PRINCIPLES OF INFORMATION SYSTEM SECURITY

UNIT - 2
PUBLIC KEY CRYPTOGRAPHY AND RSA
Information Security
BY
Khushbu Garg
Assistant Professor
Jecrcu

Plain Text to Cipher Text Conversion
Techniques

Introduction
Encryption is the process of converting
plaintext into ciphertext to secure
information.
Importance:
• Used in communication, data security, and
cryptography.

Substitution Ciphers
Caesar Cipher:
• Each letter shifted by a fixed number.
• Example: HELLO → KHOOR (Shift = 3)
Vigenère Cipher:
• Uses a keyword for multiple shifts.
• Example (Keyword = 'KEY'): HELLO →
RIJVS

Transposition Ciphers
Rail Fence Cipher:
• Letters written diagonally, read row-wise.
• Example: HELLO WORLD →
HLOWRDELLOOL (Depth = 2)
Columnar Transposition:
• Text written in a grid and rearranged by
key order.
• Example (Key = 4312): HELLO → OHLLE

PRIVATE-KEY CRYPTOGRAPHY
🞭 traditional private/secret/single
keycryptography uses one key
🞭 shared by both sender and receiver
🞭 if this key is disclosed communications
are compromised
🞭 also is symmetric, parties are equal
🞭 hence does not protect sender from receiver
forging a message & claiming is sent by sender

PUBLIC-KEY CRYPTOGRAPHY
🞭 probably most significant advance in the
3000 year history of cryptography
🞭 uses two keys – a public & a private key
🞭 asymmetric since parties are not equal
🞭 uses clever application of number
theoretic concepts to function
🞭 complements rather than replaces private
keycrypto

WHY PUBLIC-KEY CRYPTOGRAPHY?
🞭 developed to address two key issues:
🞭 key distribution – how to have secure
communications in general without having to
trust a KDC with your key
🞭 digital signatures – how to verify a
message comes intact from the claimed
sender
🞭 public invention due to Whitfield Diffie
&Martin Hellman at Stanford Uni in
1976
🞭 known earlier in classified community

PUBLIC-KEY CRYPTOGRAPHY
🞭 public-key/two-key/asymmetric cryptography
involves the use of two keys:
🞭 a public-key, which may be known by anybody, and can
be used to encrypt messages, and verify signatures
🞭 a private-key, known only to the recipient, used to
decrypt messages, and sign (create) signatures
🞭 is asymmetric because
🞭 those who encrypt messages or verify signatures
cannot decrypt messages or create signatures

PUBLIC-KEY CHARACTERISTICS
🞭 Public-Key algorithms rely on two keys where:
🞭 it is computationally infeasible to find decryption
keyknowing only algorithm & encryption key
🞭 it is computationally easy to en/decrypt messages when
the relevant (en/decrypt) key is known
🞭 either of the two related keys can be used for
encryption, with the other used for decryption (for some
algorithms)

PUBLIC-KEY APPLICATIONS
🞭 can classify uses into 3 categories:
🞭 encryption/decryption (provide secrecy)
🞭 digital signatures (provide authentication)
🞭 key exchange (of session keys)
🞭 some algorithms are suitable for all
uses, others are specific to one

RSA
🞭 by Rivest, Shamir & Adleman of MIT in 1977
🞭 best known & widely used public-key scheme
🞭 based on exponentiation in a finite (Galois) field
over integers modulo a prime
🞭 nb. exponentiation takes O((log n)3) operations (easy)
🞭 uses large integers (eg. 1024 bits)
🞭 security due to cost of factoring large numbers
🞭 nb. factorization takes O(e log n log log n) operations (hard)

RSA KEY SETUP
🞭 each user generates a public/private key pair
by:
🞭 selecting two large primes at random - p, q
🞭 computing their system modulus n=p.q
🞭 note ø(n)=(p-1)(q-1)
🞭 selecting at random the encryption key e
🞭 where 1<e<ø(n), gcd(e,ø(n))=1
🞭 solve following equation to find decryption key
d
🞭 e.d=1 mod ø(n) and 0≤d≤n
🞭 publish their public encryption key: PU={e,n}
🞭 keep secret private decryption key: PR={d,n}

