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Caesar cipher wrap-around with MOD 26

Caesar wrap-around rule: with A=0 to Z=25, add the shift then take MOD 26 so letters past Z loop back to A. Example: X (23) + 5 = 28, MOD 26 = 2, which is C.

Part of: ACARA Digital Technologies Years 9–10 — Cyber Security Threats and Vulnerabilities — Module 106, QCE Digital Solutions Unit 4 — Digital Methods for Exchanging Data — Module 12, QCE Digital Solutions Unit 4 — Prototype Digital Data Exchanges — Module 105

Understanding Caesar cipher wrap-around with MOD 26 in ACARA Digital Technologies Years 9–10

What it does: Adding the shift can go past 25 (Z). MOD 26 wraps the number back into the range 0 to 25, so Z loops round to A. 23 + 5 = 28, which is past Z (25). 28 MOD 26 = 2, and position 2 is C. The alphabet has 26 letters, so we use MOD 26. The remainder is always between 0 and 25, which matches the positions A to Z. 24 + 3 = 27. 27 MOD 26 = 1, and position 1 is B. 20 + 6 = 26. 26 MOD 26 = 0, which is A. The alphabet has looped all the way round.

Common mistakes: Capital letters start at ord 65, not 97 (that is for lowercase). Subtracting 97 gives a negative position. It should be ord('X') - 65 so that X = 23. The alphabet has 26 letters, so the wrap-around uses MOD 26. MOD 25 gives the wrong result: Z (25) + 3 = 28, and 28 MOD 26 = 2 (C). It should be % 26.

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Learning

Duo

A Caesar cipher uses shift 5. What does the letter X (position 23) become?

C28

Matches

newPos = (pos + shift) MOD 25

↓

The alphabet has 26 letters, so the wrap must use MOD 26, not MOD 25. With MOD 25, Z (25) would wrongly wrap to 0 and letters would be miscalculated.

Applying

Meanings

Term

(position + shift) MOD 26

Meaning

Caesar wrap-around rule: with A=0 to Z=25, add the shift then take MOD 26 so letters past Z loop back to A. Example: X (23) + 5 = 28, MOD 26 = 2, which is C.

Practice

position = ord("X") - 65

Testing

Gaps

(23 + 5) 26 = 2

Check Answer

Spots

1pos = ord('X') - 97

2shift = 5

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Quick Questions

1. A Caesar cipher uses shift 5. What does the letter X (position 23) become?

A. 28

B. C ✓

23 + 5 = 28, which is past Z (25). 28 MOD 26 = 2, and position 2 is C.

2. Which formula wraps a shifted letter back around the alphabet (A=0 to Z=25)?

A. (position + shift) MOD 26 ✓

B. (position + shift) MOD 25

The alphabet has 26 letters, so we use MOD 26. The remainder is always between 0 and 25, which matches the positions A to Z.

3. Shift 3: what does Y (position 24) become?

A. B ✓

B. A

24 + 3 = 27. 27 MOD 26 = 1, and position 1 is B.

4. True or false: 26 MOD 26 = 0, so a letter shifted from position 20 by 6 lands on A.

A. True ✓

B. False

20 + 6 = 26. 26 MOD 26 = 0, which is A. The alphabet has looped all the way round.

5. Why do we need MOD 26 in a Caesar cipher?

A. So the shift is always 26

B. So letters past Z loop back to A ✓

Without MOD 26, a result like 28 would not match any letter. MOD 26 wraps the number back into the range 0 to 25.

Match the Pairs

(23 + 5) MOD 26 → Encrypts X with a shift of 5: 28 wraps around to 2, which is the letter C

(2 - 5 + 26) MOD 26 → Decrypts C with a shift of 5 by moving back 5 places, wrapping round to position 23, which is X

To encrypt, add the shift to the letter's position, then use MOD 26 so the result stays between 0 and 25 → newPos = (pos + shift) MOD 26

To decrypt with a negative shift, subtract the shift, add 26 so it is not negative, then wrap with MOD 26 → newPos = (pos - shift + 26) MOD 26

newPos = (pos + shift) MOD 25 → The alphabet has 26 letters, so the wrap must use MOD 26, not MOD 25. With MOD 25, Z (25) would wrongly wrap to 0 and letters would be miscalculated.

newPos = pos + shift → There is no MOD 26, so X (23) shifted by 5 gives 28, which is not a letter. The result never wraps back to the start of the alphabet.

Key Term

(position + shift) MOD 26

Caesar wrap-around rule: with A=0 to Z=25, add the shift then take MOD 26 so letters past Z loop back to A. Example: X (23) + 5 = 28, MOD 26 = 2, which is C.

Fill in the Blanks

1. (23 + 5) MOD 26 = 2

Hint: Type the operator that gives the remainder, so the total loops back into the range 0 to 25.

2. (23 + 5) MOD 26 = 2

Hint: Type the number of letters in the alphabet, which the total is wrapped around.

3. newPos = (pos + shift) MOD 26

Hint: Type the operator that moves a letter forward when you encrypt.

4. =MOD(2+-5,26)

Hint: Type the negative shift that undoes an encryption shift of five. C (2) should go back to X.

5. X = 23 23 + 5 = 28 28 MOD 26 = 2

Hint: Type the wrapped position after the MOD step. It is the position of the letter C.

Spot the Error

Question 1

A pos = ord('X') - 97

B shift = 5

C new = (pos + shift) % 26

D print(chr(new + 65))

Line A — Capital letters start at ord 65, not 97 (that is for lowercase). Subtracting 97 gives a negative position. It should be ord('X') - 65 so that X = 23.

Question 2

A pos = 25

B new = (pos + 3) % 25

C print(new)

Line B — The alphabet has 26 letters, so the wrap-around uses MOD 26. MOD 25 gives the wrong result: Z (25) + 3 = 28, and 28 MOD 26 = 2 (C). It should be % 26.

Question 3

A shift = 5

B pos = (23 * shift) % 26

C letter = chr(pos + 65)

D print(letter)

Line B — A Caesar shift is added to the position, not multiplied. X (23) + 5 = 28, and 28 MOD 26 = 2 (C). It should be (23 + shift) % 26.

Question 4

A pos = 2

B shift = -5

C new = (pos - shift) % 26

D print(chr(new + 65))

Line C — Decrypting uses a negative shift, so you add it: 2 + (-5) = -3, and -3 MOD 26 = 23 (X). Subtracting a negative shift adds 5 instead and gives the wrong letter. It should be (pos + shift) % 26.

Question 5

A letter = 'C'

B pos = ord(letter) - 65

C shift = -5

D new = pos + shift % 26

E print(chr(new + 65))

Line D — MOD is worked out before the addition, so only shift is wrapped and the total can go outside 0-25. Brackets are needed: (pos + shift) % 26.

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