CRYPTOGRAPHY

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CRYPTOGRAPHY

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Title: CRYPTOGRAPHY

1
CRYPTOGRAPHY

Presented by
Debi Prasad Mishra
Institute of Technical Education Reaserch
Electronics Telecommunication Engineering
Section - A
7th Semester
Regd. No. - 0301212148

2
Talk Flow

Terminology
Secret-key cryptographic system
Block cipher
Stream cipher
Requirement of secrecy
Information theoretic approach
Perfect security
Diffusion 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.

4
Terminology

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

5
Services offered by Cryptography

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

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 ones
own with a private component a public
component
The private component must be kept secret for
secure communication.

7
Secret-key Cryptography
Let X -gt Plaintext message Y -gt Cryptogram
Z -gt Key F -gtInvertible transformation
producing the cryptogram Y F (X, Z) FZ (X) Let
F-1 -gtInverse transform of F to recover original
message F-1 (Y, Z) Fz-1 (Y) FZ-1 (FZ (X))
X
8
Secret-key Cryptographycontinued
Here Y -gtfraudulent message modified by an
interceptor or eavesdropper
9
Block 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.

10
Stream ciphers

Whereas block ciphers operate on large data on a
block-by-block basis, stream ciphers operate on
individual bits.

Let xn -gt Plaintext bit y -gtciphertext bit z
-gtkeystream bit at nth instant For encryption
yn xn zn, n1, 2, , N For decryption
xn yn zn, n1, 2, , N
11
Stream ciphers
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.

12
Requirement of Secrecy

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

13
Requirement of Secrecy
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 attack
Submit 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.

14
Information theoretic approach

In Shannon model of cryptography (published in
Shannons 1949 landmark paper on
information-theoretic approach to secrecy
systems)
ASSUMPTION-
Enemy cryptanalyst has unlimited time computing
power.
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

15
Information theoretic approach
(continued)

Let X X1, X2, , XN -gtN-bit plaintext
message,
Y Y1, Y2, ,YN -gtN-bit cryptogram
Secret key Z is assumed to be determined by
some probability distribution
Let H (X) -gtuncertainty about x
H (X Y) -gtuncertainty about X given
knowledge of Y
Now, mutual information between X Y,

I (XY) H (X) H(X Y)
represents a basic measure of security in the
Shannon model.
16
Perfect Security

Assuming that an enemy cryptanalyst can observe
only the cryptogram Y, for perfect security X Y
should be statistically independent.
I (XY)0 gtH (X) H (XY)
.......(1)
Given the secret key Z H (XY) H (X ZY)
H (ZY) H (XY,Z) (2)
H(XY,Z)0 iff Y Z together uniquely
determine X
Equation 2 can be rewritten as H(XY)
H(ZY)
H(Z) (3)
With equation 3 equation 1 becomes
H(Z) H(X)
..(4)
Is called Shannons fundamental bound for perfect
security.

Result The key must be at least as long as the
plaintext.
17
Diffusion 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 Cipher

For a cipher to be of practical value
It must be difficult to be broken by enemy
cryptanalyst.
It must be easy to encrypt decrypt with
knowledge of secret key.

18
Substitution cipher
Each 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),.
19
Transposition 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.

20
Data Encryption Standard(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-1FP(X)
where, X-gtplaintext
P-gtcertain permutation
F-gtcertain 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.

21
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

22
Key Computation

The 64-bit key is permuted according to the
following table 56-bit key is calculated from
it.

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
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
23
Key Computation continued

With C0 and D0 defined, we now create sixteen
blocks Cn and Dn, 1ltnlt16. 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.
24
Key Computation continued

We now form the keys Kn, for 1ltnlt16, by
applying the following permutation table to each
of the concatenated pairs CnDn.

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.
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
Thus the 16, 48-bit subkeys are obtained.
25
Encoding 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

26
Encoding Data
continued

We now proceed through 16 iterations, for
1ltnlt16, 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.

27
Encoding Data
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 .

E-table
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
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
28
Encoding Data
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
K1E(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)
29
Encoding Data
continued
S1 Box
Column number
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
R o w n u m b e r
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
30
Encoding Data
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)
P
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
From S 0101 1100 1000 0010 1011 0101 1001
0111 f 0010 0011 0100 1010 1010 1001 1011 1011

31
Encoding Data
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

32
Encoding Data
continued
IP-1
LET R16L16 00001010 01001100 11011001 10010101
01000011 01000010 00110010 00110100 IP-1
10000101 11101000 00010011 01010100 00001111
00001010 10110100 00000101 which in hexadecimal
format is 85E813540F0AB405.
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
Thus the encrypted form of M 0123456789ABCDEF
namely, C 85E813540F0AB405
33
Decryption

Decryption is simply the inverse of
encryption, following the same steps as above,
but reversing the order in which the subkeys are
applied.

34
Disadvantages ofSecret-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.

35
Public-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.

36
Diffie-Hellman 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 aX mod p, for 1 X
p-1
Discrete logarithm X logaY mod p, for
1 Yp-1
All users are assumed to know both a, 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 aXi mod p is
made public.

37
Diffie-Hellman Public-key Distribution
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
(aXj)Xi mod p
aXjXi 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.

38
Rivest-Shamir-Adleman(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.

39
RSA algorithm
Encryption C Pe n Decryption P Cd n x
y means the remainder of x divided by y

Key Generation
Generate two large prime numbers, p and q
Let n pq
Let m (p-1)(q-1)
Choose a small number e, coprime to m
Find d, such that de m 1

To be secure, very large numbers must be
used for p and q - 100 decimal digits at the very
least.
Publish e and n as the public key. Keep d and n
as the secret key.
40
RSA 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 7q19
Let n pq 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 gt gcd(e, 108) 2 (no) e 3
gt gcd(e, 108) 3 (no)e 4 gt gcd(e, 108) 4
(no) e 5 gt gcd(e, 108) 1 (yes!)
Find d, such that de m 1
n 0 gt d 1 / 5 (no) n 1 gt
d 109 / 5 (no)n 2 gt d 217 / 5 (no)
n 3 gt d 325 / 5 65 (yes!)

41
RSA An Illustration
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

42
Digital Signature A 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.

43
Reference