RSA Numbersby: Brandon Hacay & Conrad Allen

History of RSA Numbers• The letters in “RSA” are simply the initials of the people

who are credited as having developed the concept

o Ron Rivest, Adi Shamir and

Leonard Adleman at MIT

o First published and made publicly

available in 1977

History of RSA Numbers• Ronald Rivest

o Created MD5 hash function, as well as MD2, MD4, MD6, RC2, RC4, RC5, RC6

• Avi Shamiro Co-creator of differential cryptanalysis

• Leonard Adlemano Created DNA/biomolecular computing

History of RSA Numbers• A man named Clifford Cocks also

described the same system a few years earlier in 1973

o Developed in the U.K. for the GCHQ, it would have

required computing power deemed to expensive at

the time, so it was never actually implemented or testedo Due to its top-secret nature, Cocks’ papers and ideas

were not made publicly available until 1998

Basics of Encryption Keys• RSA is an example of public-key, or

asymmetric cryptologyo There is a public key (the encryption key) and a

private key (the decryption key)o The encryption key is publicly available and used to

create the encrypted message to be transmitted, while a privately-held (but mathematically related) decryption key is used to decipher it

Public Key Creation• Find the product of two distinct prime

numbers.o n = p*qo n is used as the modulus for both the public and

private keys.

• Solve Euler’s totient functiono φ(n) = (p-1)(q-1)

Public Key Creation• Choose an integer e such that:

o 1 < e < φ(n)o e and φ(n) must be coprime.

• e is used as the public key exponent to encode messages.

Encoding and Decoding• To encode a message M solve:

o C = Memod(n)o M < n

• Solve for the decoding exponent: o d = e-1mod(φ(n))

• To decode a message M solve:o M = Cdmod(n)

Example● User Y selects p and q.

○ p = 23, q = 41

● n = p*q = (23)*(41) = 943

● φ(n) = (p-1)(q-1) = (22)*(40) = 880

● e and φ(n) must be coprime and 1 < e < φ(n)

○ e = 7

Example● n and e are the public key so User

X know their values.○ n = 943, e = 7

● User X wants to send a message to User Y.

○ M = 35

● C = Memod(n) = 357mod(943)C = 545

● The encoded message 545 is sent to User Y.

Example● d = e-1mod(φ(n))

d= 7-1mod(880) = 503

● M = Cdmod(n) = 545503mod(943)M = 35

Exponent Algorithm• RSA can use very large

exponents.

• M = Cdmod(n) = 545503mod(943)

• Running time = O(e)

• total operation in example:o 503 operations

Repeated Squaring Algorithm• Using this algorithm you get:

545503mod943 = (545*[(545*545)mod943]251)mod943

= (545*923251)mod943

= (545*(923*[(923*923)mod943]150)mod943))mod943

= (545*[(923*400150)mod943]mod943)

• Running time = O(log2(e))

• Total operations for example:o 9 operations

Potential Risks• As with any encryption system, the private

key used to decrypt the message can still be vulnerable to social engineering or careless storage of the private key information

• Vulnerable with small exponent (“e”) values and small message values (“m”) for m^e

Potential Risks• Vulnerable if the same clear text message is

sent to “e” or more people with different “N” values (“Chinese Remainder Theorem”)

• Vulnerable if not padded since RSA is not “semantically secure”o Attacker can guess at the potential messages being sent, encrypt it

using RSA and the public key, and compare the encrypted messages if the message isn’t padded first

The “RSA Problem”• Can a message encrypted using RSA be

efficiently decrypted while only knowing the public key? (n, e)

• RSA Foundation has created the RSA Factoring Challenge to spur research into crackingRSA and integer factorization

RSA Factoring Challenge• The problem: you are given a number “n”

that is the product of two prime numbers, “p” and “q”. Find these factors.

• Some cash rewards reached tens of thousands of dollars

• Largest potential reward was $100,000

RSA Factoring Challenge• Example of a RSA number and its factors:

• Amount of computing needed was the equivalent of 75 years of computing on a 2.2GHz single-core processor

RSA Factoring Challenge• Largest RSA number in the challenge is RSA-2048,

which is not expected to be solved anytime soon without significant advances in integer factorization

Questions?