Introduction To Cryptography

Traditional use of cryptography:

plaintext >>>>>>>>>> ciphertext >>>>>>>>> plaintext
(encryption) (decryption)

cryptographer: invent clever secret codes.

cryptanalyst: attempt  to break these codes.

Fundamental Tenet of Cryptography:

If lots of smart people failed to solve a problem,
then it probably won't be solved (soon).

Ø The time required to break  a code should be  longer than
the time the encrypted data must remain secret.

Ø The value of most data decreases overtime.

Cryptographic System: Algorithm + Key

It is perfectly OK to let everyone know the algorithm because knowledge of the algorithm without the key does not help unmangle the information.

Publishing the algorithm provides an enormous amount of free consulting to uncover weaknesses.

Computational Difficulty:

Example: combination lock

Ø Typically require 3 numbers between 1 and 40.

If it takes 10 seconds for a good guy,
it would take 10*(40**3) seconds or about 1 week for the bad guy.

Ø By requiring 4 numbers,

If  it takes 13 seconds for the good guy,
it would take  13*(40**4) seconds or about 1 year for the bad guy!

In general, increasing the key length by 1 bit makes the good guy's job just a little bit harder, but makes the bad guy's job twice as hard!

Example of Secret Codes:

Caesar cipher: substitute each letter with another letter which is 3 letters away in the alphabet (with wrap around).

E.g., dozen >>> grchq.

Extension: Instead of 3 use any number  n between 1 and 25.

E.g.,  for n=1, HAL >>> IBM.

Monoalphabetic cipher: arbitrary map one letter to another.
There are 26!=4*(10**26) possibilities.

Ø If each possibility takes 1 microsecond it would take 10 trillion years to try all possibilities.

Ø However statistical analysis of  language makes it much easier to break.

Breaking An Encryption Scheme

Ciphertext Only: Because it is important for Trudy to be able to differentiate plaintext from gibberish, this attack sometime known as recognizable plaintext attack.
It is also essential for Trudy to have enough ciphertext, e.g., there is no way to know the plaintext corresponding to XYZ is THE or CAT or HAT.

Known Plaintext: Knowing <plaintext, ciphertext> pairs is helpful for Trudy,
e.g., in monoalphabetic cipher Trudy learn the mapping.

How to obtain  plaintext? secret data might not remain secret forever,
e.g., which city to attack might not be secret after that attack!

Chosen Plaintext: Trudy can choose any plaintext he wants and get the system to provide corresponding ciphertext, e.g.,  telegraph company can encrypt and send any message you provide.

Example: If Trudy knows that Alice  encrypted message (azsxopfq) is either (surrender) or (fight on), he can ask the system to encrypt both messages and find out the correct one.

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Types of Cryptographic Functions:

• Secret Key Cryptography (symmetric cryptography)

(encryption)
plaintext >>>>>>>>> ciphertext
|
key
|
ciphertext >>>>>>>> plaintext
(decryption)

Can be used for:

• Transmission Over an Insecure Channel:

An eavesdropper will only see unintelligible data.

• Secure Storage on Insecure Media:

Forgetting the key makes the data irrevocably lost.

• Authentication: Alice authenticating Bob:

Alice Bob

challenge: r  >>>>>>>      r

response:       K{r}        <<<<<<<     K{r}

  r         is a random number,

K{r}  is the secret key encryption of r using shared  key K.

• Public Key Cryptography (asymmetric cryptography)

Each individual has two keys:

private key (not revealed to anyone)
public key (make it known to everyone)

(encryption)
plaintext >>>>>>>>>> ciphertext
|
public key

private key
|
ciphertext >>>>>>>>> plaintext
(decryption)

The reverse process is called digital signature:

(signing)
plaintext >>>>>>>>> ciphertext
|
                    private key

public key
|
ciphertext >>>>>>>> plaintext
(verification)

Ø Public key cryptographic algorithms are orders of magnitude slower than the best known secret key cryptographic algorithms.

Ø Thus they normally used to establish temporary shared secret key for use during a session.

