Stallings_8e_Accessible_fullppt_09.pdf

Cryptography and Network Security:

Principles and PracticeEighth Edition

Chapter 9

Public Key Cryptography and R S A

Table 9.1 Terminology Related to Asymmetric

Encryption

Asymmetric Keys

Two related keys, a public key and a private key, that are used to perform

complementary operations, such as encryption and decryption or signature generation

and signature verification.

Public Key Certificate

A digital document issued and digitally signed by the private key of a Certification

Authority that binds the name of a subscriber to a public key. The certificate indicates

that the subscriber identified in the certificate has sole control and access to the

corresponding private key.

Public Key (Asymmetric) Cryptographic Algorithm

A cryptographic algorithm that uses two related keys, a public key and a private key.

The two keys have the property that deriving the private key from the public key is

computationally infeasible.

Public Key Infrastructure (PKI)

A set of policies, processes, server platforms, software and workstations used for the

purpose of administering certificates and public-private key pairs, including the ability to

issue, maintain, and revoke public key certificates.

Source: Glossary of Key Information Security Terms, NISTIR 7298.

Misconceptions Concerning Public-

Key Encryption

• Public-key encryption is more secure from cryptanalysis

than symmetric encryption

• Public-key encryption is a general-purpose technique that

has made symmetric encryption obsolete

• There is a feeling that key distribution is trivial when using

public-key encryption, compared to the cumbersome

handshaking involved with key distribution centers for

symmetric encryption

Principles of Public-Key Cryptosystems

• The concept of public-key cryptography evolved from an attempt

to attack two of the most difficult problems associated with

symmetric encryption:

• Key distribution

– How to have secure communications in general without

having to trust a K D C with your key

• Digital signatures

– How to verify that a message comes intact from the claimed

sender

• W hitfield Diffie and Martin Hellman from Stanford University

achieved a breakthrough in 1976 by coming up with a method

that addressed both problems and was radically different from

all previous approaches to cryptography

Public-Key Cryptosystems

• A public-key encryption scheme has six ingredients:

• Plaintext

– The readable message or data that is fed into the algorithm as input

• Encryption algorithm

– Performs various transforma-tions on the plaintext

• Public key

– Used for encryption or decryption

• Private key

– Used for encryption or decryption

• Ciphertext

– The scrambled message produced as output

• Decryption algorithm

– Accepts the ciphertext and the matching key and produces the original plaintext

Figure 9.1 Public-Key Cryptography (1 of 2)

Figure 9.1 Public-Key Cryptography (2 of 2)

Table 9.2 Conventional and Public-

key Encryption

Conventional Encryption Public-Key Encryption

Needed to Work:

1. The same algorithm with the same key is used for encryption and decryption.

2. The sender and receiver must share the

algorithm and the key.

Needed to Work:

1. One algorithm is used for encryption and a related algorithm for decryption with a pair of keys, one for encryption and one for

decryption.2. The sender and receiver must each have one

of the matched pair of keys (not the same one).

Needed for Security:

1. The key must be kept secret.2. It must be impossible or at least impractical

to decipher a message if the key is kept

secret.3. Knowledge of the algorithm plus samples of

ciphertext must be insufficient to determinethe key.

Needed for Security:

1. One of the two keys must be kept secret.2. It must be impossible or at least impractical

to decipher a message if one of the keys is

kept secret.3. Knowledge of the algorithm plus one of the

keys plus samples of ciphertext must be insufficient to determine the other key.

Public-Key Cryptosystem: Confidentiality

Figure 9.2 Public-Key Cryptosystem: Confidentiality

Public-Key Cryptosystem: Authentication

Figure 9.3 Public-Key Cryptosystem: Authentication

Public-Key Cryptosystem:

Authentication and Secrecy

Figure 9.4 Public-Key Cryptosystem: Authentication and

Secrecy

Applications for Public-Key

Cryptosystems

• Public-key cryptosystems can be classified into three

categories:

• Encryption/decryption

– The sender encrypts a message with the recipient’s public

key

• Digital signature

– The sender “signs” a message with its private key

• Key exchange

– Two sides cooperate to exchange a session key

• Some algorithms are suitable for all three applications, whereas

others can be used only for one or two

Table 9.3 Applications for Public-Key

Cryptosystems

Algorithm Encryption/Decryption Digital

Signature

Key Exchange

RSA Yes Yes Yes

Elliptic Curve Yes Yes Yes

Diffie–Hellman No No Yes

DSS No Yes No

Public-Key Requirements (1 of 2)

