Week 7 - Public Key Cryptography

A Session Key Distribution Scheme (SKDS)

-Involves a trusted authority and two users - Alice and Bob who wish to communicate -Objective is the production of a sesison key known only to Alice and Bob (maybe the TA) -Session keys are used to encrypt information for a specified short period fo time -Often each network user will also share a different long lived key with the TA

What is a one way function

A function from X to Y so that for all x it is easy to compute f(x) but for all y it is computaitonally infeasible to find x such that f(x)=y. private key encryption does not provide a one way fucntion because if you know how to encrpyt you can know to decrypt. If you have they key then you can use it for both purposes.

A field

A ring where every element except additive identity 0 has an inverse

A group

A set where the binary operation * is defined so: - G is clsoed - Assocative -Identity -Inverse

A ring

A set with two operations + and . where following properties satisfied: -Abelina group wrt to + with an adiditve identiy 0 -R is clsoed for . and . is associative . is distibutive over + for example: x(y+z) = xy + xZ

A signature Scheme

Alice wants to authenticate/sign her message = she can just use here own decryption algorithm with her secret key to get y = y is the decryption under alices key of her message x. She then sends y and presumably the message x. Bob can apply the ecnryption side of the algorthm to y, and since theyre inverses, we get back the value x. He knows alice created the value y, since she is the only person who knew the secret key. Publicly verfiable.

Strong Primes

Defeats some of the basic factoring algo's but cant defeat the eliptic curve method

Public Key Crypto diagram

Doesnt depend on any secure channel. Bob tells Alice how to encrypt. He does so on an unsercure channel. But can derive decryption form that

Fermat's Little Theorem

For every integer a and every prime p, a^p = a mod p. This makes RSA work!

An Abelian Group

If oepration * is commuative x*y = y*x

Public Key Cryptosystem

Infeasible to derive the decryption from the encyrption function

Can a public key cryptosystem be unconditionally secure?

No because we have given away the encryption function. Always gives the possibility of breaking the cipher. Oscar can always launch a chosen plaintext attack because he has the encrytpion function.

Chinese remainder theorem

Suppose n1,n2,....,nk are integers which are pairwise relatively prime. Then for any integers a1, a2,...,ak, there is a solution x to the system of equations x ≡ a1 (mod n1) x≡ a2(mod n2)....... x≡ ak(mod nk) x = a1• N1•x1 + a2•N1•xd mod N where, N = n1•n2 N1= N/n1 N2=N/n2 x1= inverse of N1 mod n1 x2 = inverse of N2 mod n2

Trap-door one-way function

The trap door allows us to reverse the function. I.e. the decryption key. We wll want our trap door one way fucntions to be invertible.

General Purpose Algorithms

They have asymptotic runtimes

A key predistribution scheme (PKS)

Trsted authority distributes keying information in advance. Keyeing information is distributed using secure channels Keying information might be long lived keys or secret information that can later be used to produce keys