CRYPTOGRAPHY

CRYPTOGRAPHCRYPTOGRAPHYY

Presented by:Presented by:

Debi Prasad MishraDebi Prasad Mishra

Institute of Technical Education & ReaserchInstitute of Technical Education & Reaserch

Electronics & Telecommunication EngineeringElectronics & Telecommunication Engineering

Section - ASection - A

77thth Semester Semester

Regd. No. - 0301212148Regd. No. - 0301212148

Talk FlowTalk Flow Terminology Secret-key cryptographic system Block cipher Stream cipher Requirement of secrecy Information theoretic approach

Perfect securityDiffusion and confusion

Practicability of cipher Substitution cipher Transposition cipher Data Encryption Standard (DES) algorithm Public-key cryptographic system

Diffie-Hellman key distribution Rivest-Shamir-Adleman (RSA) algorithm Digital Signature: A hybrid approach

Cryptology is the term used to describe the science of secret communication.

Derived from Greek words kryptos (hidden) & logos (word).

Divided into two parts. Cryptography:- transforms message into

coded form and recovers the original signal. Cryptanalysis:- deals in how to undo

cryptographic communication by breaking coded signals tht may be accepted as genuine.

TerminologyTerminology Plaintext:- The original message to be encoded

Enciphering or Encryption:- The process of

encoding

Ciphertext or Cryptogram:- The result produced

by encryption

Cipher:- The set of data transmission used to do

encryption

Key:- parameters of transformation

Services offered by Services offered by CryptographyCryptography

Secrecy, which refers to the denial of access to information by unauthorised users

Authenticity, which refers to the validation of the source of message

Integrity, which refers to the assurance that a message was not modified by accidental or deliberate means in transit

Cryptography

Secret-key (Single-key)

Cryptography

Public-key (Two-key)

Cryptography

•A conventional Cryptographic system relies on use of a single piece of private and necessarily secret key.

•Key is known to sender & receiver, but to no others.

•Each user is provided with key material of one’s own with a private component & a public component

•The private component must be kept secret for secure communication.

Secret-key CryptographySecret-key Cryptography

Let X -> Plaintext message; Y -> Cryptogram; Z -> Key

F ->Invertible transformation producing the cryptogram

Y = F (X, Z) =FZ (X)

Let F-1 ->Inverse transform of F to recover original message

F-1 (Y, Z) = Fz-1 (Y) = FZ

-1 (FZ (X)) = X

Secret-key CryptographySecret-key Cryptographycontinued…continued…

Here Y’ ->fraudulent message modified by an interceptor or eavesdropper

Block CiphersBlock Ciphers

•Block ciphers are normally designed in such a way that a small change in an input block of plaintext produces a major change in the resulting output.

•This error propagation property of block ciphers is valuable in authentication in that it makes it improbable for an enemy cryptanalyst to modify encrypted data, unless knowledge of key is available.

Stream ciphersStream ciphers

Whereas block ciphers operate on large data on a block-by-block

basis, stream ciphers operate on individual bits.

Let xn -> Plaintext bit; y ->ciphertext bit; z ->keystream bit at nth instant

For encryption: yn = xn zn, n=1, 2, …, N

For decryption: xn = yn zn, n=1, 2, …, N

Stream ciphersStream ciphers continued… continued…

A binary additive stream cipher has no error propagation; the decryption of a distorted bit in the ciphertext affects only the corresponding bits of the resulting output.

Stream ciphers are generally better suited for secure transmission of data over error – prone communication channels; they are used in application where high data rates are a requirement (as in secure video) or when a minimal transmission delay is essential.

Requirement of SecrecyRequirement of Secrecy

ASSUMPTION:-

An enemy cryptanalyst has knowledge of the entire mechanism used to perform encryption, except for the secret key.

Requirement of SecrecyRequirement of Secrecy continued…continued…

Attacks employed by enemy cryptanalyst: Ciphertext-only attack

Access to part or all of the ciphertext Known-plaintext attack

Knowledge of some ciphertext:-plaintext pairs formed with the actual secret key

Chosen-plaintext attackSubmit any chosen plaintext message and receive in

return the correct ciphertext for the actual secret key. Chosen-ciphertext attack

Choose an arbitrary ciphertext and find the correct result for its decryption.

