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

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

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

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

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

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

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

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

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

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

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

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

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