COSMO · STUDY + TEACHING DECK

Number Theory

December 6, 2025

A question worth sharing

Introductory Cryptography

Use modular inverses and one-way arithmetic to separate encryption from decryption.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use plaintext, ciphertext, modular inverse to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Encrypt letter value 7 with affine rule E(x)=5x+8 mod 26.

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

plaintext

original message

ciphertext

encoded message

modular inverse

number undoing multiplication mod n

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

E(x)\equiv ax+b\pmod m,\qquad D(y)\equiv a^{-1}(y-b)\pmod m

The idea

Use modular inverses and one-way arithmetic to separate encryption from decryption.

What to remember

Encryption must be easy to apply, while decryption requires the right inverse or secret.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Find the inverse of 5 mod 26.

Predict each next step

Use the Euclidean algorithm or inspection.
5(21)=105.
105 = 1 mod 26.
The inverse is 21.

Which step needs a reason, not just a calculation?

See the structure

The structure behind introductory cryptography

E(x)\equiv ax+b\pmod m,\qquad D(y)\equiv a^{-1}(y-b)\pmod m

An affine cipher has a unique decryption exactly when a is coprime to the modulus.

Read the representation

Use modular inverses and one-way arithmetic to separate encryption from decryption.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate introductory cryptography

Your shared task

Pairs exchange short affine-cipher messages and justify each inverse used.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

Encryption must be easy to apply, while decryption requires the right inverse or secret.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Decrypt y=17 from E(x)=5x+8 mod 26.

Predict each next step

Subtract 8 to get 9.
Multiply by inverse 21.
21(9)=189.
189 mod 26 is 7.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

An affine cipher ax+b mod m is decryptable exactly when gcd(a,m)=1.

Build the argument

Decryption needs a multiplicative inverse for a.
Such an inverse exists exactly for residues coprime to m.
Then x= $a^{-1}$ (y-b) mod m.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Explain why multiplying by 13 cannot be used in an affine cipher mod 26.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Modern cryptography uses the same modular ideas at much larger scale with carefully studied security assumptions.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

Encrypt letter value 7 with affine rule E(x)=5x+8 mod 26.

Question 2

Explain why this claim is true: An affine cipher ax+b mod m is decryptable exactly when gcd(a,m)=1.

Question 3

Explain why multiplying by 13 cannot be used in an affine cipher mod 26.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

Encryption must be easy to apply, while decryption requires the right inverse or secret.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.