Information theoretic Information theoretic approachapproach

• In Shannon model of cryptography (published in Shannon’s 1949 landmark paper on information-theoretic approach to secrecy systems)

ASSUMPTION:-

1. Enemy cryptanalyst has unlimited time & computing power.

2. But the enemy is presumably restricted to ciphertext-only attack.

• The secrecy of the system is said to be broken when decryption is performed successfully, obtaining a unique solution to the cryptogram

Information theoretic Information theoretic approach approach (continued…)(continued…)

Let X = {X1, X2, …, XN} ->N-bit plaintext message, Y = {Y1, Y2, …,YN} ->N-bit cryptogram

Secret key Z is assumed to be determined by some probability distribution

Let H (X) ->uncertainty about x H (X | Y) ->uncertainty about X given knowledge of Y

Now, mutual information between X & Y,

I (X;Y) = H (X) – H(X | Y)

represents a basic measure of security in the Shannon model.

Perfect SecurityPerfect SecurityAssuming that an enemy cryptanalyst can observe only the

cryptogram Y, for perfect security X & Y should be statistically independent.

I (X;Y)=0 =>H (X) = H (X|Y) …………….......(1)Given the secret key Z; H (X|Y) ≤ H (X; Z|Y) = H (Z|Y) + H (X|Y,Z) …(2)H(X|Y,Z)=0; iff Y & Z together uniquely determine XEquation 2 can be rewritten as H(X|Y) ≤ H(Z|Y) ≤ H(Z) …………(3)With equation 3 equation 1 becomes H(Z) ≥ H(X) ……………………………..(4)Is called Shannon’s fundamental bound for perfect security.

Result: The key must be at least as long as the plaintext.

Diffusion & ConfusionDiffusion & Confusion In diffusion, statistical nature of the plaintext is hidden by

spreading out the influence of single bit in plaintext over large number of bits in ciphertext.

In confusion, the data transformations are designed to complicate the determination of the way in which the statistics of ciphertext depend on that of the plaintext.

Practicability of CipherFor a cipher to be of practical value 1. It must be difficult to be broken by enemy cryptanalyst.2. It must be easy to encrypt & decrypt with knowledge of

secret key.

Substitution cipherSubstitution cipherEach letter of plaintext is replaced by a fixed substitute.

For plaintext X = {x1,x2,x3,x4,…)

ciphertext Y ={y1,y2,y3,y4,,…)

={f(x1),f(x2),f(x3),f(x4),….}

Transposition cipherTransposition cipher•The plaintext is divided into groups of fixed period d & the same permutation is applied to each group.

•The particular permutation rule being determined by the secret key.

Data Encryption StandardData Encryption Standard(DES)(DES)

It is the most widely used secret-key cryptalgorithm. It operates on 64-bit plaintext and uses 56-bit key. The overall procedure can be given as

P-1{F[P(X)]}

where, X->plaintext

P->certain permutation

F->certain transposition & substitution

F is obtained by cascading a certain function f, with each stage of cascade referred as around.

There are 16 rounds employed here.

How DES works?How DES works?

DES operates on 64-bit of data. Each block of 64 bits is divided into two blocks of 32 bits each, a left half block L and a right half R.

M = 0123456789ABCDEF

M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

L = 0000 0001 0010 0011 0100 0101 0110 0111

R = 1000 1001 1010 1011 1100 1101 1110 1111

Key ComputationKey Computation The 64-bit key is permuted according to the following table & 56-

bit key is calculated from it.

