RSA ALGORITHM

THE RSA ALGORITHM

BY,
SHASHANK SHETTY
ARUN DEVADIGA

INTRODUCTION
 By Rivest, Shamir & Adleman of MIT in 1977.
 Best known & widely used public-key scheme.
uses large integers (eg. 1024 bits)
 Based on exponentiation in a finite field over integers
modulo a prime

Plaintext is encrypted in blocks, with each block
having the binary value less than some number n.
Security due to cost of factoring large numbers.

1) 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. To find random primes, we start at a random number and
go up ascending odd numbers until we find a prime. Lets have:
p=7
q = 19
2) Let n = pq
n = 7 * 19
= 133
3) Let m = (p - 1)(q - 1)
m = (7 - 1)(19 - 1)
= 6 * 18
= 108

4) Choose a small number, e coprime to m (
 e coprime to m, means that the largest number that can exactly
divide both e and m (their greatest common divisor, or gcd) is 1.
Euclid's algorithm is used to find the GCD of two numbers.

e = 2 => GCD(108,e) = 2 (no)
e = 3 => GCD(108,e) = 3 (no)
e = 4 => GCD(108,e) = 4 (no)
e = 5 => GCD(108,e) = 1 (yes!)

5) Find d, such that {de mod ɸ(n) = 1}
This is equivalent to finding d which satisfies de = 1 + km where k is
any integer. We can rewrite this as d = (1 + km) / e. Now we work
through values of k until an integer solution for e is found:
k = 0 => d = 1 / 5 (no)
k = 1 => d = 109 / 5 (no)
k = 2 => d = 217 / 5 (no)
k = 3 => d = 325 / 5
= 65 (yes!)
To do this with big numbers, a more sophisticated algorithm called
extended Euclid must be used.

Public Key

Secret Key

n = 133
e=5

m =108

d = 65

Communication
6) Encryption
The message must be a number less than the smaller of p and q. However, at
this point we don't know p or q, so in practice a lower bound on p and q must
be published. This can be somewhat below their true value and so isn't a major
security concern. For this example, lets use the message "6".
Cipher = (message)e mod n
= 65 mod 133
= 7776 mod 133
= 62

7) Decryption
This works very much like encryption, but involves a larger
exponentiation, which is broken down into several steps.
message = (cipher)d mod n
= 6265 mod 133
= 62 * 6264 mod 133
= 62 * (622)32 mod133
= 62 * 384432 mod 133
= 62 * (3844 mod133)32 mod 133
= 62 * 12032 mod 133
We now repeat the sequence of operations that reduced 6265 to 12032 to
reduce the exponent down to 1.
= 62 * 3616 mod 133
= 62 * 998 mod 133
= 62 * 924 mod 133
= 62 * 852 mod 133
= 62 * 43 mod 133
= 2666 mod 133 = 6

Fermat's Theorem
ap-1 mod p = 1
where p is prime and gcd(a,p)=1

also known as Fermat’s Little Theorem
useful in public key and primality testing

EULER’S THEOREM & TOTIENT FUNCTION Ø(n)

•

Euler’s totient function (ɸ(n)), defined as the number of
positive integers less than n and relatively prime to n.

– for p (p prime)
– for p.q (p,q prime)

ø(p) = p-1
ø(p.q) = (p-1)(q-1)

• For Example: DETERMINE ɸ (37) and ɸ(35).
• Because 37 is prime, all of the positive integers from 1
through 36 are relatively prime to 37.Thus ɸ (37)=36.
• To determine ɸ(35), we list all of the positive integers less
than 35 that are relatively prime to it:
1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18
19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34
• There are 24 numbers on the list, so ɸ(35) = 24.

