James C. Bradas, Ph.D. Engineering & Analysis Operation 18 June 2009 Public Key Encryption.

James C. Bradas, Ph.D.

Engineering & Analysis OperationEngineering & Analysis Operation

18 June 2009

Public Key Encryption

A Bit of History

• In mid 1970s, the primary means of encrypting data was via “single-key encryption”. This uses a single mathematical formula or key to encrypt data or messages and requires the sender and receiver to have the key. If you want to share information with multiple users, all have to have the key. The more users, the less secure the key becomes.

• Whitfield Diffie & Martin Hellman at Stanford University proposed a solution in 1976 - Asymmetric “Public Key Encryption”.

• This system was perfected by three MIT students in 1977 – Ronald Rivest, Adi Shamir & Leonard Adleman. Their system has come to be known as RSA Public-Key Cryptography and is now the standard method of data encryption used today.

• An interesting article on the history of RSA encryption is contained at the website: http://www.fundinguniverse.com/company-histories/RSA-Security- Inc-Company-History.html

• RSA Encryption makes use of the properties of Prime Numbers to develop a unique and “break-proof” key for each sender.

http://www.fundinguniverse.com/company-histories/RSA-Security-

ALL Integers Can Be Built From Prime Numbers

.

9333

82222

77

632

55

4222

33

22

2

3

2

etc

Prime Numbers are the “Elementary Particles”

of Mathematics

What Are Prime Numbers?

Prime Numbers are numbers divisible only by themselves and 1

The Fundamental Theorem of Arithmetic

ANY integer greater than 1 can be written as a UNIQUE product of prime numbers

864

32 27

4 8 3 9

2 32 2 2 2 3 3

35 32864

Integer Factorization of 864

A semi-prime number is the product of two (not necessarily distinct)prime numbers (not raised to a power greater than 1)

As of September 2008, the largest known semi-prime is:

2609,113,43 12 (has over 25 million digits)

This number is the square of the largest prime number discovered so far

White Light

SpectrometerColors of the Spectrum

Semi-Prime NumberPrime Number 1

Prime Number 2

There is no “Integer Factorization Spectrometer”

Cryptography, and in particular, RSA Public Key Encryption, makes useof the fact that it is EXTREMELY EASY to multiply two numbers togetherand EXTREMELY DIFFICULT to factor a semi-prime number into the product of prime numbers.

Factoring Semi-Prime Numbers is particularly challenging because there areonly TWO unique factors

RSA Uses Prime Numbers

RSA Encryption Scheme

I choose 2 extremely large prime numbers, p & q*.

This number, combined with modular arithmetic and the RSA scheme, provides secure data encryption.

If I use 128-bit encryption, then I choose a prime number between 2 &

38128 103.4 ,........938,920,366,282,34012

How many prime numbers are there in that range?

qpn

Gauss’ formula for estimatingthe number of prime numbers in

2 - N integers

How Large Are These Two Prime Numbers?

)ln(N

NN p

* In reality, much goes into the selection of the prime numbers to be used. They should not be tooclose together (p-q should not be less than 2n ). If p-1 or q-1 have only small prime factors, n canbe factored quickly by Pollard’s p-1 algorithm. There is some art to this.

1/4

RSA Uses Prime Numbers

RSA Encryption Scheme

I choose 2 extremely large prime numbers, p & q.

This number, combined with modular arithmetic and the RSA scheme, provides secure data encryption.

If I use 128-bit encryption, then I choose a prime number between 1 &

38128 103.4 ,........938,920,366,282,34012

36

128

128

108.32ln

2pN

How many prime numbers are there in that range?

qpn

How Large Are These Two Prime Numbers?

36

128

128

108.32ln

2pN

In order to determine p & q, I might use trial & error. In other words, pick a prime number p & divide it into n to see if I get the other prime number q.

To get an idea of the size of the problem, suppose I use trial & error.