• Conditions that these algorithms must fulfill:

– It is computationally easy for a party B to generate a pair

(public-key P Ub, private key P Rb)

– It is computationally easy for a sender A, knowing the public

key and the message to be encrypted, to generate the

corresponding ciphertext

– It is computationally easy for the receiver B to decrypt the

resulting ciphertext using the private key to recover the

original message

– It is computationally infeasible for an adversary, knowing the

public key, to determine the private key

– It is computationally infeasible for an adversary, knowing the

public key and a ciphertext, to recover the original message

– The two keys can be applied in either order

Public-Key Requirements (2 of 2)

• Need a trap-door one-way function

– A one-way function is one that maps a domain into a range such

that every function value has a unique inverse, with the condition

that the calculation of the function is easy, whereas the calculation

of the inverse is infeasible

▪ Y = f(X) easy

▪ X = f–1(Y) infeasible

• A trap-door one-way function is a family of invertible functions fk, such

that

– Y = fk(X) easy, if k and X are known

– X = fk–1(Y) easy, if k and Y are known

– X = fk–1(Y) infeasible, if Y known but k not known

• A practical public-key scheme depends on a suitable trap-door one-

way function

Public-Key Cryptanalysis

• A public-key encryption scheme is vulnerable to a brute-force attack

– Countermeasure: use large keys

– Key size must be small enough for practical encryption and decryption

– Key sizes that have been proposed result in encryption/decryption speeds that are too slow for general-purpose use

– Public-key encryption is currently confined to key management and signature applications

• Another form of attack is to find some way to compute the private key given the public key

– To date it has not been mathematically proven that this form of attack is infeasible for a particular public-key algorithm

• Finally, there is a probable-message attack

– This attack can be thwarted by appending some random bits to simple messages

Rivest-Shamir-Adleman (R S A)

Algorithm

• Developed in 1977 at M I T by Ron Rivest, Adi Shamir &

Len Adleman

• Most widely used general-purpose approach to public-key

encryption

• Is a cipher in which the plaintext and ciphertext are

integers between 0 and n – 1 for some n

– A typical size for n is 1024 bits, or 309 decimal digits

R S A Algorithm

• RSA makes use of an expression with exponentials

• Plaintext is encrypted in blocks with each block having a binary

value less than some number n

• Encryption and decryption are of the following form, for some

plaintext block M and ciphertext block C

C = Me mod n

M = Cd mod n = (Me)d mod n = Med mod n

• Both sender and receiver must know the value of n

• The sender knows the value of e, and only the receiver knows

the value of d

• This is a public-key encryption algorithm with a public key of

PU={e,n} and a private key of PR={d,n}

Algorithm Requirements

• For this algorithm to be satisfactory for public-key

encryption, the following requirements must be met:

1. It is possible to find values of e, d, n such that Med mod

n = M for all M < n

2. It is relatively easy to calculate Me mod n and Cd mod n

for all values of M < n

3. It is infeasible to determine d given e and n

Figure 9.5 The R S A Algorithm

Example of R S A Algorithm

Figure 9.6 Example of R S A Algorithm

Figure 9.7 R S A Processing of

Multiple Blocks (1 of 2)

Figure 9.7 R S A Processing of

Multiple Blocks (2 of 2)

Exponentiation in Modular Arithmetic

• Both encryption and decryption in RSA involve raising an

integer to an integer power, mod n

• Can make use of a property of modular arithmetic:

[(a mod n) x (b mod n)] mod n =(a x b) mod n

• With RSA you are dealing with potentially large exponents

so efficiency of exponentiation is a consideration

Figure 9.8 Algorithm for Computing

ab mod n

Note: The integer b is expressed as a binary number bkbk − 1…b0

Table 9.4 Result of the Fast Modular

Exponentiation Algorithm for ab mod n, where a

= 7, b = 560 = 1000110000, and n = 561

I 9 8 7 6 5 4 3 2 1 0

Bi 1 0 0 0 1 1 0 0 0 0

C 1 2 4 8 17 35 70 140 280 560

F 7 49 157 526 160 241 298 166 67 1

Efficient Operation Using the Public

Key

• To speed up the operation of the R S A algorithm using the

public key, a specific choice of e is usually made

• The most common choice is 65537 (216 + 1)