EULER’S THEOREM & TOTIENT FUNCTION Ø(n) (cont…)
(Generalization of Fermat’s Little Theorem)
In a more general sense what is (p-1)? Ø(p).
Ø(p) = Number of integers a < p such that (a, p) = 1.
This is obviously (p-1) because p is prime.
So we can say
akø(p)+1 ≡ a (mod p)
More generally, let n = p • q p prime, q prime.
akø(n)+1 ≡ a (mod n)
p = 3, q = 5, n = 15, Ø(n) = (3-1) • (5-1) = 8
a = 7:
78+1 = 7 (mod 15)
78k+1 = 7 (mod 15)
a = 5:
58+1 = 5 (mod 15)
58k+1 = 5 (mod 15)

CARMICHAEL’S THEOREM
( A Refinement of Euler’s Theorem)

 Is also called as reduced totient function.
Ø(n) = (p-1) • (q-1)
λ(n) = LCM{(p-1), (q-1)}
akλ(n)+1 ≡ a (mod n)
p = 3, q = 5, n = 15, Ø(n) = (3-1) • (5-1) = 8
λ(n) = LCM{(p-1), (q-1)} = LCM{2, 4} = 4
a = 7:
74+1 = 7 (mod 15)
74k+1 = 7 (mod 15)
a = 5:
54+1 = 5 (mod 15)
54k+1 = 5 (mod 15)

RSA EXAMPLE 2
Choose p = 19, q = 37, n = 703
Ø(n) = 648,
λ(n) = 36,
GCD{(p-1),(q-1)} = 18
Choose e = 5
CARMICHAEL d = 29
EULER d = 389
Both d’s will work, but Carmichael gives a much simpler d; in
this case security is reduced.

Factoring Problem
 Mathematical approach takes 3 forms:
– factor N=p.q, hence find ø(N) and then .
– determine ø(N) directly and find d
– find d directly

FERMAT FACTORIZATION
 Fermat factorization attempts to factor a number by
representing it as the difference of two squares.
• Proposition: Let n be an odd integer. There is a one-to-one
correspondence .

• Proof: If n=ab, n is odd so a and b are odd. Then a+b and a-b
are even, so (a+b)/2 and (a-b)/2 are integers. Now

expresses n as a difference of two squares. Conversely,
suppose n is written as as difference of squares:
Then n=(s+t) (s-t) is a factorization of n.

QUADRATIC SIEVE FACTORIZATION
Our goal is to find a nontrivial factorization of n:
• Consider the value n=1649, a composite number but not
divisible by any prime up to its logarithm.

and so on, with no squares in immediate sight.
• Note that while neither 32 nor 200 is a square, their
product is a square: 6400 = 802. Thus, since

•

we have

that is,

QUADRATIC SIEVE FACTORIZATION (cont…)

We have found a solution,

So, GCD (a+b, n) and GCD (a-b, n) must find the non
trivial factors of n.
 Hence the,
GCD( 114+80,1649)
GCD (194,1649)
GCD (194, 97)
GCD(97,0)
=97

and
and
and
and
and

GCD(114-80, 1649 )
GCD(34,1649)
GCD(34,17)
GCD(17,0)
=17

 So, 17 and 97 are the non trivial factor of 1649.

RSA Challenges:
•
•
•
•
•

RSA - 640 November 2, 2005
RSA $200,000 Challenge
RSA - DES Challenge
RSA - 576 Challenge
Cracking RSA
Ref: On the Cost of Factoring RSA-1024
Adi Shamir and Eran Tromer

What are RSA factoring Challenges???
 The RSA Factoring Challenge was a challenge put
forward by RSA Laboratories on March 18, 1991.
 to encourage research into computational number theory
and the practical difficulty of factoring large integers
and cracking RSA keys used in cryptography
 They published a list of semiprimes (numbers with
exactly two prime factors) known as the RSA numbers,
with a cash prize for the successful factorization of some
of them

RSA - 640 November 2, 2005
The factoring research team of F. Bahr, M. Boehm, J. Franke, T. Kleinjung
continued its productivity with a successful factorization of the challenge
number RSA-640, reported on November 2, 2005. The factors [verified by
RSA Laboratories] are:
16347336458092538484431338838650908598417836700330
92312181110852389333100104508151212118167511579
And
1900871281664822113126851573935413975471896789968
515493666638539088027103802104498957191261465571
The effort took approximately 30 2.2GHz-Opteron-CPU years according to
the submitters, over five months of calendar time. (This is about half the
effort for RSA-200, the 663-bit number that the team factored in 2004.)

