BA-02 · LESSON 01

Modular arithmetic

Modular arithmetic wraps numbers around a fixed boundary, like hours on a clock. Bitcoin's signature mathematics uses this kind of arithmetic.

35 min Difficulty 1/5 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 · cryptographyFrom secret material to a verifiable commitmentSelect any state to inspect its exact role.

RELATIONSHIP MAPOrdered mechanism flow

01
INPUTMessage or scalar
The exact byte string or private scalar is the starting state.
02
DOMAINTag and encoding
Domain separation and serialization decide which bytes the primitive commits to.
03
PRIMITIVEModular arithmetic
This lesson isolates the exact operation used by Modular arithmetic.
04
ARTIFACTDigest, point, or signature
The result is fixed-size and can be passed to a verifier.
05
CHECKSame bytes, same rule
A changed message, domain, key, or encoding must fail the intended verification.

Rule shown: One-way operations expose a checkable result without exposing the secret input.

STAGE 1 OF 5INPUT: Message or scalar

The exact byte string or private scalar is the starting state.

Governing relationship: One-way operations expose a checkable result without exposing the secret input. Complete path: 1. INPUT, Message or scalar. The exact byte string or private scalar is the starting state. 2. DOMAIN, Tag and encoding. Domain separation and serialization decide which bytes the primitive commits to. 3. PRIMITIVE, Modular arithmetic. This lesson isolates the exact operation used by Modular arithmetic. 4. ARTIFACT, Digest, point, or signature. The result is fixed-size and can be passed to a verifier. 5. CHECK, Same bytes, same rule. A changed message, domain, key, or encoding must fail the intended verification.

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

BEFORE

Solve 3x = 4 mod 7.

MECHANISM

With modulus 7, the allowed remainders are 0 through 6. Any integer maps to exactly one of those values.

AFTER

An inverse lets modular equations be solved without ordinary fractions.

Text equivalent: Starting from Solve 3x = 4 mod 7. The mechanism begins when With modulus 7, the allowed remainders are 0 through 6. Any integer maps to exactly one of those values. The resulting lesson is An inverse lets modular equations be solved without ordinary fractions.

2. Mechanism pathSelect a stage to inspect its job

STAGE 1 OF 5

Choose a modulus

With modulus 7, the allowed remainders are 0 through 6. Any integer maps to exactly one of those values.

Complete path: 1. Choose a modulus: With modulus 7, the allowed remainders are 0 through 6. Any integer maps to exactly one of those values. 2. Reduce after addition: Add normally, then take the remainder. Five plus four is nine, and nine mod seven is two. 3. Reduce after multiplication: Multiplication works the same way. Five times four is twenty, and twenty mod seven is six. 4. Use multiplicative inverses: An inverse of a is a value b for which a times b equals one modulo n. Under modulus 7, the inverse of 3 is 5 because 15 mod 7 is 1. 5. Connect to curve operations: Bitcoin's secp256k1 arithmetic uses very large finite fields and a finite group of curve points. The numbers are huge, but the same closure and inverse ideas apply.

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

ACCEPTED PATH

The inverse must multiply to one modulo n, and substituting the solved x must reproduce the equation's right-hand remainder.

CHANGED CONDITION

A value sharing a factor with the modulus has no inverse, and non-whole inputs are outside this finite-group exercise.

Important distinction: The mod operation discards information randomly. It deterministically maps every integer to a remainder class. Cryptographic security comes from carefully chosen hard problems, not from unpredictable remainder calculation.

Text equivalent: The accepted path is enforced by The inverse must multiply to one modulo n, and substituting the solved x must reproduce the equation's right-hand remainder. If the required condition changes, A value sharing a factor with the modulus has no inverse, and non-whole inputs are outside this finite-group exercise.

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Learn the idea

Modular arithmetic keeps only the remainder after division by a fixed number called the modulus. Written a mod n, two integers are equivalent when they leave the same remainder after division by n. Closure means that applying the group operation to allowed values produces another allowed value. Cryptographic groups use wraparound arithmetic and closure so operations stay inside a finite set.

GUIDED EXPLANATION1/5 ideas inspected

Choose a modulus

With modulus 7, the allowed remainders are 0 through 6. Any integer maps to exactly one of those values.

The worked example follows the explanation

Inspect every idea above to open it.

Important distinction“The mod operation discards information randomly.”

It deterministically maps every integer to a remainder class. Cryptographic security comes from carefully chosen hard problems, not from unpredictable remainder calculation.

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.