1. Cryptography & Number Theory
COMP4690 Tutorial
Cryptography
&
Number Theory

2. Outline
DES Example
Number Theory
RSA Example
Diffie-Hellman Example

3. DES Some remarks DES works on bits
DES works by encrypting groups of 64 bits, which is the same as 16 hexadecimal numbers
DES uses keys which are also apparently 64 bits long. However, every 8th key bit is ignored in the DES algorithm, so the effective key size is 56 bits.
If the length of the message to be encrypted is not a multiple of 64 bits, it must be padded. E.g.:
The plaintext message "Your lips are smoother than vaseline" is, in hexadecimal, "596F C D 6F6F E C696E650D0A".
We then pad this message with some 0s on the end, to get a total of 80 hexadecimal digits: "596F C D 6F6F E C696E650D0A0000".
Then apply DES.

4. Key generation example
Let K be the hexadecimal key K = BBCDFF1. This gives us as the binary key :
K =
16 subkeys (48-bit) will be generated from K.

5. Key generation example
Based on table PC-1 (Permuted Choice 1), we get the 56-bit permutation
K+ =
Next, split this key into left and right halves, C0 and D0, where each half has 28 bits.
C0 = D0 =

6. Key generation example
we now create sixteen blocks Cn and Dn, 1<=n<=16. Each pair of blocks Cn and Dn is formed from the previous pair Cn-1 and Dn-1, respectively, for n = 1, 2, ..., 16, using a “schedule of left shifts".
Round number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Bits rotated

7. Key generation example
C0 = D0 =
C1 = D1 =
C2 = D2 =
C3 = D3 = ……

8. Key generation example
We now form the subkeys Kn, for 1<=n<=16, by applying the table PC-2 (Permutation Choice Two) to each of the concatenated pairs CnDn.
For the first subkey, we have
C1D1 =
After we apply the permutation PC-2:
K1 =

9. Modular Arithmetic
Two integers a and b are said to be congruent modulo n, if :
(a mod n) = (b mod n)
This is written as a≡b mod n
Define Zn as the set of nonnegative integers less than n: Zn={0,1,…,(n-1)}

10. Modular Arithmetic
Properties of modular arithmetic

11. Modular Arithmetic
Define Zp as the set of nonnegative integers less than a given prime number p: Zp={0,1,…,(p-1)}
Because p is prime, all of the nonzero integers in Zp are relatively prime to p.
There exists a multiplicative inverse for all of the nonzero integers in Zp :
For each nonzero w in Zp, there exists a z in Zp such that w x z ≡ 1 mod p. z is called the multiplicative inverse of w. Or, z = w-1.

12. Number Theory Fermat’s Little Theorem: ap-1 ≡ 1 mod p E.g.
where p is prime and gcd(a,p)=1
E.g.
a = 7, p = 19
72=49≡11 mod 19
74≡121≡7 mod 19
78≡49≡11 mod 19
716≡121≡7 mod 19
ap-1=718=716x72≡7x11=77≡1 mod 19

13. Number Theory An alternative form of Fermat’s Little Theorem:
ap ≡ a mod p
where p is prime and a is any positive integer
E.g.
p=5,a=3,35=243≡3 mod 5
p=5,a=10,105=100000≡10 mod 5≡0

14. Number Theory Euler’s Totient Function ø(n) For prime number p,
The number of positive integers less than n and relatively prime to n
For prime number p,
ø(n)= p – 1
For n = pq where p and q are two different prime numbers
ø(n)= (p – 1) (q – 1)

15. Number Theory Example: ø(21) From 1 to 21, totally 21 numbers
21 = 3x7, 3 and 7 are prime
3’s multiples:
3, 6, 9, 12, 15, 18, 21
7’s multiples:
7, 14, 21
Other numbers are all relatively prime to 21
= (3-1)x(7-1)

16. Number Theory Euler’s Theorem aø(n) ≡ 1 mod n E.g. where gcd(a,n)=1
hence 34 = 81 ≡ 1 mod 10
a=2;n=11; ø(11)=10;
hence 210 = 1024 ≡ 1 mod 11

17. Number Theory The powers of an integer a, modulo n
a, a2, a3, … (mod n)
If a and n are relatively prime, based on Euler’s theorem, we have aø(n) ≡ 1 mod n
a, a2, a3, … will have a repeated pattern
E.g., ø(5)=4, 3ø(5)=81≡1 mod 5
3, 4, 2, 1, 3, 4, 2, 1, …
There may exist lots of m such that am ≡ 1 mod n
The least positive exponent m such that am ≡ 1 mod n is referred to as
the order of a (mod n)
the exponent to which a belongs (mod n)
the length of the period generated by a

18. Number Theory

19. Number Theory Primitive root Property of primitive root
If a number’s order (mod n) is ø(n), this number is called a primitive root of n
Property of primitive root
If a is a primitive root of n, then its powers a,a2, a3,…, aø(n) are distinct (mod n), and are all relatively prime to n.
In particular, for a prime number p, if a is a primitive root of p, then a,a2, a3,…, ap-1 are distinct (mod p).
From the previous table, we can see that prime number 19’s primitive roots are 2, 3, 10, 13, 14, and 15.

20. RSA Example Select primes: p=17 & q=11 Compute n = pq =17×11=187
Select e: gcd(e,160)=1; choose e=7
Determine d: de=1 mod 160 and d < 160
d=23 since 23×7=161= 10×160+1
Publish public key KU={7,187}
Keep secret private key KR={23,17,11}

21. RSA Example given message M = 88 encryption: decryption:
C = 887 mod 187 = 11
decryption:
M = 1123 mod 187 = 88

22. RSA Example Fast Modular Exponentiation To calculate 887 mod 187
882 mod 187 = 7744 mod 187 = 77
884 mod 187 = 772 mod 187 = 132
887 mod 187 = mod 187 = 132x77x88 mod 187 = 894,432 mod 187 = 11
To calculate 1123 mod 187
111 mod 187 = 11
112 mod 187 = 121
114 mod 187 = 14,641 mod 187 = 55
118 mod 187 = 552 mod 187 = 33
1116 mod 187 = 332 mod 187 = 154
1123 mod 187 = mod187 = 154x55x121x11 mod 187 = 11,273,570 mod 187 = 88

23. Diffie-Hellman Key Exchange

24. Diffie-Hellman Key Exchange
users Alice & Bob who wish to swap keys:
agree on prime q=7 and α=5
select random secret keys:
A chooses xA=3, B chooses xB=2
compute public keys:
yA=53 mod 7 = 6 (Alice)
yB=52 mod 7 = 4 (Bob)
compute shared session key as:
Alice: KAB= yBxA mod 7 = 43 mod 7 = 1
Bob: KAB= yAxB mod 7 = 62 mod 7 = 1

25. Diffie-Hellman Key Exchange
users Alice & Bob who wish to swap keys:
agree on prime q=353 and α=3
select random secret keys:
A chooses xA=97, B chooses xB=233
compute public keys:
yA=397 mod 353 = 40 (Alice)
yB=3233 mod 353 = 248 (Bob)
compute shared session key as:
Alice: KAB= yBxA mod 353 = mod 353 = 160
Bob: KAB= yAxB mod 353 = mod 353 = 160