57 49 41 33 25 17 9

1 58 50 42 34 26 18

10 2 59 51 43 25 27

19 11 3 60 52 44 36

63 55 47 39 31 23 15

7 62 54 46 38 30 22

14 6 61 53 45 37 29

21 13 5 28 20 12 4

LET

K = 00010011 00110100 01010111 01111001

10011011 10111100 11011111 11110001

The 56-bit permutation:

K+ = 1111000 0110011 0010101 0101111

0101010 1011001 1001111 0001111

From the permuted key K+, we get

C0 = 1111000 0110011 0010101 0101111 D0 = 0101010 1011001 1001111 0001111

Key Computation Key Computation continued…continued…

With C0 and D0 defined, we now create sixteen blocks Cn and Dn, 1<=n<=16. Each

pair of blocks Cn and Dn is formed from the previous pair Cn-1 and Dn-1, respectively, for

n = 1, 2, ..., 16, using the following schedule of "left shifts" of the previous block.

Iteration Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of Left Shifts

1 1 2 2 2 2 2 2 1 2 2 2 2 2 2 1

C0 = 1111000011001100101010101111D0 = 0101010101100110011110001111

C1 = 1110000110011001010101011111D1 = 1010101011001100111100011110

C2 = 1100001100110010101010111111D2 = 0101010110011001111000111101 and so on upto C16 & D16.

Key Computation Key Computation continued…continued…

We now form the keys Kn, for 1<=n<=16, by applying the following

permutation table to each of the concatenated pairs CnDn.

14 17 11 24 1 5

3 28 15 6 21 10

23 19 12 4 26 8

16 7 27 20 13 2

41 52 31 37 47 55

30 40 51 45 33 48

44 49 39 56 34 53

46 42 50 36 29 32

C1D1 = 1110000 1100110 0101010 1011111

1010101 0110011 0011110 0011110

K1 = 000110 110000 001011 101111

111111 000111 000001 110010

Similarly,

K2 = 011110 011010 111011 011001

110110 111100 100111 100101

K3 = 010101 011111 110010 001010

010000 101100 111110 011001

and so on upto K16.

Thus the 16, 48-bit subkeys

are obtained.

Encoding DataEncoding Data There is an initial permutation, IP of the 64 bits of the message

data, M. This rearranges the bits according to the following table.

58 50 42 34 26 18 10 2

60 52 44 36 28 20 12 4

62 54 46 38 30 22 14 6

64 56 48 40 32 24 16 8

57 49 41 33 25 17 9 1

59 51 43 35 27 19 11 3

61 53 45 37 29 21 13 5

63 55 47 39 31 23 15 7

M = 0000 0001 0010 0011 0100 0101 0110 0111

1000 1001 1010 1011 1100 1101 1110 1111

IP = 1100 1100 0000 0000 1100 1100 1111 1111

1111 0000 1010 1010 1111 0000 1010 1010

Next divide the permuted block IP into a left half L0 of 32 bits, and a

right half R0 of 32bits. L0 = 1100 1100 0000 0000 1100 1100 1111 1111

R0 = 1111 0000 1010 1010 1111 0000 1010 1010

Encoding DataEncoding Data continued… continued…

We now proceed through 16 iterations, for 1<=n<=16, using a

function, f which operates on two blocks - a data block of 32 bits

and a key Kn of 48 bits - to produce a block of 32 bits.

Ln = Rn-1

Rn = Ln-1 f(Rn-1, Kn)

For n = 1, we have

K1 = 000110 110000 001011 101111 111111 000111 000001 110010 L1 = R0 = 1111 0000 1010 1010 1111 0000 1010 1010 R1 = L0 + f(R0, K1)

It remains to explain how the function f works.

Encoding DataEncoding Data continued… continued…

To calculate f, we first expand each block Rn-1 from 32 bits to 48 bits.

This is done by using a selection table called E-table that repeats some of the bits in Rn-1 .

32 1 2 3 4 5

4 5 6 7 8 9

8 9 10 11 12 13

12 13 14 15 16 17

16 17 18 19 20 21

20 21 22 23 24 25

24 25 26 27 28 29

28 29 30 31 32 1

E-table

We calculate E(R0) from R0 as follows:

R0 = 1111 0000 1010 1010 1111 0000 1010 1010 E(R0) = 011110 100001 010101 010101 011110 100001 010101 010101

Encoding DataEncoding Data continued… continued…

Next in the f calculation, we XOR the output E(Rn-1) with the key Kn:

For K1 , E(R0), we have

K1 = 000110 110000 001011 101111 111111 000111 000001 110010

E(R0) = 011110 100001 010101 010101 011110 100001 010101 010101

K1+E(R0) = 011000 010001 011110 111010 100001 100110 010100 100111

We now use each group of six bits as addresses in tables called "S boxes".

