Lecture 3.1: Handling Remote Access: RADIUS Motivation

Birthday paradox again
Digest = D bits
Number of messages = K
1. How many messages
K to observe to get
50% probability to
have my same digest?
2. How many messages
K to observe to get
50% probability to
have two same
digest?
e
e
k
Giuseppe Bianchi
e
= e
Giuseppe Bianchi
= e
⎛ 1
⎜1−
⎝ 2
D ( 2 )
n!
/ ( n − k)!
n n −1
n − ( k −1)
= ⋅ ⋅⋅⋅
=
k
n n n n
k −1
⎛ 1 ⎞⎛
2 ⎞ ⎛ k −1⎞
⎛
= ⎜1−
⎟⎜1−
⎟⋅
⋅⋅
⎜1−
⎟ = ∏⎜1
−
⎝ n ⎠⎝
n ⎠ ⎝ n ⎠ i=
1 ⎝
≈
k −1
i=
1
2
−k
/ 2n
⇒
∏
−i
/ n
2
=
∑ −
= e
1
2
⇒
k−1
i
i=
1
n
= 2log(
2)
⋅n
2
−k
/ 2n
−k
( k−1)
/ 2n
⇒
D
K
⎞
⎟
⎠
=
1
2
⇒
2
K ≈
2
D D
2 ! / ( 2 − K)!
1
D D
= ⇒ K ≈ 1.
177 2 ≈ 2
K
2
≈ e
−log(
2)
2
−k
/ 2n
⇒
k = 1.
177
n
i
n
⎞
⎟ ≈
⎠
Ricordando che
per x piccolo
1-x approx e^(-x)
D
/ 2
12