What are Cryptographic communications?
- Throughout history there’s often been the need to securely transmit information through insecure channels
What is the general idea behind encryption schemes?
- A message M (the plaintext) is to be sent is encrypted before transmission into an unrecognisable string of characters C
- the ciphertext C is then trasmitted to the recipient who uses a key to decrypt C to recover the original plaintext M
What is symmetric key encryption
- This is where the sender and receiver use the same key as each other to both encrypt and decrypt messages
What are substitution ciphers?
- they use symmetric encryption
- the key is a permutation of the alphabet
- so each letter in the plaintext is replaced by another letter or a number
What is a Caesar cipher
Also known as a shift ciphers
- since we just shift the letters in the alphabet by a certain number
- so if the key = 3, a is now x
Why don’t we use substitution ciphers?
- they are not secure as they can be easily broken using frequency analysis
What is a one time pad?
- secure symmetric ciphers do exist - such as the one time pad
- for this encryption scheme, the sender and recipient share a random bit string K (a series of 0’s and 1’s) that’s randomly generate and is as long as any message that they are going to send
- this key k is the symmetric key that is used for the encryption/decryption process
How is encryption achieved using a one time pad:
- to encrypt the message M, alice computes C = M XOR K
What is the problem with symmetric key encryption and how can we securely distribute keys?
The problem is symmetric key distribution is not anywhere near as secure as asymmetric encryption but asymmetric encryption is extremely space and time consuming
- so the compromise is to exchange the shared key used for symmetric encryption using asymmetric encryption so that the shared key is computationally secure but then we can use symmetric encryption which is a lot less time consuming to send and decrypt
What is a one way function?
- Where E(D(M)) = M
- It means that knowledge of the encryption method gives no information about the decryption scheme. Anyone can send an encrypted message to the holder of D but only that person can decrypt the cybertext
What is a digital signature?
It allows a recipient to verify that a message was truly sent by the sender that claims they sent it
What does the RSA encryption scheme use to ensure computational security?
- uses difficulty of factoring large numbers and maths concepts, as it’s unknown how to do factoring of large numbers in polynomial time
What is the rule called of a divides b and b divides c meaning a also divides c?
Known as transitivity
- a|b and b|c so a|c
what does a|b and a|c mean for all integers i and j
a|(ib and jc)
What is the greatest common divisor?
- where a and b are positive integers, the GCD of a and b denotes GCD(a,b) is the largest integer that divides both a and b
- if GCD(a,b) = 1 we say a and b are relatively prime
What can we apply the logic of the extended Euclidean algorithm to RSA?
- In RSA we can publish the encryption function but using the extended Euclidean algorithm, only the recipient can
Why is it so difficult to break RSA?
- Because even knowing e doesn’t allow us to figure out d unless we know Φ(n)
- and Φ(n) is not easily found if it’s a large number as factoring large numbers can’t be done in polynomial time
- as the ability to factor large numbers improves, we simply just increase the size of p and q, so that p * q = n is bigger
Why do digital signatures work?
- A signature is applied to the message by doing S = M^d mod n, when alive gets the message she can check bob sent it by doing M = S^e mod n
- since bob is the only one who knows the decryption function, this will verify bob sent M.
What is fast exponentiation?
- ## A method for calculating values raised to an exponent (power) - x^k