Divide n by each prime number between 2 and

Approximate number of prime numbers using 128 bits

12101

12128

If my computer operates at Teraflop speed, I can do divisions per second

There are primes to try. 3610

12

36

10

10 Number of primes to try

Number of tries per secondsecs 1024

=

Size of the Problem

Since the universe is only seconds old, it will take a while!1810

What About Some Sort of a Lookup Table?

With 128-bit numbers, there are ~ prime numbers. If I take 2 at atime, there are possible combinations. The number ofelementary particles in the visible universe is around . There isn’tenough memory capacity to store all of the prime numbers between 1and , much less the number of possible combinations available.

Because of the number of primes, “table lookup” cannot be done. Theencryption algorithm must generate prime numbers “on the fly”.

3610723636 101010

8410

mli

kji

ppn

ppn

1

List of semi-prime integers

X

1st Prime 2nd Prime

3610

So What About Faster Computers?

Top500 – 2x per year – the 500 fastest known computer systems

The Top Three (As of June 2009)

1

2

3

1105 TFLOPs ( 1.105 PFLOPs )

1059 TFLOPs

825 TFLOPs ( )

IBM Roadrunner System @ LANL

Cray XTS Jaguar @ DOEs ORNL

IBM BlueGene/P @ FZJ in Germany12101

Even with Zettaflop performance, time is still seconds or about1 billion years.

1510

YottaFLOPS 10

taFLOPS Zet10

ExaFLOPS 10

PetaFLOPS 10

TeraFLOPS 10

GigaFLOPS 10

24

21

18

15

12

9

Today

# Floating Point Operations Per

Second Abbreviation

2007

~2008

So What About Faster Computers?

Top500 – 2x per year – the 500 fastest known computer systems

The Top Three (As of June 2009)

1

2

3

1105 TFLOPs ( 1.105 PFLOPs )

1059 TFLOPs

825 TFLOPs ( )

IBM Roadrunner System @ LANL

Cray XTS Jaguar @ DOEs ORNL

IBM BlueGene/P @ FZJ in Germany

by 2019?

12101

Even with Zettaflop performance, time is still seconds or about1 billion years.

1510

YottaFLOPS 10

taFLOPS Zet10

ExaFLOPS 10

PetaFLOPS 10

TeraFLOPS 10

GigaFLOPS 10

24

21

18

15

12

9

(That’s a Lotta FLOPS!)

Today

# Floating Point Operations Per

Second Abbreviation

by 2030?

Of course, no one actually tries every prime number by brute force.

There are algorithms that make it possible to make a reasonable “guess”based on the value of the number to be factored. However, given thesheer number of prime numbers, the task of finding the correct two primenumbers is still quite formidable.

A number of semi-prime numbers published by RSA in 1991 (54) have beensuccessfully factored (12) by networked computers. However, it took months to factor just one semi-prime number.

There is still no computer system/algorithm that can factor semi-primenumbers quickly (days – hours – minutes) which would put RSA encryptionat risk.

So Now That We’ve Convinced Ourselves About the Viability of Prime Numbers,

Let’s Look At How RSA Works

First, we need some basic modular arithmetic

)(modnba Two numbers a & b are congruent modulo nif their difference ( a – b ) is an integer multipleof n (the integer can be positive or negative).

knba We can write this as:

7

73

21

3

425

)3(mod425

k

So, for example:

)2(mod24

)7(mod122

)4(mod133

Other Examples

1

3

8

k

k

k

“a” divided by “n” is equal to some integer x “n” + “b”. “b” is the remainder

orn

bak

Note: the sign is NOT the same thing as an “=“ sign.

In our example, replace “25” by “x”. What values of x make thestatement true?