– Two other popular choices are e=3 and e=17

– Each of these choices has only two 1 bits, so the

number of multiplications required to perform

exponentiation is minimized

– With a very small public key, such as e = 3, R S A

becomes vulnerable to a simple attack

Efficient Operation Using the Private

Key

• Decryption uses exponentiation to power d

– A small value of d is vulnerable to a brute-force attack

and to other forms of cryptanalysis

• Can use the Chinese Remainder Theorem (C R T) to speed

up computation

– The quantities d mod (p – 1) and d mod (q – 1) can be

precalculated

– End result is that the calculation is approximately four

times as fast as evaluating M = Cd mod n directly

Key Generation

• Before the application of the public-key cryptosystem each

participant must generate a pair of keys:

– Determine two prime numbers p and q

– Select either e or d and calculate the other

• Because the value of n = pq will be known to any potential

adversary, primes must be chosen from a sufficiently large

set

– The method used for finding large primes must be

reasonably efficient

Procedure for Picking a Prime

Number

• Pick an odd integer n at random

• Pick an integer a < n at random

• Perform the probabilistic primality test with a as a

parameter. If n fails the test, reject the value n and go to

step 1

• If n has passed a sufficient number of tests, accept n;

otherwise, go to step 2

The Security of R S A

• Five possible approaches to attacking RSA are:

– Brute force

▪ Involves trying all possible private keys

– Mathematical attacks

▪ There are several approaches, all equivalent in effort to factoring the product of two primes

– Timing attacks

▪ These depend on the running time of the decryption algorithm

– Hardware fault-based attack

▪ This involves inducing hardware faults in the processor that is generating digital signatures

– Chosen ciphertext attacks

▪ This type of attack exploits properties of the RSA algorithm

Factoring Problem

• We can identify three approaches to attacking RSA

mathematically:

– Factor n into its two prime factors. This enables

calculation of ø(n) = (p – 1) x (q – 1), which in turn

enables determination of d = e-1 (mod ø(n))

– Determine ø(n) directly without first determining p and

q. Again this enables determination of d = e-1 (mod

ø(n))

– Determine d directly without first determining ø(n)

Timing Attacks

• Paul Kocher, a cryptographic consultant, demonstrated

that a snooper can determine a private key by keeping

track of how long a computer takes to decipher messages

• Are applicable not just to RSA but to other public-key

cryptography systems

• Are alarming for two reasons:

– It comes from a completely unexpected direction

– It is a ciphertext-only attack

Countermeasures

• Constant exponentiation time

– Ensure that all exponentiations take the same amount of time before returning a result; this is a simple fix but does degrade performance

• Random delay

– Better performance could be achieved by adding a random delay to the exponentiation algorithm to confuse the timing attack

• Blinding

– Multiply the ciphertext by a random number before performing exponentiation; this process prevents the attacker from knowing what ciphertext bits are being processed inside the computer and therefore prevents the bit-by-bit analysis essential to the timing attack

Fault-Based Attack

• An attack on a processor that is generating R S A digital

signatures

– Induces faults in the signature computation by reducing the

power to the processor

– The faults cause the software to produce invalid signatures

which can then be analyzed by the attacker to recover the

private key

• The attack algorithm involves inducing single-bit errors and

observing the results

• W hile worthy of consideration, this attack does not appear to be

a serious threat to R S A

– It requires that the attacker have physical access to the

target machine and is able to directly control the input power

to the processor

Chosen Ciphertext Attack (C CA)

• The adversary chooses a number of ciphertexts and is

then given the corresponding plaintexts, decrypted with the

target’s private key

– Thus the adversary could select a plaintext, encrypt it

with the target’s public key, and then be able to get the

plaintext back by having it decrypted with the private

key

– The adversary exploits properties of R S A and selects

blocks of data that, when processed using the target’s

private key, yield information needed for cryptanalysis

• To counter such attacks, R S A Security Inc. recommends

modifying the plaintext using a procedure known as

optimal asymmetric encryption padding (O A E P)

Figure 9.9 Encryption Using Optimal

Asymmetric Encryption Padding

(O A E P)

Summary

• Present an overview of the basic principles of public-key

cryptosystems

• Explain the two distinct uses of public-key cryptosystems

• List and explain the requirements for a public-key cryptosystem

• Present an overview of the R S A algorithm

• Understand the timing attack

• Summarize the relevant issues related to the complexity of

algorithms

This work is protected by United States copyright laws and is

provided solely for the use of instructors in teaching their

courses and assessing student learning. Dissemination or sale of

any part of this work (including on the World Wide Web) will

destroy the integrity of the work and is not permitted. The work

and materials from it should never be made available to students

except by instructors using the accompanying text in their

classes. All recipients of this work are expected to abide by these

restrictions and to honor the intended pedagogical purposes and

the needs of other instructors who rely on these materials.