--- title: "Agreeing a key in the open" subtitle: "Secret messages · Trust · Lesson 6" --- # Agreeing a key in the open **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Every time you open a secure web page, your computer and a server agree a key with a stranger, in public, in less than a second. This lesson is the idea behind it, with numbers small enough to follow by hand. :::teacher-note **In one sentence:** Students agree a key over an open channel with a Diffie–Hellman style exchange using small numbers, then use it as the key for XOR. **Students should be able to say:** I can explain why a key cannot simply be sent by radio. I can say which numbers cross the air and which stay private in a key exchange. I can use an agreed number as the key for a cipher. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict ![The program as blocks](https://studio.learningwaypoints.com/tutorials/_pictures/makecode/code/ef9830d64217.png?v=3bb74d45cbdf0417) :::keep-together **One micro:bit has private number 6 (this program) and the other has 15. Both press A, so each hears the other's mix. What does each micro:bit show?** ☐ 6 on one and 15 on the other ☐ 2 on both ☐ 8 on one and 19 on the other ☐ 23 on both My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 2 on both. The micro:bit with 6 receives 19 and shows 19⁶ mod 23 = 2; the one with 15 receives 8 and shows 8¹⁵ mod 23 = 2. ::: ## Run Run the program in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Investigate ![The program as blocks](https://studio.learningwaypoints.com/tutorials/_pictures/makecode/code/ef9830d64217.png?v=3bb74d45cbdf0417) :::keep-together ### Mixes and keys **A micro:bit with private number 3 sends its mix: 5³ mod 23. What number goes out? (5³ = 125.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10 125 ÷ 23 is 5 with remainder 10, so the mix is 10. ::: :::keep-together ### What goes across **Which of these cross the air, and which stay on one micro:bit?** Groups: Crosses the air, Stays on one micro:bit - P and G: {{field-15}} - The mix (8 or 19): {{field-15}} - A private number (6 or 15): {{field-15}} - The shared key (2): {{field-15}} ::: :::teacher-note **Answer:** P and G → Crosses the air; The mix (8 or 19) → Crosses the air; A private number (6 or 15) → Stays on one micro:bit; The shared key (2) → Stays on one micro:bit ::: :::keep-together ### Why the toy numbers are weak **With P = 23 anyone who hears the mixes could simply try every private number from 1 to 22. Why is the real thing safe?** ☐ It sends the mixes twice ☐ It uses a secret radio group ☐ The real P and private numbers have hundreds of digits, so there are far too many to try ☐ Only the first mix is public ::: :::teacher-note **Answer:** The real P and private numbers have hundreds of digits, so there are far too many to try This is the same lesson as the Caesar cipher: a small number of keys can be tried one by one. Real systems use numbers so large that trying them all would take longer than the age of the universe. ::: :::typical ## Modify First press **A** on both micro:bits, so each works out **shared**. Then let **B** send the number 7 locked with **shared**, and show it unlocked on the other. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} :::teacher-note **One possible solution.** Other correct programs are fine; check that the program runs and does what the task asks. ![The program as blocks](https://studio.learningwaypoints.com/tutorials/_pictures/makecode/code/8e4d1408161f.png?v=8cdff3ae1aecc675) ::: ::: ## Make Put it together: **A** sends your mix and each micro:bit works out the shared key and shows a tick; **B** sends a PIN locked with it; the other micro:bit unlocks and shows the PIN. The key was never sent. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} :::teacher-note **One possible solution.** Other correct programs are fine; check that the program runs and does what the task asks. ![The program as blocks](https://studio.learningwaypoints.com/tutorials/_pictures/makecode/code/0f8a5df4c891.png?v=cc7b59639bfeda05) ::: ## Exit ticket :::keep-together **Which of the numbers go across the air in a key exchange?** {{lines-3}} ::: :::teacher-note **Answer:** P, G and the two mixes. The private numbers and the shared key never do. :::