1. Appendix 5: Cryptography p 640-644
Lewis Westfall
The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang

2. Types of Cryptographic Systems
Two common cryptographic Systems
Symmetric systems use the same key to encrypt and decrypt
Examples: DES and AES
Asymmetric Sytems use one key to encrypt and another, but related, key to decrypt
Examples: Diffie-Hellman-Merkle, RSA
Quantum Compuring

3. Symmetric Key Cryptography
Same key (k) used to encrypt and decrypt
f(m,k) = e, f(e,k) = m
Efficient
Key must be secret
Secure key distribution problem
Quantum Compuring

4. Asymmetric Key Cryptography
One key (k1) to encrypt
Another key (k2) to decrypt
f(m,K1) = e, f(e,K2) = m, f(e,k1) ≠ m
Public key and private key
Not as efficient as symmetric key cryptography
Public key is published. Anyone can know it.
Private key is secret.
Either key can be the private or the public key
Quantum Compuring

5. Asymmetric Key Cryptography
Alice wants to send a message to Bob
Alice encrypts the message using Bob’s public key
Alice sends the encrypted message to Bob
Bob decrypts the message using his private key
Bob wants to send a message to Alice
Bob encrypts the message using Alice’s public key
Bob sends the message to Alice
Alice decrypts the message with her private key
Quantum Compuring

6. Asymmetric Key Cryptography
Message/Signature authentication
Bob wants to send a message to Alice that she knows is from him
Bob calculates the message checksum
Bob encrypts the message with Alice’s public key
Bob encrypts the checksum with his private key
Bob sends the message and checksum to Alice
Quantum Compuring

7. Asymmetric Key Cryptography
Alice decrypts the message using her private key
Alice calculates the checksum
Alice decrypts the checksum using Bob’s public key
Alice compares the two checksums
If the checksums match, the message was from Bob
Quantum Compuring

8. RSA Cryptography Ronald Rivest, Adi Shamir, and Leonerd Adleman
Public Key and Private key are related mathematically
The relationship is very complex
The security is in the difficulty to resolve the relationship.
Quantum Compuring

9. RSA RSA uses the product of two prime numbers to create the keys
Each prime number can have 200 or more digits
The prime numbers are obtained
Picking a possible prime number at random
Primality test such as Miller-Rabin
The security is that factoring large numbers is hard
Quantum Compuring

10. Cracking RSA
The most efficient algorithm for factoring large number is the number field sieve
Quantum Fourier Transform
An exponential improvement
There are other points of attack
Quantum Compuring

11. Using Asymmetric Cryptography
Symmetric cryptography is efficient
Problem of secure key distribution
Asymmetric cryptography is less efficient
No key distribution problem
Use asymmetric cryptography to send the symmetric key
Quantum Compuring

12. RSA and Quantum Computing
RSA keys are hard to factor using classical computers
Quantum computers may be able to factor the product of two primes in a reasonable time frame
Quantum cryptography may create stronger cryptography
Quantum Compuring

13. References
Quantum Computation and Quantum Information by Nielsen & Chuang
Quantum Compuring