the next computation.

Next the Rkey is copied into the HASH_LEFT register. And the leaf value is computed by using HASH1 as, \(\text{HASH1}\big(\text{HASH}\_ {\text{LEFT}}\\text{HASH}\_ {\text{RIGHT}}\big)\). i.e., The value of the leaf is, \(\text{HASH1}\big( \text{RKey}\\text{HashedValue}\big)\). The leaf value is then copied into another register called NEW_ROOT.

Climbing the SMT
Check if the path bit that led to the leaf is 0 or 1, by using the GetNextKeyBit() function.
Case 1: If the path bit (called ‘key bit’ in the code) is 0, then the corresponding sibling is on the right. Therefore, using ‘jump if zero’ JMPZ, the code jumps to the SU_SiblingIsRight routine.

The leaf value in NEW_ROOT is pushed into the HASH_LEFT register.

The hash value of the sibling node is fetched, using the GetSiblingHash() function. And it is pushed into the HASH_RIGHT register.

The hash value of the parent node is computed using HASH0 as follows, \(\text{HASH0}\big(\text{HASH}\_ {\text{LEFT}} \\text{HASH}\_{\text{RIGHT}}\big)\).
The parent node is \(\text{POSEIDON}\big(0\0\0\0\\text{LeafValue}\\text{SiblingHash}\big)\).
Case 2: If the path bit is 1, then the corresponding sibling is on the left. The routine SU_SiblingIsRight is then executed.

The leaf value in NEW_ROOT is pushed into the HASH_RIGHT register.

The hash value of the sibling node is fetched, using the GetSiblingHash() function. And it is pushed into the HASH_LEFT register.

The hash value of the parent node is computed using HASH0 as follows, \(\text{HASH0}\big(\text{HASH}\_ {\text{LEFT}}\\text{HASH}\_ \text{RIGHT}\big)\).
The parent node is \(\text{POSEIDON}\big(0\0\0\0\\text{SiblingHash}\\text{LeafValue}\big)\).
 Check if tree top has been reached
The code uses the function GetTopTree() to check is the top of the tree has been reached.
Case 1. If GetTopTree() returns 1, then Step 2 is repeated. But this time using the hash value of the corresponding sibling at the next level (at leaf level  1).
Case 2. If GetTopTree() returns 0, then the code jumps to the SU_Latch routine.
The SU_Latch is an overall routine for the entire Set_UPDATE Assembly code. It is here where,

Equality between the reconstructed key and the original key is checked.

Equality between the computed old root value and the original old root is checked.
Once consistency is established both between the keys and the old roots, then all new values; the new root, the new hash value, and the new leaf value are set using LATCH_SET.
Remaining secondary assembly codes¶
The Assembly codes for the other seven SMT Actions to a certain extent follow a similar pattern except for a few cases where especially adjusted routines are used.
Actions such as:
 The Set_InsertFound (or SIF) may involve a change in the topology of the SMT by extending a branch once or several times.
In cases where a branch has been extended, the SIF Assembly code, when computing the new root, uses another routine called SIF_ClimbBranch just for updating values along the newly extended branch. This is done in addition to the SIF_ClimbTree, which is the exact same routine as the aforementioned SU_ClimbTree of the Set_UPDATE case.
It is for the same reason that SIF Assembly utilizes special registers: the SIBLING_VALUE_HASH and SIBLING_RKEY.
 The opposite SMT Action, the Set_DeleteFound or SDF, may entail a previously extended branch being reversed.
As in the SIF case, if a branch had been extended but now the extension needs to be reversed due to a deleted leaf value, a special routine called SDF_ClimbBranch is used when updating values of nodes along the newly shortened branch. This SDF_ClimbBranch routine is the exact same routine as the SIF_ClimbBranch. Similarly, the SDF Assembly code uses the SDF_ClimbTree as in the Set_UPDATE Assembly.
Note also that there is only one Get Assembly code for the READ SMT Action, and the rest of the secondary Assembly codes are Set Assembly codes differing according to their respective SMT Actions. So Get uses LATCH_GET at the end of a run, while the Set codes use LATCH_SET.