Ref: RSA Laboratories - RSA-640 is factored!

RSA $200,000 Challenge
 RSA Security is running a factoring challenge that offers would-be code breakers a
prize of up to $200,000 for finding the two numbers of the kind used to create ultrasecure 2048-bit encryption key.
RSA-2048
Status: Not Factored
Decimal Digits: 617
25195908475657893494027183240048398571429282126204
03202777713783604366202070759555626401852588078440
69182906412495150821892985591491761845028084891200
72844992687392807287776735971418347270261896375014
97182469116507761337985909570009733045974880842840
17974291006424586918171951187461215151726546322822
16869987549182422433637259085141865462043576798423
38718477444792073993423658482382428119816381501067
48104516603773060562016196762561338441436038339044
14952634432190114657544454178424020924616515723350
77870774981712577246796292638635637328991215483143
81678998850404453640235273819513786365643912120103
97122822120720357
Decimal Digit Sum: 2738

RSA - DES Challenge
Identifier: DES-Challenge-III
Cipher: DES
Start: January 18, 1999 9:00 AM PST
Prize: $10,000

Plaintext: See you in Rome (second AES Conference, March 22-23, 1999)
Ciphertext:
bd 0d de 91 99 60 b8 8a 47 9c b1 5c 23 7b 81 18 99 05
45 bc de 82 01 ab 53 4d 6f 1c b4 30 63 3c ee cd 96 2e
07 c6 e6 95 99 9c 96 46 5a 95 70 02 02 70 98 bd 41 c2
88 a9 f0 2f 8b e5 48 20 d2 a8 a0 6b bf 93 de 89 f6 e2
52 fd 8a 25 eb d0 7d 96 83 ee a4 2d c8 8d 1b 71

REF: RSA Laboratories - DES Challenge III

RSA - 576 Challenge
 On December 3, 2003, a team of researchers in Germany and
several other countries reported a successful factorization of
the challenge number RSA-576. According to the
announcement by J. Franke:
 The factors [verified by RSA Laboratories] are
3980750864240649373971255005503864911990643623425267
08406385189575946388957261768583317
and
4727721461074353025362230719730482246329146953020971
16459852171130520711256363590397527

 Lattice sieving was done by J. Franke and T. Kleinjung
using Hardware of the Scientific Computing Institute and
the Pure Mathematics
REF: RSA Laboratories - RSA-576 is factored!

RSA EXAMPLE 3
Encryption

Decryption
Ciphertext

Plaintext

11
7

88 mod 187 = 11

11

23

Plaintext
mod 187 = 88

88

88

KU = 7, 187

KR = 23, 187

Figure 1. Example of RSA Algorithm

RSA Security
Three approaches to attacking RSA:
brute force key search (infeasible given size
of numbers)

mathematical attacks (based on difficulty of
computing ø(N), by factoring modulus N)
timing attacks (on running of decryption)

Timing Attacks
• developed in mid-1990’s
• exploit timing variations in operations
– eg. multiplying by small vs large number
– or IF's varying which instructions executed
• infer operand size based on time taken
• RSA exploits time taken in exponentiation
• countermeasures
– use constant exponentiation time
– add random delays
– blind values used in calculations

References
[1] William Stallings, “The cryptography and network
security”.
[2] RSA Laboratories, http://www.emc.com/emcplus/rsa-labs/historical/the-rsa-factoring-challengefaq.htm.
[3] Dr. Herong Yang, “Cryptography Tutorials - Herong's
Tutorial Example”,
http://www.herongyang.com/Cryptography/.

RSA ALGORITHM

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