General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Preparation: The person who anticipates receiving messages first creates both a public key
and an associated private key, and publishes the public key.
Application: Confidential Messaging: To send a confidential message the sender encrypts
it using the intended recipient’s public key; to decrypt the message, the recipient uses the
private key.
Application: Digital Signatures: A message signed with a sender’s private key can be verified
by anyone who has access to the sender’s public key, thereby proving that the sender had
access to the private key (and therefore is likely to be the person associated with the public
key used), and the part of the message that has not been tampered with.
c○: Michael Kohlhase 381
The confidential messaging is analogous to a locked mailbox with a mail slot. The mail slot is
exposed and accessible to the public; its location (the street address) is in essence the public key.
Anyone knowing the street address can go to the door and drop a written message through the
slot; however, only the person who possesses the key can open the mailbox and read the message.
An analogy for digital signatures is the sealing of an envelope with a personal wax seal. The
message can be opened by anyone, but the presence of the seal authenticates the sender.
Note: For both applications (confidential messaging and digitally signed documents) we have only
stated the basic idea. Technical realizations are more elaborate to be more efficient. One measure
for instance is not to encrypt the whole message and compare the result of decrypting it, but only
a well-chosen excerpt.
Let us now look at the mathematical foundations of encryption. It is all about the existence of
natural-number functions with specific properties. Indeed cryptography has been a big and somewhat
unexpected application of mathematical methods from number theory (which was perviously
thought to be the ultimate pinnacle of “pure math”.)
Encryption by Trapdoor Functions
Idea: Mathematically, encryption can be seen as an injective function. Use functions for
which the inverse (decryption) is difficult to compute.
Definition 576 A one-way function is a function that is “easy” to compute on every input,
but “hard” to invert given the image of a random input.
In theory: “easy” and “hard” are understood wrt. computational complexity theory, specifically
the theory of polynomial time problems. E.g. “easy” ˆ= O(n) and “hard” ˆ= Ω(2 n )
Remark: It is open whether one-way functions exist ( ˆ≡ to P = NP conjecture)
In practice: “easy” is typically interpreted as “cheap enough for the legitimate users” and
“prohibitively expensive for any malicious agents”.
Definition 577 A trapdoor function is a one-way function that is easy to invert given a
piece of information called the trapdoor.
Example 578 Consider a padlock, it is easy to change from “open” to closed, but very
difficult to change from “closed” to open unless you have a key (trapdoor).
c○: Michael Kohlhase 382
Of course, we need to have one-way or trapdoor functions to get public key encryption to work.
Fortunately, there are multiple candidates we can choose from. Which one eventually makes it
into the algorithms depends on various details; any of them would work in principle.
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