1. Cryptology Design Fundamentals
Grundlagen des kryptographischen Systementwurfs
Module ID: ET-IDA-048
, v10
Prof. W. Adi
Tutorial-11
Messy–Omure lock for shamir 3-pass Protocol over GF(2m)

2. Massey-Omura Lock for: Shamir’s 3-Pass Protocol
Secrecy without Authenticity
User B
User A **
Arithmetic in GF(2m)
Eb = secret key
Db = Eb (mod 2m -1)
Ea = secret key
Da = Ea-1 (mod 2m -1)
gcd (Eb , 2m -1) = 1
gcd (Ea , 2m -1) = 1 1
= M Ea Ea M
( ) Eb M
Ea Eb M 2 Da
( )
Ea Eb Db 3
( ) M Eb Eb M
= M M
** For high security, (2m -1) must have a large prime factors or be a prime, ( ) is a Mersenne prime
# of keys = φ(2m-1) 2

3. Example : Set up Messy –Omura lock for Shamir 3-pass protocol using GF(27), which is generated by the irreducible polynomial P(x) = ( x7+ x6 +1 ). The secret keys for users A and B are 57 and 73 respectively. Send Message M = from user A to B
Solution 4:
If P(x)= x7+ x6 +1 is the modulus then x7+ x6 +1 = 0,
thus x7= x the exponents of x in GF(27) are:
x = x
x2= x
x7= x
x8= x + x7= x + x
x9 = x2 + x7 + x = x2 + x7 + x = x2 + x6 +1+x
x10 = x3 + x7 + x + x2 = x3 + x6 +1 +x+ x
x11 = x4 + x7+x + x2+x3 = x4 +x6 +1+x+x2 +x
x12= x5+x7+x + x2+x3 +x4= = x5 +x6+1+x+x2 +x3 +x
x13=x6+x7+x+x2 +x3 +x4+x5= x5 +1+x+x2 +x3+x
x20= (x10)2= x6 + x12 +1+x2 +x4 = x5 + x3 + x
x40= (x20)2 = x10 + x6 + x2 = x3 + x
x51= x40 . x11 = x6 + x4 + x3 + x2 + x
The order of any element is a divisor of
27-1=127 3

4. ( ) ( ) ( ) User B User A Solution : M=x11 = 1011111 **
Arithmetic in GF(27)
Eb = 73
Db = Eb-1 =-40 mod 127 = 87
Ea = 57 (1..127)
Da = Ea-1 =-49 mod 127 =78
X mod 127=x119 mod p(x) 11
119 x
( ) 73
= M 57
= x
x11 51 x
X mod 127=x51 78
( ) 51 87 3
( ) 11 x 41 41 x
= x x
x mod 127=x41
Note: x41=(x11)73 mod 127= M73 4