RSA USE
🞭 to encrypt a message M the sender:
🞭 obtains public key of recipient PU={e,n}
🞭 computes: C = Me mod n, where 0≤M<n
🞭 to decrypt the ciphertext C the owner:
🞭 uses their private key PR={d,n}
🞭 computes: M = Cd mod n
🞭 note that the message M must be smaller
than the modulus n (block if needed)

WHY RSA WORKS
🞭 because of Euler's Theorem:
🞭 aø(n)mod n = 1 where gcd(a,n)=1
🞭 in RSA have:
🞭 n=p.q
🞭 ø(n)=(p-1)(q-1)
🞭 carefully chose e & d to be inverses mod ø(n)
🞭 hence e.d=1+k.ø(n) for some k
🞭 hence :
Cd = Me.d = M1+k.ø(n) = M1.(Mø(n))k
= M1.(1)k = M1 = M mod n

RSA EXAMPLE - KEY SETUP
1. Select primes: p=17 & q=11
2. Compute n = pq =17 x 11=187
3. Compute ø(n)=(p–1)(q-1)=16 x 10=160
4. Select e: gcd(e,160)=1; choose e=7
5. Determine d: de=1 mod 160 and d < 160 Value
is d=23 since 23x7=161= 10x160+1
6. Publish public key PU={7,187}
7. Keep secret private key PR={23,187}

RSA EXAMPLE - EN/DECRYPTION
🞭 sample RSA encryption/decryption
is:
🞭 given message M = 88 (nb.
88<187)
🞭 encryption:
C = 887 mod 187 = 11
🞭 decryption:
M = 1123 mod 187 = 88

EXPONENTIATION
🞭 can use the Square and Multiply Algorithm
🞭 a fast, efficient algorithm for exponentiation
🞭 concept is based on repeatedly squaring
base
🞭 and multiplying in the ones that are needed
tocompute the result
🞭 look at binary representation of exponent
🞭 only takes O(log2 n) multiples for number n
🞭 eg. 75 = 74.71 = 3.7 = 10
mod 11
= 3128.31
🞭 eg. 3129 = 5.3 = 4 mod 11

EXPONENTIATION
c = 0; f = 1
for i = k downto 0
do c = 2 x c
f = (f x f) mod n
if bi == 1 then
c = c + 1
f = (f x a) mod n
return f

EFFICIENT ENCRYPTION
🞭 encryption uses exponentiation to power e
🞭 hence if e small, this will be faster
🞭 often choose e=65537 (216-1)
🞭 also see choices of e=3 or e=17
🞭 but if e too small (eg e=3) can attack
🞭 using Chinese remainder theorem & 3
messages with different modulii
🞭 if e fixed must ensure gcd(e,ø(n))=1
🞭 ie reject any p or q not relatively prime to e

EFFICIENT DECRYPTION
🞭 decryption uses exponentiation to power d
🞭 this is likely large, insecure if not
🞭 can use the Chinese Remainder Theorem
(CRT) to compute mod p & q separately. then
combine to get desired answer
🞭 approx 4 times faster than doing directly
🞭 only owner of private key who knows values
ofp & q can use this technique

RSA KEY GENERATION
🞭 users of RSA must:
🞭 determine two primes at random - p, q
🞭 select either e or d and compute the other
🞭 primes p,q must not be easily derived
from modulus n=p.q
🞭 means must be sufficiently large
🞭 typically guess and use probabilistic test
🞭 exponents e, d are inverses, so use
Inverse algorithm to compute the other

RSA SECURITY
🞭 possible approaches to attacking RSA are:
🞭 brute force key search (infeasible given size
ofnumbers)
🞭 mathematical attacks (based on difficulty
ofcomputing ø(n), by factoring modulus n)
🞭 timing attacks (on running of decryption)
🞭 chosen ciphertext attacks (given properties of
RSA)

