Aki Nakao

1
Public-key Crypto-system

• Aki Nakao
• For Information Processing class at the
  University of Tokyo
• 2005 Summer

2
Shared-key Crypto-system

• Use the same key for encryption/decryption

decryption
encryption
Alice
Bob
Problem Hard to securely distribute a key
3
Public-key Crypto-system

• Use different keys for encryption/decryption

encryption (public key)
decryption (private key)
Alice
Bob

• They (Alice, Eve, and others) can only encrypt a
  secret
• Only I can decrypt the secret

4
Email and Digital Signature

• Email

Email
encryption (Bs public key)
decryption (Bs private key)
Alice
Bob

• Digital Signature

Signature (MD message digest)
Encrypted MD (a)
MD (a)
encryption (As private key)
decryption (As public key)
MD (b)
MD (a)
Alice
Bob
Bob compares the decrypted MD (a) and an MD (b)
generated from the received email
5
Message Digest

• Message Digest Small data crunched down from the
  data by a process called "hashing"
• It is not possible to change an MD back into the
  original data
• A slight change in the original data results in a
  significant change in its MD

Hey guys, I have really a cool idea. Lets start
up a company on Monday. My idea is blah .
hashing
HNFmsEm6Un BejhhyCGKOK
e.g. MD5
Signature
Email
6
Man-In-the-Middle Attack

• Eve the eavesdropper in the middle

Bobs public-key
Eves public-key
Bobs secret-key
Eves secret-key
Alice
Bob
Eve the eavesdropper

• Eve intercepts Bobs public key and pass her
  public key to Alice.
• Alice encrypts her message with Eves public
  key, thinking that its Bobs.
• Eve decrypts Alices message and encrypts it with
  Bobs public key.
• A fingerprint (a hash value generated from a
  public key) defeats this attack

Bobs public-key
2628 487D F786 29C4 A368
(fingerprint)
Hash
7
RSA
Pick (d, e) for given prime numbers (p, q)
ed 1 mod n
gcd(e,n)1,
n pq ,
n (p-1)(q-1),
encryption c me mod n
public-key (e, n)
decryption c cd mod n m mod n
private-key (d, n)
xp-1 mod p 1 (for all x lt p) ?Fermats Little
Theorem
x mq-1 mod p ? mn mod p 1

mn mod n 1
mn mod q 1
cd mod n (me mod n)d mod n
med mod n mkn1 mod n m mod n
8
Fermats Little Theorem
If p is a prime number, xp-1 mod p 1 , for a
natural numer x lt p
Suppose pxp-x (i.e., p divides xp-x),
(x1)p xp pC1 xp-1 pCp-1 x1
Binomial theorem
(x1)p -(x1) xp -xpC1 xp-1 pCp-1 x
pCi p(p-1)(p-i1)/i ! and p is a prime number,
so ppCi (i lt p)
Therefore, p (x1)p -(x1),
and by induction, pxp-x.
When gcd(x,p)1, xkpm1 for some k, m ?Euclids
k(xp-x) kx(xp-1-1) (1-pm) (xp-1-1)
Since pk(xp-x), pxp-1-1 follows.