ALittleBitofCalculus

Theorem

on $(0,\infty ).$

Proof

:

Since

for $x\preceq 1$

and

for $x\succeq 1$

Proof

Proof

Proof

Proof

Hence

MATH

or

graphics/littlecalc__10.png

Theorem

For , let and MATH

- MATH

Proof

:

Hence

or

MATH

or

- MATH

Hence the result follows from the fact that and $\log _{e}(2)>0.$

Application:

Definition

A binary cipher is said to be an instantaneous code if no ciphertext is a prefix of any other.

Here is an example of an instantaneous code

MATH

Here is an example of code that is not instantaneous.

MATH

It turns out that this second code is uniquely decipherable and the result we are about to consider holds for this class as well.

Lemma

Let () be an instantaneous code, where and . Let then

MATH

Proof

$\vspace{1pt}$

The proof can be visualized by looking at a tree diagram of the code

On can quickly prove that an instantaneous code has the property that all of its plaintext elements occur on the leafs of its tree. The proof is a simple induction on the depth of the tree.

Theorem

: Let () be an instantaneous code, where and . Let . Finally let $p_{i}$ be the probability that the plaintext $a_{i}$ occurs in a message. Then

- MATH

Proof

:

Choose such that

MATH . By the previous lemma we know that $k\geq 1$ . Setting we have

$\vspace{1pt}$

- MATH

MATH

MATH since $\log _{2}(k)\geq 0$