Information Security
Diffie – Hellman Key Exchange &
Elliptic Curve Cryptography

Diffie-HellmanKey Exchange
• first public-key type scheme proposed
• by Diffie & Hellman in 1976 along with the
exposition of public key concepts
• note: now know that Williamson (UK CESG)
secretly proposed the concept in 1970
• is a practical method for public exchange of a
secret key
• used in a number of commercial products
[ Continue…]

• a public-key distribution scheme
• cannot be used to exchange an arbitrary message
• rather it can establish a common key
• known only to the two participants
• value of key depends on the participants (and
their private and public key information)
• based on exponentiation in a finite (Galois) field
(modulo a prime or a polynomial) - easy
• security relies on the difficulty of computing
discrete logarithms (similar to factoring) – hard
[ Continue…]

Diffie-HellmanSetup
• all users agree on global parameters:
• large prime integer or polynomial q
• a being a primitive root mod q
• each user (eg. A) generates their key
• chooses a secret key (number): xA < q
A
• compute their public key: y = a
xA mod q
• each user makes public that key yA
[ Continue…]

• shared session key for users A & B is KAB:
KAB
= axA.xB
mod q
xB
B
= y
xA
= yA mod q
mod q
(which B can compute)
(which A can compute)
• KAB is used as session key in private-key
encryption scheme between Alice and Bob
• if Alice and Bob subsequently communicate,
they will have the same key as before, unless
they choose new public-keys
• attacker needs an x, must solve discrete log
[ Continue…]

Diffie-Hellman Example
• users Alice & Bob who wish to swap keys:
• agree on prime q=353 and a=3
• select random secret keys:
• A chooses xA=97, B chooses xB=233
• compute respective public keys:
A
• y =3
97
B
• y =3
233
mod 353 = 40
mod 353 = 248
(Alice)
(Bob)
• compute shared session key as:
AB B
• K = y xA 97
AB A
• K = y
xB
mod 353 = 248
mod 353 = 40
233
= 160
= 160
(Alice)
(Bob)
[ Continue…]

KeyExchangeProtocols
• users could create random private/public D-H
keys each time they communicate
• users could create a known private/public D-H
key and publish in a directory, then consulted
and used to securely communicate with them
• both of these are vulnerable to a meet-in-the-
Middle Attack
• authentication of the keys is needed

Introduction to Encryption
• Encryption is used to secure information by
converting plaintext into ciphertext.
• Symmetric encryption uses the same key
for both encryption and decryption.
• DES and AES are two widely known
symmetric encryption algorithms.

DES (Data Encryption Standard)
• Developed by IBM in the 1970s, adopted
by NIST.
• Uses a 56-bit key and encrypts data in 64-
bit blocks.
• 16 rounds of Feistel structure encryption.
• Vulnerable to brute-force attacks due to
small key size.
• Considered obsolete and replaced by AES.

AES (Advanced Encryption Standard)
• Developed by Vincent Rijmen and Joan
Daemen, adopted in 2001.
• Supports 128, 192, or 256-bit key sizes.
• Encrypts data in 128-bit blocks.
• Uses 10, 12, or 14 rounds of substitution-
permutation encryption.
• Highly secure and widely used in modern
encryption.

DES vs AES: Key Differences
• Key Size: DES (56-bit) vs AES (128, 192,
256-bit).
• Block Size: DES (64-bit) vs AES (128-bit).
• Rounds: DES (16 rounds) vs AES (10, 12,
or 14 rounds).
• Security: DES is weak against brute-force
attacks; AES is highly secure.
• Usage: DES is obsolete, AES is widely used
in modern encryption applications.

Conclusion
• DES was a pioneer in symmetric
encryption but is now outdated.
• AES is the modern standard due to its
enhanced security and flexibility.
• AES is used in secure communication, data
protection, and cybersecurity.
• Understanding these algorithms helps in
choosing the right encryption method.

PRINCIPLES OF INFORMATION SYSTEM SECURITY

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