BA-06 · LESSON 02

Pushes, arithmetic, and control flow

KEY DEFINITIONS

Terms used in this lesson

OP_IF: A Bitcoin Script conditional operation that executes one branch when the consumed condition is true.
OP_VERIFY: A Bitcoin Script operation that consumes a value and immediately fails if it is false.
Witness: Signatures, scripts, or other data supplied to satisfy a Bitcoin output’s spending condition.
OP_ELSE: The marker for the alternate branch of a Bitcoin Script conditional.
OP_ENDIF: The marker that ends a Bitcoin Script conditional block.
OP_SUB: A Bitcoin Script operation that removes two script numbers and pushes the first minus the second.
OP_EQUALVERIFY: A Bitcoin Script operation that compares two values and immediately fails if they differ.
OP_EQUAL: A Bitcoin Script operation that compares two byte strings and pushes true or false.
UTXO: Unspent transaction output: one discrete chunk of bitcoin that can be used as a transaction input.

Data pushes, numeric encodings, conditionals, and verification opcodes compose spending policies.

45 min Difficulty 2/5 Not started

VISUAL MODEL

See the whole mechanism before going step by step

Use the diagrams to orient yourself. Then the guided explanation below teaches every part in depth.

4 explanatory views

CONCEPT-SPECIFIC MAP · p2pkh executionP2PKH checks both the public-key hash and signatureSelect any state to inspect its exact role.

RELATIONSHIP MAPIndependent peer relationships

01
SCRIPT SIGSignature and matching public key
The spender supplies two stack items.
02
DUP HASH160Hash A
The public key is duplicated and HASH160-hashed.
03
EQUALVERIFYhash A = committed A
A mismatch aborts before signature verification.
04
CHECKSIGVerify sig under pubkey
The signature commits to transaction fields selected by the sighash type.
05
RESULTTrue stack
Both identity-of-key and authorization checks must pass.

Rule shown: The supplied public key must hash to the committed value and its signature must verify the transaction message.

STAGE 1 OF 5SCRIPT SIG: Signature and matching public key

The spender supplies two stack items.

Governing relationship: The supplied public key must hash to the committed value and its signature must verify the transaction message. Complete path: 1. SCRIPT SIG, Signature and matching public key. The spender supplies two stack items. 2. DUP HASH160, Hash A. The public key is duplicated and HASH160-hashed. 3. EQUALVERIFY, hash A = committed A. A mismatch aborts before signature verification. 4. CHECKSIG, Verify sig under pubkey. The signature commits to transaction fields selected by the sighash type. 5. RESULT, True stack. Both identity-of-key and authorization checks must pass.

1. State transitionWhat exists, what changes, and what results

BEFORE

Initial witness provides true. Script is OP_IF 7 2 OP_SUB OP_ELSE 9 OP_ENDIF 5 OP_EQUAL.

MECHANISM

Small integer opcodes directly push values from negative one through sixteen. Larger script numbers are byte vectors whose sign uses the high bit of the final byte.

AFTER

The condition chooses which code executes, but both the branch structure and the executed stack operations must remain valid.

Text equivalent: Starting from Initial witness provides true. Script is OP_IF 7 2 OP_SUB OP_ELSE 9 OP_ENDIF 5 OP_EQUAL. The mechanism begins when Small integer opcodes directly push values from negative one through sixteen. Larger script numbers are byte vectors whose sign uses the high bit of the final byte. The resulting lesson is The condition chooses which code executes, but both the branch structure and the executed stack operations must remain valid.

2. Mechanism pathSelect a stage to inspect its job

STAGE 1 OF 5

Push minimally encoded numbers

Small integer opcodes directly push values from negative one through sixteen. Larger script numbers are byte vectors whose sign uses the high bit of the final byte.

Complete path: 1. Push minimally encoded numbers: Small integer opcodes directly push values from negative one through sixteen. Larger script numbers are byte vectors whose sign uses the high bit of the final byte. 2. Consume operands in order: Binary arithmetic pops the top item as one operand and the next item as the other. Subtraction order matters: OP_SUB computes second-from-top minus top. 3. Select a branch: OP_IF removes a condition from the stack. A true value executes the first branch; false skips to OP_ELSE or OP_ENDIF. 4. Verify rather than retain: OP_VERIFY consumes the top value and fails if it is false. Operations such as OP_EQUALVERIFY combine comparison and immediate verification. 5. End with a valid structure: Every conditional must be balanced, executed operations must have enough operands, and resulting values must satisfy the script's final rules.

3. Rule and failure boundaryWhat accepts the result and where it can stop

ACCEPTED PATH

Script numbers use minimally encoded signed-magnitude little-endian bytes, condition values obey minimal-if rules, and only the selected balanced branch executes.

CHANGED CONDITION

Negative zero is false, a redundant sign byte fails when minimal encoding is required, and an unbalanced conditional or VERIFY false stops execution.

Important distinction: OP_IF reads a permanent variable stored in the UTXO. It consumes a stack item supplied or computed during this spend. Script has no mutable variable store attached to an output.

Text equivalent: The accepted path is enforced by Script numbers use minimally encoded signed-magnitude little-endian bytes, condition values obey minimal-if rules, and only the selected balanced branch executes. If the required condition changes, Negative zero is false, a redundant sign byte fails when minimal encoding is required, and an unbalanced conditional or VERIFY false stops execution.

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Learn the idea

Script data pushes place byte strings on the stack. Script numbers use a signed-magnitude little-endian byte encoding with minimal-encoding rules in relevant contexts. Conditional opcodes OP_IF, OP_ELSE, and OP_ENDIF choose which branch executes based on a stack boolean, while skipped branches are parsed but most contained operations are not executed.

GUIDED EXPLANATION1/5 ideas inspected

Push minimally encoded numbers

Small integer opcodes directly push values from negative one through sixteen. Larger script numbers are byte vectors whose sign uses the high bit of the final byte.

The worked example follows the explanation

Inspect every idea above to open it.

Important distinction“OP_IF reads a permanent variable stored in the UTXO.”

It consumes a stack item supplied or computed during this spend. Script has no mutable variable store attached to an output.

FINISH LEARNING FIRST

The questions unlock after every required learning activity

Inspect every guided idea, open the worked example, rebuild its mechanism, and complete the deterministic lesson tool. Your progress is saved automatically.