The RSA Algorithm
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The RSA Algorithm
Public Key Cryptography● Diffie-Hellman is the first use of asymmetric key.
Public Key Cryptography● Diffie-Hellman is the first use of asymmetric key.● It has some impracticalities:
Public Key Cryptography● Diffie-Hellman is the first use of asymmetric key.● It has some impracticalities:
○ Setup phase: Alice has to exchange public information with each Bob.
Public Key Cryptography● Diffie-Hellman is the first use of asymmetric key.● It has some impracticalities:
○ Setup phase: Alice has to exchange public information with each Bob.
● These impracticalities are avoided by the Rivest-Shamir-Adleman (RSA) algorithm.
Public Key Model● The key is broken into a public and private part.
Public Key Model● The key is broken into a public and private part.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys -- for all.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys -- for all.● Alice encrypts “hello” using Bob’s public key.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys -- for all.● Alice encrypts “hello” using Bob’s public key.● Alice sends the encrypted “g#bzr&*h” to Bob.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys -- for all.● Alice encrypts “hello” using Bob’s public key.● Alice sends the encrypted “g#bzr&*h” to Bob.● Bob decrypts with his private key.
Public Key Model● The key is broken into a public and private part.● Bob and Alice publish their public keys -- for all.● Alice encrypts “hello” using Bob’s public key.● Alice sends the encrypted “g#bzr&*h” to Bob.● Bob decrypts with his private key.● Bob reads “hello”.
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)e: public exponent
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)e: public exponentN: public modulus
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)e: public exponentN: public modulusc: encrypted message
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)e: public exponentN: public modulusc: encrypted message
Easy: Compute me mod N
Rivest Shamir Adleman (RSA)● Based on a one-way function -- i.e., a function that
is easy in one direction and hard in the other.
me mod N → c
m: message (a number)e: public exponentN: public modulusc: encrypted message
Easy: Compute me mod N Hard: Find m given (e,N)
Rivest Shamir Adleman (RSA)● Basic structure of RSA
med mod N → c
publicmodulus
privatekeypublic
key
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N○ E.g: N= 20 20 = 2 x 2 x 5
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N○ E.g: N= 20 20 = 2 x 2 x 5○ E.g: N= 45 45 = 3 x 3 x 5
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N○ E.g: N= 20 20 = 2 x 2 x 5○ E.g: N= 45 45 = 3 x 3 x 5
● Easy for small values of N
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N○ E.g: N= 20 20 = 2 x 2 x 5○ E.g: N= 45 45 = 3 x 3 x 5
● Easy for small values of N● Intractable for large values of N
Rivest Shamir Adleman (RSA)● Prime factorization problem:
○ Find the prime factors of N○ E.g: N= 20 20 = 2 x 2 x 5○ E.g: N= 45 45 = 3 x 3 x 5
● Easy for small values of N● Intractable for large values of N● RSA uses very large numbers for N : > 300 digits
RSA Summary
RSA Summary● Solves the key-exchange problem.
RSA Summary● Solves the key-exchange problem.
● Public keys can be published and distributed.
RSA Summary● Solves the key-exchange problem.
● Public keys can be published and distributed.
● Secured by the intractability of the prime factorization problem -- i.e., the problem of discovering the prime factors of a large number.
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