Public Key and RSA.pdf

Public Key Cryptography and RSA
© 2017 Pearson Education, Ltd., All rights reserved.

Table 9.1
Terminology Related to Asymmetric Encryption
Source: Glossary of Key Information Security Terms, NIST IR 7298 [KISS06]

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

• The concept of public-key cryptography evolved from
an attempt to attack two of the most difficult
problems associated with symmetric encryption:
• Whitfield 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
Principles of Public-Key
Cryptosystems
• How to have secure communications in general without having to
trust a KDC with your key
Key distribution
• How to verify that a message comes intact from the claimed sender
Digital signatures

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
ciphertex
t and the
matching
key and
produces
the
original
plaintext

Comparison of SK and PK
DISTINCT
FEATURES
SECRET KEY PUBLIC KEY
NUMBER OF
KEYS
Single key. Pair of keys.
TYPES OF KEYS Key is secret. One key is
private, and one
key is public.
SIZE OF KEY 50-250 bits 500-2500 bits
RELATIVE
SPEEDS
Faster. Slower.

Table 9.2
Conventional and Public-Key Encryption

Public-Key Cryptosystem: Secrecy

Public-Key Cryptosystem: Authentication

Public-Key Cryptosystem:
Authentication and Secrecy

Applications for Public-Key
Cryptosystems
• Public-key cryptosystems can be classified into
three categories:
• Some algorithms are suitable for all three
applications, whereas others can be used only for
one or two
•The sender encrypts a message
with the recipient’s public key
Encryption/decryption
•The sender “signs” a message
with its private key
Digital signature
•Two sides cooperate to
exchange a session key
Key exchange

Table 9.3
Applications for Public-Key Cryptosystems
Table 9.3 Applications for Public-Key Cryptosystems

Public-Key Requirements
• Conditions that these algorithms must fulfill:
• It is computationally easy for a party B to generate a pair
(public-key PUb, private key PRb)
• 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
• 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

Important Mathematical
Ideas
• Prime Numbers
• Multiplication vs. Factorization
• Greatest Common Divisor/Euclidean Algorithm
• Relatively Prime Numbers
• Modular Arithmetic/Modular Inverse
• Euler’s Theorem
• Multiplicative functions

Prime Numbers ...
• A prime number, or prime, is a number that is
evenly divisible by only 1 and itself.
• For instance 10 is not prime because it is evenly
divisible by 1, 2, 5 and 10. But 11 is prime, since
only 1 and 11 evenly divide it.
• The numbers that evenly divide another
number are called factors. The process of
finding the factors of a number is called
factoring.

Factoring a Number ...
• For example, factoring 15 is simple, it is 3 * 5.
But what about 6,320,491,217?
• Now how about a 155-digit number? Or 200
digits or more? In short, factoring numbers
takes a certain number of steps, and the
number of steps increases subexponentially as
the size of the number increases. That means
even on supercomputers, if a number is
sufficiently large, the time to execute all the
steps to factor it would be so great that it
could take years to compute.

Relatively Prime
• Two numbers are relatively prime if they
share only one factor, namely 1.
• For example, 10 and 21 are relatively prime.
Neither is prime, but the numbers that
evenly divide 10 are 1, 2, 5 and 10, whereas
the numbers that evenly divide 21 are 1, 3, 7
and 21.
• The only number in both lists is 1, so the
numbers are relatively prime.

Greatest Common
Divisor
• If two numbers are relatively prime their GCD
is 1.
• m and n are relatively prime means gcd(m, n) =
1
• There is a simple algorithm to calculate the gcd
of two integers – Euclidean Algorithm

Example of Euclidean
Algorithm
• Calculate the GCD of 1156 and 112
Divisor Dividend Quotient Remainder
112 1156 10 36
36 112 3 4
4 36 9 0
When you get a
zero remainder,
the remainder
before it is the
GCD

GCD of 1156 and 112
1156 = 22 × 172 = 2 × 2 × 17 × 17
112 = 24 × 71 = 2 × 2 × 2 × 2 × 7
1156
2 578
2 289
17 17
112
2 56
2 28
2 14
2 7

Example of Euclidean
Algorithm
• Calculate the GCD of 2428 and 60
Divisor Dividend Quotient Remainder
60 2428 40 28
28 60 2 4
4 28 7 0
When you get a
zero remainder,
the remainder
before it is the
GCD

Modular Arithmetic
• a = b mod (m) means that when a is divided by
m the remainder is b.
• Examples
• 11 = 1 mod (5)
• 20 = 2 mod (6)
• Modular Math and Prime Numbers
• Prime numbers possess various useful
properties when used in modular math.
• The RSA algorithm takes advantage of these
properties.