Uses of Public Key Cryptography:

• Transmission Over an Insecure Channel:

Alice Bob
                   {K}_eB >>>>>>>>>             [K]_{dB
K{mB}        >>>>>>>>> K{mB}
K{mA}        <<<<<<<<<   K{mA}}

_{K is a random session key.}

_{<eB, dB> is the <public-key , private-key> pair for Bob.}

^{m}_eB ^{is the public key encryption of message m with eB.}

^[m]_dB ^{is the public key decryption of message m with dB.}

^{K{m} is the symmetric key encryption of message m with K.}

_{mB  message for Bob.}

_{mA message for Alice.}

• Secure Storage on Insecure Media:

_Alice generates a random key _{K and save:}

_{F= K{File}}

 ^{KF= {K}}_eA

_{To restore  the file:}

^{K= [KF]}_{dA
File = K{F}}

• Authentication: Alice authenticating Bob:

Alice Bob

challenge:       c = { r }_eB   >>>>>           c

response:          r                   <<<<<          r = [c]_dB

   r        is a random number

• Hash Algorithms

(also known as message digest /fingerprint, one-way functions )

The hash of a message m, h=H(m) has the following properties:

• Given m, it is easy to compute h.
• Given h, it is hard to compute m.
• Given m, it is hard to find another m' such that H(m) = H(m').
• It is hard to find m1 and m2 such that H(m1) = H(m2).

Uses of Hash Algorithms:

• MAC/MIC (Message Authentication/Integrity Code)

Using Secret Key:

Alice sends                        Bob receives

m,h where h = H(m|K)     >>>>>      m,h , OK if h = H (m|K)

K is the shared secret between Alice and Bob

Ø Bob is sure that Alice sent the message, since she knows K.

Ø Bob can NOT prove to any one  that Alice sent him message m, since he also knows K.

Using Public Key:

Alice sends                         Bob receives

m,sh where sh = [H(m)]_dA >>  m,sh , OK if H (m) = {sh}_eA

<eA, dA>  is the public/private key pair of Alice

Ø Bob is sure that Alice sent the message, since she knows dA.

Ø He can also prove to any one  that Alice sent him message m,  since he does not know dA.

This property of public key cryptography is known as non-repudiation,

where the sender should not be able to falsely deny that he sent a message.

• Password Hashing:

OS like UNIX stores the hash of passwords instead of  storing the actual passwords.

Ø For each user U, there is a tuple <U, h>

where h = H(P)  is the hash of  password  P of user U.

Ø When a user U types a password, P, the OS compute H(P) and
if it is equal to the saved value h  in the tuple <U,h>, the user is OK.

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The magic of XOR:

A Simple XOR symmetric algorithm:

(from    Bruce Shneier textbook)

0 ® 0 = 0
0 ® 1 = 1
1 ® 0 = 1
1 ® 1 = 0

Note that:

 x  ® x = 0
 x  ® 0 = x

The following program is a very simple symmetric algorithm.
(see  /home/cs772/public_html/demos/xor )

Ø To encrypt: the plaintext P is XORed with a key K to produce a ciphertext C.

Ø To decrypt: the ciphertext C  is XORed with a key K to produce a plaintext P.

Why?
P ® K = C
C ® K = P

Since C ® K = (P ® K) ® K = P ® (K ® K) = P ® 0 = P

This is a very insecure algorithm that can be easily broken.
It is good enough to keep "your kid sister" from reading your files!

crypto

Usage:

 % crypto <key> <input_file> <output_file>

Example:

Ø To encrypt infile:

% crypto "odu computer science" infile outfile

Ø To decrypt outfile:

% crypto "odu computer science" outfile tmpfile

tmpfile is identical to infile

Code

void main (int argc, char *argv[])

{
     FILE *fi, *fo;
     char *cp;
     int c;

     if ((cp = argv[1]) && *cp!='\0')  {
         if ((fi = fopen(argv[2], “rb”)) != NULL)  {
             if ((fo = fopen(argv[3], “wb”)) != NULL)  {
while ((c = getc(fi)) != EOF)  {
if (!*cp) cp = argv[1];
c ^= *(cp++);
putc(c,fo);
                  }
                  fclose(fo);
             }
             fclose(fi);
         }
     }
}