Cracking a Caesar cipher with letter frequencies

Plaintext	L	E	A	V	E		A	T		D	A	W	N
Ciphertext	Q	J	F	A	J		F	Y		I	F	B	S

(b) W is position 22. (22 + 5) MOD 26 = 27 MOD 26 = 1, which is B. V also wraps round: (21 + 5) MOD 26 = 26 MOD 26 = 0, which is A.

THE KEY IS UNDER THE MAT

(b) Anyone who picks up the note gets the ciphertext and the key, so they can decrypt it as easily as Nia can. The key must be shared a different way, for example agreed in person beforehand.

Key tried	PHHW becomes	Real word?
1	OGGV	No
2	NFFU	No
3	MEET	Yes
4	LDDS	No

(b) Key 3: MEET AT NOON.

(c) 25. There are 26 letters, so there are only 26 possible shifts, and a shift of 0 (or 26) leaves the message unchanged. Trying 25 keys by hand takes a few minutes; a computer does it in far less than a second.

(a) Counts (lines 1 and 2 add up to 94; lines 3 and 4 add up to 90)

Letter	A	B	C	D	E	F	G	H	I	J	K	L	M
Count	22	1	2	7	1	6	0	16	4	3	7	33	4
Letter	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
Count	5	11	10	0	1	8	4	13	8	2	0	8	8

(b) Total = 184. Line by line: lines 1, 2, 3 and 4 contain 60, 34, 25 and 65 letters. Small slips of 1 or 2 in a tally do not change the answer.

(c) Percentages

Letter	Count	Working	%
L	33	33 ÷ 184 × 100 = 17.93…	17.9
A	22	22 ÷ 184 × 100 = 11.95…	12.0
H	16	16 ÷ 184 × 100 = 8.69…	8.7
U	13	13 ÷ 184 × 100 = 7.06…	7.1
O	11	11 ÷ 184 × 100 = 5.97…	6.0

(d) A bar chart with letters L, A, H, U, O along the horizontal axis (labelled “Ciphertext letter”) and “Percentage of letters” up the vertical axis from 0 to 18 or 20, bars of equal width with gaps, heights 17.9, 12.0, 8.7, 7.1 and 6.0.

(e) L is by far the most common ciphertext letter, just as E is by far the most common English letter, so L most likely stands for E. E is position 4 and L is position 11. 11 − 4 = 7, so the key is 7.

(f) With key 7, A decrypts to T (position 0 − 7 + 26 = 19) and H decrypts to A (7 − 7 = 0). T and A are the 2nd and 3rd most common English letters, so the pattern of the three tallest bars matches the English chart moved 7 places along. The word AOL appears 8 times and decrypts to THE, the most common English word. Three separate pieces of evidence agree, so I am confident the key is 7.

(g) Line 1: THE TEAM WILL MEET AT THE GATE OF CONWY CASTLE AT SEVEN ON TUESDAY EVENING.

The whole message: THE TEAM WILL MEET AT THE GATE OF CONWY CASTLE AT SEVEN ON TUESDAY EVENING. BRING THE MAP, THE TORCH AND THE SPARE KEYS. TELL NOBODY WHERE WE ARE GOING. IF THE WEATHER IS BAD WE WILL MEET IN THE LIBRARY INSTEAD AND SET OFF THE NEXT DAY.

(h) The English chart is an average of a huge amount of text, but this message is only 184 letters, so a few repeated words change the percentages a lot. This message uses THE 8 times and words such as MEET, SEVEN and EVENING, which push E up to 17.9%, well above 12.7%. Nia’s message in Step 3 has only 10 letters, which is far too small a sample: no letter stands out reliably, so brute force is the better method there.

What is being guessed	Number of keys	Working	Longest time
Caesar cipher key	25	25 ÷ 1,000,000,000	0.000000025 s (2.5 × 10−8 s): instant
8 lower-case letters	208,827,064,576	208,827,064,576 ÷ 1,000,000,000 = 208.8 s; 208.8 ÷ 60	about 3.5 minutes
128-bit AES key	3.4 × 1038	3.4 × 1038 ÷ 109 = 3.4 × 1029 s; ÷ 31,500,000	about 1.1 × 1022 years

For comparison, the universe is about 13,800,000,000 (1.38 × 10¹⁰) years old, so trying every AES key would take nearly a trillion times longer than that.

Situation	Encrypt?	Reason
1. Medical form for the school nurse	Yes	Health information is sensitive personal data. If someone intercepted it they could learn private facts about the child.
2. Concert date on the public website	No	It is meant for everyone to read, so there is nothing to keep secret. (The website connection may still use https, which protects people from changed pages.)
3. Pupils’ reports on a USB stick	Yes	USB sticks are easily lost. If it is encrypted, whoever finds it cannot read the pupils’ personal information without the key.
4. Bank card number on a shop website	Yes	Anyone who intercepts a card number could spend money. Only type it into a page that uses https.
5. Puzzle answer everyone has finished	No	It is no longer secret and has no value to anyone who intercepts it.

(b) Model explanation for a Year 7 pupil: When a web address starts with https and shows a padlock, everything you send to that website, such as a password, is encrypted on the way, so anyone who intercepts it only sees scrambled data. A Caesar cipher is not safe because it has only 25 keys, which a computer can try in a fraction of a second, and because its letter patterns give the key away: I found the key in our message just by counting letters. Modern encryption such as AES has about 3.4 × 10³⁸ keys, which would take about 10²² years to try, and it hides letter patterns. However, the padlock only protects the data while it travels. It does not tell you the website is honest, because a scam website can have a padlock too.

• Pupils’ own messages vary. Longer messages are easier to crack by frequency analysis.
• With the keys 3, 1, 4 repeating, the letter E is turned into H, F or I depending on its place, so the E count is spread across 3 different ciphertext letters and no single tall bar appears. You would first have to work out how long the repeating key is, then do a separate frequency analysis on every 3rd letter.