Each group of six bits will give us an address in a different S box. Located at

that address will be a 4 bit number.

This 4 bit number will replace the original 6 bits.

The net result is that the eight groups of 6 bits are transformed into eight

groups of 4 bits (the 4-bit outputs from the S boxes) for 32 bits total.

Kn E(Rn-1)

Encoding DataEncoding Data continued… continued…

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 14 4 13 1 3 15 11 8 3 10 6 12 5 9 0 7

1 0 15 7 4 14 2 13 1 10 6 12 11 9 5 3 8

2 4 1 14 8 13 6 2 11 15 12 9 7 3 10 5 0

3 15 12 8 2 4 9 1 7 5 11 3 14 10 0 6 13

Column numberRow number

S1 Box

Here S1(011011) = 0101

Similarly, there exists S1, S2,…, S8 For the first round, we obtain as the output of the eight S boxes:

K1 + E(R0) = 011000 010001 011110 111010 100001 100110 010100 100111. S = 0101 1100 1000 0010 1011 0101 1001 0111

Encoding DataEncoding Data continued… continued…

The final stage in the calculation of f is to do a permutation P of the S-box output to obtain the final value of f:

The permutation P is defined in the following table. P yields a 32-bit output from a 32-bit input by permuting the bits of the input block.

f = P(S)

16 7 20 21

29 12 28 17

1 15 23 26

5 18 31 10

2 8 24 14

32 27 3 9

19 13 30 6

22 11 4 25

P

From

S = 0101 1100 1000 0010 1011 0101 1001 0111

f = 0010 0011 0100 1010 1010 1001 1011 1011

Encoding DataEncoding Data continued… continued…

R1 = L0 f(R0, K1)

Proceeding like this we obtain L1R1, L2R2,…, L16R16.

At the end of the sixteenth round we have the blocks L16 and R16. We then reverse the order of

the two blocks into the 64-bit block R16L16 and apply a permutation IP-1.

= 1100 1100 0000 0000 1100 1100 1111 1111 0010 0011 0100 1010 1010 1001 1011 1011 = 1110 1111 0100 1010 0110 0101 0100 0100

Encoding DataEncoding Data continued… continued…

40 8 48 16 56 24 64 32

39 7 47 15 55 23 63 31

38 6 46 14 54 22 62 30

37 5 45 13 53 21 61 29

36 4 44 12 52 20 60 28

35 3 43 11 51 19 59 27

34 2 42 10 50 18 58 26

33 1 41 9 49 17 57 25

IP-1

LETR16L16 = 00001010 01001100 11011001 10010101 01000011 01000010 00110010 00110100 IP-1 = 10000101 11101000 00010011 01010100 00001111 00001010 10110100 00000101 which in hexadecimal format is 85E813540F0AB405.

Thus the encrypted form of M = 0123456789ABCDEF:

namely, C = 85E813540F0AB405

DecryptionDecryption

Decryption is simply the inverse of encryption, following

the same steps as above, but reversing the order in

which the subkeys are applied.

Disadvantages ofDisadvantages ofSecret-key CryptographySecret-key Cryptography

Use of physical secure channel

Courier service or registered mail for key distribution is costly, inconvenient & slow

Requirement of large network

For n user channels required n*(n-1)/2

This large network leads to use of insecure channel for key distribution & secure message transmission.

Public-key CryptographyPublic-key Cryptography It contains two components.

Private component, known to the authorised user only

Public component, visible to everybody Each pair of keys must have two basic properties.

Whatever message encrypted with one of the keys can be decrypted by the other key.

Given knowledge of the public key, it is computationally infeasible to compute the private key.

