# The State of Random.

*From*: adacrypt <austin.obyrne@xxxxxxxxxxx>*Date*: Sat, 16 Oct 2010 05:20:11 -0700 (PDT)

Anybody who has ever rolled a die on the kitchen table as a child in a

board game like “Snakes ‘n Ladders” has an intuitive understanding of

what random means and anybody who has rattled the cup and then slowly

tipped out a six knows what an unbiased retrieval system means also.

What I am leading up to is the use of a random key-set to secure

cryptographic ciphers and I have to make the point quickly that random

is far from being a universally understood thing even at high levels

of cryptography.

Many books on random are available from amazon on the internet but

these are nearly always pretty uninformative about how to achieve a

random state and most of the print is spent on repeating and

reinforcing already well-known and accepted cases of a haphazard state

by means of sensational anecdotes, they spend their time on ‘preaching

to the already converted’ reader without ever giving out useful

methodology on how to achieve and deploy a proper random state to

solve any real life problem.

Note well that I am using the word ‘random’ rather carefully when

randomness would sound better – there is a reason for this, the word

‘randomness’ wrongly implies shades of the state or property of random

when in fact random is a binary property that does not have degrees of

quality. You either have a random state or not at all. It’s a black

or white issue that does not have shades of grey.

In my experience there is a huge misunderstanding of what random

really means and there is good reason for this. What most people mean

when they say some event is random is that it is a chance occurence

i.e. haphazard outcome that is sometimes way beyond the control of man

albeit that is always not the case. This is often the basis of much

conjecture and is a much loved theme in matters of human interest like

the ball unexpectedly striking the pin in golf and players theorising

about the probability of it happening again say (I hope not to open a

can of worms with this example – it nearly always does that when any

discussion gets under way regarding random). In passing, one

historian reckons that the Sumerians (3000 BC) invented mathematics to

get some of the unwanted randomness out of their daily lives.

I am going to call this colloquial randomness (taking licence with

the word randomness after all, simply for ease of expression). It can

be a pleasant or unpleasant experience, a sometimes quirky thing in

our lives that will always remain as a chance event but a favourite

element of day-to-day living that will never disappear and long may it

live as such in my view but this randomness is not a workable property

in mathematics. I will call it romantic randomness to distinguish it

from scientific randomness that is indeed workable in mathematics.

Randomness is mathematics is what I am calling scientific randomness

and is defined in mathematical dictionaries as a set of elements that

all have equal probability. The question begs “ probability of

what”. The answer to that question in cryptography is this, equal

probability of being the next one to be called in any unbiased

retrieval system at decryption time.

In order to have equal probability the elements must be equal in

number (frequency) because clearly, if that is not the case then some

numbers have greater probability than their neighbours and the set is

not random.

It is amazing how this simple fact is misunderstood at even the

highest levels in cryptography. The former notion i.e. haphazard

randomness is the one that has taken hold and there is a spate of duff

cryptography that is based on the wrong idea that a haphazard

collection system makes for a bona fide set of random data. It seems

to be believed that the data will inherit some kind of mystical

property from the haphazard collection process that created it and

that this will later cause confusion to any illegal cryptanalyst who

intercepts the ciphertext that is in transit. It seems to be thought

that this will happen over and above the rules of any mathematical

rigour that may be relevant. Clearly, that is not true.

The way scientific randomness works is that it causes total

uncertainty to an illegal cryptanalyst who may have in his possession

ciphertext data that he has intercepted and is now trying to illegally

decrypt. His problem is that the key-set that protects the ciphertext

is random and therefore each elemental key is equally likely to be the

correct one to use for his nefarious purposes and he can only guess at

any string of plaintext that might evolve being the correct one from a

larger space of permutations of similar strings, when all of these

have been decrypted experimentally (Appendix A refers).

In essence, randomness translates as total uncertainty to an illegal

cryptanalyst, uncertainty is a property in the subject “Theory of

Information” that has obvious meaning here also.

Imagine next for example, this discussion model:

Suppose I stop one hundred people on the pavement and ask them to

verbalise the first integer that comes to mind, will this give me a

random set of one hundred integers?

Clearly, the experiment is spontaneous, unrehearsed, totally unbiased

and is haphazard. That, by most ideas of understanding would surely

seem to be a random exercise that would be sure to yield a random set

of 100 integers. But that would be true only if there were no repeats

of any integer in the set of collected data. If there is a repeat of

any integer then that integer has greater probability than the others

and the set cannot be random by a scientific meaning of what

constitutes random data.

The appraisal then is that there are two kinds of randomness, 1) a

haphazard randomness and 2) a proper scientific randomness that is

mathematically defined. Of those two random types it would seem

obvious that cryptographers would go down the latter road embracing

this proper, mathematically workable random, as the means of

underpinning cipher research but no, even at the highest levels of

expertise, haphazard randomness was the one they followed. I quote

one of the best cryptography writers in the business, “ the best

random keys are created by harnessing natural physical processes, such

as radioactivity, which is known to exhibit truly random

behaviour” (this is in fair proper context of his quote). I contend

that he should be saying “ truly haphazard but unusable

behaviour”instead.

The reader is referred to the simple definition in the Penguin

Dictionary of Mathematics for the word ‘random’ – basically it says

random means having equal probability between the elements of a random

set. It is acknowledged that outside of mathematics and cryptography,

random is synonymous with haphazard but not within cryptography and it

is amazing that any cryptographer would make the mistake of thinking

it still applies within intensely number-theoretic cipher design

theory. This massively entrenched fallacy that random data equals

haphazard data, has stymied all progress in cryptography for the past

half century and I suspect has led cryptographers to shy clear of

number-theoretic cryptography and instead go down the complexity-

theoretic road instead with poor results.

That situation is changing now, I have invented two forms of

cryptography namely, “Vector Cryptography” and “ Scalar Cryptography”

that use 1) A mathematical one-way function and 2) Randomness, as the

securing means respectively. These are described on http://www.adacrypt.com

and http://www.scalarcryptography.co.uk respectively.

Appendix – A.

“Handbook of Applied Cryptography” – P. 20 ?

Quote: “ That is, if a cryptanalyst has a cipher text string encrypted

using a random key string that has been used only once, the

cryptanalyst can do no better than guess at the plaintext being any

binary string of length ‘t’ i.e. (t-bit binary strings are equally

likely as plaintext). It has been proven that to realize an

unbreakable system requires a random key of the same length as the

message.

Unquote.

This is probably the most respected reference and information source

in existence in the field of cryptography to day but it is becoming

obsolete now with anachronistic information in my view. The above

extract was written in relation to the one-time pad cipher but is

generally true of all ciphers. A scientific understanding of ‘random’

is implicit in this quote.

.

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