# Number TheoryA question on cryptography...

#### chisigma

##### Well-known member
The recent case 'datagate' suggests me to prose to You a question I didn.t resolve completely. Let's suppose that we have a plaintext $p_{n}$ and we code it with a key $k_{n}$ generating a chipertext...

$\displaystyle c_{n} = p_{n} + k_{n}\ (1)$

... where the sum is modulo some 'large number' N. It is well known that the (1) is 'theoretically secure' if and only if...

a) the sequence $k_{n}$ is 'absolutely random'...

b) the sequence $k_{n}$ is to be use only for a single message...

It is also well known that this solution has many pratical problems, mainly the necessity to have a large number of sequence $k_{n}$ only for have a secure communication between two persons. An idea to overcome that drawback may be to use a $k_{n,m}$ for the message m and a $k_{n,m+1}$ for the message m+1 and in the message m to communicate...

$\displaystyle c_{n,m} = p_{n,m} + k_{n,m},\ h_{n,m}= k_{n,m} + k_{n,m+1}\ (2)$

What is Your opinion regarding the security of this type of cryptosystem?...

Kind regards

$\chi$ $\sigma$

#### Ackbach

##### Indicium Physicus
Staff member
I should think communicating $h_{n,m}= k_{n,m} + k_{n,m+1}$ would greatly compromise the security of the so-called One-Time Pad, as you've described it. Naturally, for a single transmission, your proposal will have the same level of security as a one-time pad normally does. However, the additional information could be decoded once several messages have been transmitted in this way, in order to figure out what the next $k$ will be.

In general, communicating much of any key information along with the ciphertext is a bad idea.

#### chisigma

##### Well-known member
May be that for the readers and for me is useful to remember some basic concept. A ciphertext $\displaystyle c_{n}$ is given by...

$\displaystyle c_{n}= p_{n} + k_{n}\ \text{mod}\ N\ (1)$

... i.e. the sum modulo N of a plaintext and a key. The main difference between a plaintext and a key is that the key is 'random' and a plaintext isn't random. Why the use of the same key to code two different plaintexts is 'a bad choice'?...

Let's suppose that the two plaintexts are monday and friday. In that case with a systematic search i find, sooner or later, the key that produces the playntext monday produces the plaintext friday and both the ciphertexts are brooken. Completely different is the situation if the same key is used to code a plaintext and another key. In that case monday produces something like jaab?+ and friday something like \gw>o@ and it's impossible to extablishes which plaintext is 'more probable'...

Am I right?...

Kind regards

$\chi$ $\sigma$

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