The key management here helps in development of large network.

Diffie-HellmanDiffie-Hellman Public-key Distribution Public-key Distribution

It uses the concept that, it is easy to calculate the discrete exponential but difficult to calculate discrete logarithm.

Discrete exponential : Y = αX mod p, for 1≤ X ≤p-1

Discrete logarithm : X = logαY mod p, for 1≤ Y≤p-1

All users are assumed to know both α, p.

A user i, selects an independent random number Xi,

uniformly from the set of integers {1, 2,…, p} that is kept private.

But the discrete exponential Yi = αXi mod p is made public.

Diffie-HellmanDiffie-Hellman Public-key Distribution Public-key Distribution

continued… continued… Now, user I & j want to communicate. To proceed, user i fetches Yj from public directory & uses the private Xi

to compute

Kji =(Yj)Xi mod p

=(αXj)Xi mod p

=αXjXi mod p In a similar way, user j computes Kij. But we have

Kij = Kji

For an eavesdropper must compute Kji from Yi & Yj applying the formula

Kji =(Yj)log Yi mod p

Since it involves discrete logarithm not easy to calculate.

Rivest-Shamir-AdlemanRivest-Shamir-Adleman(RSA) System(RSA) System

It is a block cipher based upon the fact that finding a random prime number of large size (e.g., 100 digit) is computationally easy, but factoring the product of two such numbers is considered computationally infeasible.

RSA algorithmRSA algorithm1. Key Generation

2. Generate two large prime numbers, p and q

3. Let n = p*q

4. Let m = (p-1)*(q-1)

5. Choose a small number e, coprime to m

6. Find d, such that de % m = 1

Encryption

C = Pe % n

Decryption

P = Cd % n

x % y means the remainder of x divided by y

Publish e and n as the public key.

Keep d and n as the secret key.

To be secure, very large numbers must be used for p and q - 100 decimal digits at the very least.

RSA : An IllustrationRSA : An Illustration Generate two large prime numbers, p and q

To make the example easy to follow I am going to use small numbers, but this is not secure.

Lets have: p = 7;q=19 Let n = p*q = 7 * 19 = 133 Let m = (p - 1)*(q - 1) = (7 - 1)(19 - 1) = 6 * 18 = 108 4) Choose a small number, e coprime to m

e = 2 => gcd(e, 108) = 2 (no); e = 3 => gcd(e, 108) = 3 (no);e = 4 => gcd(e, 108) = 4 (no); e = 5 => gcd(e, 108) = 1 (yes!)

Find d, such that de % m = 1

n = 0 => d = 1 / 5 (no); n = 1 => d = 109 / 5 (no);n = 2 => d = 217 / 5 (no); n = 3 => d = 325 / 5 = 65 (yes!)

RSA : An IllustrationRSA : An Illustration continued…continued…

Public Key: n = 133; e = 5 Secret Key: n = 133; d = 65

Encryption lets use the message "6" .

C = Pe % n = 65 % 133= 7776 % 133 = 62

Decryption

P = Cd % n = 6265 % 133 = 6

Digital Signature:Digital Signature: A hybrid approachA hybrid approach

The most useful requirements for a digital signature is

authenticity and secrecy.

RSA provide an effective method for key management,

but they are inefficient for bulk encryption of data.

DES provide better throughput, but require key

management.

So, a combinational approach can be considered for

practical usability, e.g., RSA may be used for

authentication and DES used for encryption.

ReferenceReference Simon Haykin, Communication Systems, 4th ed. (New York: John

Wiley & Sons, 2004) Martin A. Hellman, “An overview of public key cryptography,” IEEE

communications magazine, vol. 16, no. 6, November 1978. C. E. Shannon, “A mathematical theory of communication,” Bell

system technical journal, p. 623, July 1948. Gary C. Kessler, “An overview of cryptography,” May 1998 edited version of Handbook on Local Area Networks

(Auerbach, September 1998) http://orlingrabbe.com/ www.rsasecurity.com www.wikipedia.com www.bambooweb.com

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