The adventures of Alice, Bob & Eve in the Quantumland

1. The adventures of Alice, Bob & Eve in the Quantumland Stefano Mancini University of Camerino, Italy

2. A typical scenario in secret communications Bob Alice Eavesdropper

3. A short dictionary • To exchange sensitive info people has always desired securecommunications • Legitimate users, sender, recipient (Alice & Bob) • Eavesdroppers, enemies, third parties (Eve) • Cryptology, Cryptography & Cryptoanalysis (steganography) • Message (plain-text), cryptogram (cipher-text) • Enciphering, deciphering, keys

4. Outline • Part 1: A survey of classical cryptology • Part 2: A quantum leap into the future: Quantum Key Distribution (QKD) • Part 3: QKD protocols and security • Part 4: Beyond QKD

5. Outline (part 1) • Ancient techniques • A historical overview • The history of “Enigma” • An unbreakable cipher • The public key cryptography

6. 400 BC SPARTA Scytale Permutation of characters

7. A T T A C K T O M O R R O W DWWD F N WR P R U U R Z Ceaser cipher 50 BC ROME ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC

8. Typical Techniques • PERMUTATIONS • e.g. Scytale (400 BC) • SUBSTITUTIONS • e.g. Caesar cipher (50 BC) • PERMUTATIONS +SUBSTITUTIONS

9. Origin of Cryptoanalysis On Deciphering Cryptographic Messages Baghdad, al-Kindi (800-873)

10. Typical attack Frequency of letters in a typical English text

11. Counterexamples-Lipograms That's right - this is a lipogram - a book, paragraph or similar thing in writing that fails to contain a symbol, particularly that symbol fifth in rank out of 26 (amidst 'd' and 'f') and which stands for a vocalic sound such as that in 'kiwi'. I won't bring it up right now, to avoid spoiling it… First lipogram: Lasus of Achaia (600 BC) The most famous lipogram: Georges Perec, La Disparition(1969), 85 000 words without the letter e Tout avait l'air normal, mais tout s'affirmait faux. Tout avait l'air normal, d'abord, puis surgissait l'inhumain, l'affolant. Il aurait voulu savoir où s'articulait l'association qui l'unissait au roman : sur son tapis, assaillant à tout instant son imagination, … English translator, Gilbert Adair, in A Void, succeeded in avoiding the letter e as well

12. From monoalphabethic cipher to polialphabetic cipher The Vigenère cipher (1586) Plain a b c d e f g hi j k l m n o p q r s t u v w x y z 1 B C D E F G H I J K L M N O P Q R S T U V W X Y Z A 2 C D E F G H I J K L M N O P Q R S T U V W X Y Z A B 3 D E F G H I J K L M N O P Q R S T U V W X Y Z A B C 4 E F G H I J K L M N O P Q R S T U V W X Y Z A B C D ……………. ……………. 26 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Babbage breaks the Vigenère cipher (1854)

13. Even more complex ciphers can be broken….. First computers were constructed to break ENIGMA codes …and the man who broke it! Scrambler’s initial setting represents the key ENIGMA…

14. Code-makers vs Code-breakers Cryptanalysis Cryptography Is there a perfect cipher?

15. Where the math comes into • Text Message (M) Integers (P) • Encryption C=E(P) • Decryption P=E -1(C) • Integers (P) Text Message (M)

16. One Time Pad (Vernam Cipher, 1917) • Alice and Bob possess a quantity of secret key material (random numbers) as large as the message to be transmitted • If Alice has P={p1,….,pn} • She uses K={k1,….,kn} • To get the cryptogram C={c1,….,cn}; ci=pi+ki (mod N) • Bob receives C, then he subtracts K (mod N) and recovers P One time pad is unbreakable provided the key material is truly random and used only once (Shannon). Drawbacks: trusted channel for key distribution, huge amount of key, generation of truly random numbers.

17. One-time pad plaintext KEY cryptogram cryptogram KEY plaintext

18. …key distribution problem ? Before they can exchange a secret they must already share a secret !

19. Possible solutions • Public key cryptosystems • mathematical, security based oncomputational complexity (use of one-way functions) • Can be broken by quantumcomputers! • Quantum cryptography • Physical, security based on fundamental principles of quantum mechanics

20. The symmetric cryptosystems Publicly agree the yellow color secret color secret color + = + = + = + = key key

21. The Diffie-Hellmann scheme (1976) Alice and Bob publicly agree the values for Y and N • Alice chooses a number A, and keeps it secret. • Alice calculates YA mod N= • Bob chooses a number B, and keeps it secret. • Bob calculates YB mod N= Alice and Bob publicly exchange the values for  and  • Alice calculates A mod N • Bob calculates B mod N The used function is one-way, so whereas it was easy for Alice, Bob to turn A, B into ,  (mod exp) it is very difficult for Eve to reverse the process (discrete log), especially for large numbers

22. Example Alice and Bob publicly agree the values Y=7 and N=11 • Alice chooses a number 3, and keeps it secret. • Alice calculates 73 mod 11=343 mod 11=2 • Bob chooses a number 6, and keeps it secret. • Bob calculates 76 mod 11=117649 mod 11=4 Alice and Bob publicly exchange the values for 2 and 4 • Alice calculates 43 mod 11=64 mod 11=9 • Bob calculates 26 mod 11=64 mod 11=9

23. The asymmetric cryptosystems Private key unlocks the box Public key locks the box

24. RSA cryptosystem (1978) • Pick up two large prime numbers p,q and compute the product n=pq • Notice that (p-1)(q-1)=(n) as p and q are primes • Pick up a random integer e [1<e<(n)] coprime with  Compute the inverse d of e mod  [e d=1 mod  ] (Euclid’s algorithm) • Create the public key with {e,n} and broadcast it • Create the private key with {d,n}

25. Represent the message to be transmitted as a sequence of integers {Pi} each in the range 1 to n • Encrypt each Pi using the public key: Ei = (Pi)e mod n • The receiver decrypts the message using the private key: (Ei)d mod n = Pi • Convert {Pi} back to the original message

27. Example

28. Breaking RSA • Eve needs to know {d,n}. If she could find twofactors of n, that is p,q, she easily compute(n) and knowing e could compute d. • Thus the security of RSA relies on the assumption that factoring large numbers is computationally hard. • In 1977 a challenge was made in Scientific American: break RSA-129. The time needed was estimated in 1016 years! Nevertheless, in 1994 factorization was achieved, but with a cluster of 103 workstations working for 8 months. • Shor’s algorithm would factor RSA-129 in few seconds running on a quantum computer at the speed of a desktop PC!

29. Crypto for fun: the “Bible Code” • In 1997 M. Drosnin claimed that the Bible contains hidden messages which could be discovered by searching for equidistant letter sequences (EDLS). According to this code the Bible contains predictions about the assasination of Kennedy, Sadat….. • B. McKay demonstrated the weakness of EDLS by finding in Moby Dick statements about the assassinations of Trotsky, Gandhi…..