Encryption Coursepak little bit in chap 10 of reed

Encryption A message sent over a network can be read by any site between the source and the destination (e.g. each hop in ). Encryption is the process of encoding a message so that it is decipherable only by its intended recipient (anyone intercepting the message would be unable to interpret it)

Encryption Cryptography - study of methods to encrypt text. Cryptanalysis - study of how to decode an encypted text. Strong Encryption - infeasible to crack Weak Encryption - can be cracked

Example Substitution cipher - each letter of the alphabet is substituted with a different letter or symbol. Ceasar's method - replace every letter in the alphabet with the letter 3 places after it –A - > D –B - > E … –Y - > B –Z - > C

Encryption The sender encrypts the message The sender transmits the message to the receiver The receiver decodes the message How does the receiver decode the message? The sender needs to send the key to the receiver. BUT, now this can be intercepted.

Symmetric-Key Encryption Symmetric (Private, Single)-key encryption assumes that the sender and the recipient use a shared key. Same key is used for encryption and decryption. Introduces key security risk: –how communicate the key? Intercepted? –Imagine shopping from online catalog. Shared by too many

Symmetric-Key Encryption Problem is that the privacy of the message is only as good as the secrecy of the key. If a cracker learns the key by sniffing it in an internet transmission or being told by a traitor (e.g. an employee who was fired), the complexity of the algorithm won’t do any good.

Asymmetric (Public)-Key Encryption In 1976 Whitfield Diffie and Martin Hellman proposed public-key encryption –generate a pair of associated keys, one is public and the other is private –a message encoded with a public key requires the corresponding private key for decoding, and vice versa. Each undoes the other (in either order). –It should be computationally infeasible to derive the private key from the public key. –almost all secure communications on the Internet use asymmetric key encryption

Asymmetric (Public)-Key Encryption S’pose you want to send me a message. I have published my public key. –Website, PKI (Public Key Infrastructure) You use it to encrypt the message and send it to me. I use my private key to decrypt the message.

Digital Signature Authenticate identity of sender You create a digital signature using your private key and add it to the message. I use your public key to decrypt it. Signature is also a function of the text in the document. –Can’t be copied to another document –I can verify that document hasn’t been altered (integrity).

Asymmetric (Public)-Key Encryption allows for double encryption to encrypt text and verify the identity of the sender. sender can encode messages with his private key and the recipient’s public key, and the recipient decodes the message in reverse.

RSA Ron Rivest, Adi Shamir, and Len Adleman invented the most popular instantiation of public-key encryption, which has become known as RSA. Alice, the sender, wants to send Bob, the receiver a message that Eve, the eavesdropper, cannot decrypt. Bob has published a public key, KR, which Alice is to use to encrypt her message.

RSA Alice partitions her cleartext message T into 32- byte groups of ASCII letters, T0T1... Tn-1. For i = 0 to n-1, she sends Ci, the remainder of (Ti)3/KR. Bob's public key KR pq, where p and q are primes, each 2 greater than a multiple of 3. To decrypt the message Bob computes s = (1/3)(2(p-1)(q-1) + 1). He then decrypts each Ci by computing the remainder of (Ci)s)/KR, which we can show is Ti.

RSA Example –We can choose p = 5, q = 11, so KR = 55, s = 27. –Suppose Ti = 42 ( in binary). Then Ci is the remainder of 423/55, that is, 74088%55 = 3. To decrypt 3, Bob computes 327%55 = 7,625,597,484,987%55 = 42. –RSA is based on a theorem by Euler (1736): Suppose p and q are distinct primes, K = pq, 0 ≤ T 0. Then Tr(p-1)(q-1)) + 1%K = T. –In our example r = 2, because (T3)(1/3)(2(p-1)(q-1) + 1)) = T2(p-1)(q-1) + 1)

RSA M = the message C - the encrypted message e = the public exponent d = the private exponent n = a very large integer The message is encrypted by C = M^e mod n (mod means divide by n and keep the remainder) The message is decrypted by M = C^d mod n where n = p * q, p and q are prime numbers, d *e≡ 1 mod((p-1)*(q-1))

RSA Public key: n,e Private key: n,d

RSA If n is a large number (128 bits or 256 bits), it is computationally infeasible to find p and q. Why? –find all factors of n –determine which are prime –try all pairs of primes to find p and q

RSA These methods hinge on the fact that it is relatively easy (computationally) to multiply two large numbers, but it is quite difficult to factor a large number if the only factors are large prime numbers. Example: Try to factor See how long it takes.

RSA 53 and 61.

Attacking RSA Try to find primes p and q such that pq = KR. –In 1977, Rivest, Shamir and Adleman issued a challenge to the world: break a small cipher text encrypted with RSA129, a 129-digit key. –RSA129 was cracked with the largest computation of all time (about 1000 computers around the world working 8 months). The secret message was: THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE. –It is estimated that with 250 digit keys and current factoring algorithms, you'd need 1000 computers for about 800 million months (or 100 billion computers for 8 months). –Thus, RSA is still deemed to be strong encryption.

Asymmetric-Key Encryption Advantages One can send secure message to someone without having to share any secret keys. Only public key is distributed. The recipient can verify the identity of the sender.

Asymmetric-Key Encryption Disadvantage SLOW

PGP Pretty Good Privacy (Phil Zimmerman) Uses both Single key and Public-private- key pair. Nice compromise –Use slow public key to encrypt single key –Use quick single key to encrypt large file. –Send both

PGP (Sender) PGP compresses the message - saves transmission time and increases security We previously discussed compression of graphics files. (e.g. simple compression - replace multiple copies of a character with a single character and the number of repeats.) How would compression increase security?

PGP (Sender) PGP creates a session key that is used only once during this session. Created from randomly selected mouse movements and keystrokes. The session key used to encrypt the message.

PGP (Sender) The receiver's public key is used to encrypt the session key. The encrypted message and encrypted session key are sent to the receiver.

PGP (Receiver) The receiver uses private key to decrypt session key The session key is used to decode the message text. The text is decompressed. The session key is discarded. (diagram)

PGP Advantages: Only a very small content (the session key) is publicly encrypted The session key is used just once - hard to decode by repeated attacks Symmetric-key encryption is ~10,000 times faster than asymmetric-key encryption. (Fast session key is used on large file)