433

4

)3(mod4

kx

kx

x

onsoand

xk

xk

xk

xk

xk

if

13 then 3

10 then 2

7 then 1

4 then 0

1 then 1

:

onsoand

)3(mod413

)3(mod410

)3(mod47

)3(mod44

)3(mod41

So, the following are all true:

For RSA Encryption, we will be performing the following calculation:

)(mod

determine ,

nyx

thatsuchxnandyGiven

I claim that and y are congruent modulo n

n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven


kn

yx

means

nyx

)(mod

:Proof

n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyx

)(mod


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyx

)(mod


n

yINTnyx

insert expression for x


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyn

yINTny

)(mod


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyn

yINTny

)(mod


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyn

yINTny

)(mod


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyn

yINTny

)(mod


n

yINTnyx


)(mod

determine ,

nyx

thatsuchxnandyGiven

kn

yx

means

nyx

)(mod

:Proof

kn

yINT

kn

yny

INTny

nyn

yINTny

)(mod


n

yINTnyx

k is an integer

)(modnyx

n

yINTnyx

To solve for x given the expression

This means, divide y by n and find the remainder. x is the remainder.

Example: I want to solve for x given:

Bottom Line….

. 1 ,

1 1 3 4

)3(mod4

solutionaisxSo

remainderisbydivided

x

Proposed solution:

Alice Bob

Public Key

Private Key

Meet Alice & BobAlice & Bob want to send secure

messages to one another without fearof their messages being read

by a third party

Using RSA PKE, each person creates a “Public Key” and a “Private Key”

Their Public Key is sent to any party that wishes to send them a message

Their Private Key is kept to themselves

Alice

Private Key

Public Key

RSA Public Key Encryption SchemeAlice Creates Her Public & Private Key

1. Select two large prime numbers, p & q

)1)(1( qp

2. Compute their product – the “modulus” n:

3. Compute Euler’s “totient”

4. Choose e, 1 < e < such that the greatest common divisor (gcd) ( e , = 1

e is the “public key exponent”

( Common choices are e = 3, 17 & 65537 )

5. Compute d such that )(mod1 ed

( n , e )Public Key

( n , d )Private Key

d is the “private key exponent”

Alice)1)(1( qp

)(mod1 ed

( n , e )Public Key


)(11 qpnqppq

So Where’s the Secret?

qpn

Because Alice doesn’t publish the values of p & q –

an eavesdropper can’t determine and thus can’t determine the correct value of d which is needed for message decryption

The public knows this

The public DOESN’T know this

Alice Bob

Bob wishes to send a message to Alice

( n , e )Public Key


“M”

Alice Bob

Alice sends her public key to Bob

and keeps her private key secret


( n , e )Public Key


“M”

Alice Bob




( n , e )Public Key


( n , e )Public Key

“M”

Alice Bob




Bob first turns his message M into an integer m (or series of integers) where 0 < m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the cipher text c (for each integer in his message) via the following:

)(modnmc e

( n , e )Public Key


( n , e )Public Key

“M”

Alice Bob




Bob first turns his message M into an integer m (or series of integers) where 0 < m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the cipher text c (for each character in his message) via the following:

)(modnmc eBob sends each “c” in the now-encrypted message to Alice.

( n , e )Public Key


( n , e )Public Key

“M”

Alice Bob

Alice Receives Bob’s Encrypted Message

( n , e )Public Key


“c”?

Alice Bob

Alice uses her Private Key and reverse padding schemeto decrypt Bob’s message


( n , e )Public Key


“c”?

Alice Bob



( n , e )Public Key


)(modncm d

“c”?

Alice Bob



( n , e )Public Key


)(modncm d

Reverse Padding SchemeReverse Padding Scheme

m

“c”?

Alice Bob



( n , e )Public Key


)(modncm d

Reverse Padding SchemeReverse Padding Scheme

m

MAlice Recovers Bob’s Original Message “M”

“M”

Reverse Padding Scheme

Alice Bob

Sending Information from “B” to “A” – “Bob” and “Alice”Via RSA Public Key Encryption

EncryptedMessage

Bob’s Message

)(modnmc e

M m

M

Alice’s Public Key

)(modncm d M M

Bob’s Message is Recovered

Private Key

Public Key

Public Key

In Summary

( n , d )Alice’s Private Key

James C. Bradas, Ph.D. Engineering & Analysis Operation 18 June 2009 Public Key Encryption.

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