A brief overview to public key cryptography. Why it's important, how it works and an example.

1. PUBLIC KEY
CRYPTOGRAPHY
A brief overview
By Andy Brodie

2. PUBLIC/PRIVATE KEYS
• A normal lock, for example on a door, has one key.
• If you have a key, you can lock and unlock the door.
• This is a useful way of securing content (whatever is behind the door).
• Asymmetric keys mean that the lock has two keys.
• One key can only lock the door (a public key)
• One key can only unlock the door (a private key)
• Encryption and decryption is a metaphor for locking and unlocking the door to get
at whatever is behind the door.
• This is really, really powerful.

3. BENEFITS OF ASYMMETRIC KEYS
• If you give away the public key (that encrypts) then anyone can protect
content.
• No-one else can decrypt and read the content unless they have the private
key.
• But, there’s more… it works the other way around too!

4. BUT HOW DO THEY WORK?
• Private keys must be unguessable.
• It must be impossible to derive the private key from the public key.
• Keys are calculated from very, very large prime numbers.
• Key “strength” is measure in bits, i.e. the number of bits the key consists of.
• E.g. an 8-bit key has a range of 256 potential values (28).
• This is not very secure as it would be trivial to manufacture all 256 keys and try them
against the lock (a brute force attack)
• A 2048 bit key has a range of 3.23x10616.
• 3.23 with 616 zeros after the end.
• Even the Milliard Gargantubrain could only manage ~1x1057 in a millisecond!

5. MATHS
• Select 2 large primes: 𝑝 and 𝑞.
• Calculate product of the keys 𝑛 = 𝑝𝑞
• Calculate the totient of 𝑛.
• For any integer, 𝑥, the totient of 𝑥, written 𝜑(𝑥) is the number of integers smaller than 𝑥 that are
relatively prime to 𝑥.
• For any prime number 𝜑 𝑥 = 𝑥 − 1, therefore for the product of two primes, 𝜑(𝑛) = (𝑝 − 1)(𝑞 − 1).
• Choose any integer, 𝑒, smaller than and relatively prime to 𝜑(𝑛). 𝑒 is called the public key
exponent.
• Calculate an integer 𝑑 such that 𝑑𝑒 = 1 𝑚𝑜𝑑 𝜑 𝑛 .
• I.e.
𝑑𝑒
𝜑 𝑛
results in any number with remainder 1.
• This is done reliably using a multiplicative inverse function… which is TMI for now.
• Once calculated, 𝑑 is called the private key exponent.
• The public key pair is the modulus and the public key exponent: 𝑛, 𝑒
• The private key is the modulus and the private key exponent: (𝑛, 𝑑)

6. REAL MATHS!
• Let’s pick 2 random small primes: 𝑝 = 3, 𝑞 = 11
• Modulus of keys 𝑛 = 𝑝𝑞 = 33
• Totient 𝜑 𝑛 = 3 − 1 11 − 1 = 2 ∗ 10 = 20
• Choose 𝑒, any prime less than 20. Choices are 7, 11, 13, 17, 19. Pick 𝑒 = 7.
• Therefore public key pair = (33, 7)
• For private key exponent, calculate multiplicative inverse, i.e. 𝑑𝑒 ≡ 𝑚𝑜𝑑 𝜑(𝑛).
• 𝑑 ∗ 7 = 1 𝑚𝑜𝑑 20
• I.e. some number, multiplied by 7 and divided by 20 leaves a remainder of 1.
• In our heads, we know that 21 𝑚𝑜𝑑 20 = 1, so 𝑑 =
21
7
= 3
• Private key pair = (33, 3)

7. ENCRYPTING AND DECRYPTING
• Encrypting and decrypting is (comparatively) very, very simple.
• Our key pairs: Public 𝑛, 𝑒 = 33, 7 ; Private 𝑛, 𝑑 = 33, 3
• So, let’s encrypt the number 14
• To encrypt: 𝑝 𝑒
= 𝑝′
𝑚𝑜𝑑 𝑛
• 𝑝 is the byte to encrypt
• 𝑝’ is the encrypted byte
• 𝑛 is the modulus
• 𝑒 is the public key exponent
• Real values:
• 147
= 105413504
• 𝑝’ = 10541348 𝑚𝑜𝑑 33 = 𝟐𝟎
• To decrypt: 𝑝′ 𝑑
= 𝑝′
𝑚𝑜𝑑 𝑛
• 𝑝 is the byte to encrypt
• 𝑝’ is the encrypted byte
• 𝑛 is the modulus
• 𝑒 is the public key exponent
• Real values:
• 𝟐𝟎3
= 8000
• 𝑝 = 8000 𝑚𝑜𝑑 33 = 𝟏𝟒

8. END
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