Modular Inverse
• Another aspect of modular math is the
concept of a modular inverse.
• Two numbers are the modular inverses of each
other if their product equals 1.
• For instance, 7 * 343 = 2401, but if our modulus
is 2400, the result is:
• (7 * 343) mod 2400 = 2401 – 2400 = 1 mod 2400

Euler’s phi-function
• In the eighteenth century, the mathematician
Leonhard Euler (pronounced “Oiler”)
described j(n) as the number of numbers less
than n that are relatively prime to n.
• The character j is the Greek letter "phi" (in
math circles it rhymes with “tea” in the
academic organization Phi Beta Kappa it
rhymes with “tie”).
• This is known as Euler’s phi-function.

Euler’s phi-function
• So ϕ(6), for instance, is 2, since of all the numbers
less than 6 (1, 2, 3, 4 and 5), only two of them (1 and
5) are relatively prime with 6. The numbers 2 and 4
share with 6 a common factor other than 1, namely
2. And 3 and 6 share 3 as a common factor.
• What about ϕ(7)? Because 7 is prime, its only
factors are 1 and 7. Hence, any number less than 7
can share with 7 only 1 as a common factor.
Without even examining those numbers less than 7,
we know they are all relatively prime with 7. Since
there are 6 numbers less than 7, ϕ(7) = 6. Clearly
this result will extend to all prime numbers.
Namely, if p is prime, ϕ(p) = (p - 1).

Exponentiation
• Exponentiation is taking numbers to powers,
such as 23, which is 2 * 2 * 2 = 8. In this
example, 2 is known as the base and 3 is the
exponent. There are some useful algebraic
identities in exponentiation.
• (bx) * (by) = bx+y
• (bx)y = bxy

Exponential Period
modulo n
• Euler noticed that ϕ(n) was the “exponential
period” modulo n for numbers relatively prime
with n.
• What that means is that for any number a < n, if
a is relatively prime with n, a ϕ(n) mod n = 1.
• So if you multiply a by itself ϕ(n) times, modulo
n, the result is 1. Then if you multiply by a one
more time, you are finding the product of 1 * a
which is a, so you are starting over again.
• Hence, a ϕ(n) *a = a ϕ(n)+1 mod n = a.

Exponential Period
modulo n
• For example, if n is 5 (a prime number), then
ϕ(5) = 4. Let a be 3 and compute
• a ϕ(n) mod n = 34 = 3 * 3 * 3 * 3 mod 5
• = 81 mod 5
• = 1 mod 5

Rivest-Shamir-Adleman
(RSA) Algorithm
• Developed in 1977 at MIT 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

RSA 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

RSA Example
• p = 3
• q = 11
• n = p × q = 33 -- This is the modulus
• z = (p-1) × (q -1) = 20 -- This is the totient function φ(n). There are 20 relative primes to 33.
What are they? 1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32
• d = 7 -- 7 and 20 have no common factors but 1
• 7e = 1 mod 20
• e = 3
• C = Me (mod n)
• M = Cd (mod n)

RSA Example
Plaintext (M) Ciphertext (C) After Decryption
Symbolic Numeric Me Me(mod n) Cd Cd (mod n) Symbolic
S 19 6859 28 13492928512 19 S
U 21 9261 21 1801088541 21 U
Z 26 17576 20 1280000000 26 Z
A 01 1 1 1 01 A
N 14 2744 5 78125 14 N
N 14 2744 5 78125 14 N
E 05 125 26 8031810176 05 E

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

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