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Neuro-Amorphic Construction Algorithm
(NACA)
Egger Mielberg
egger.mielberg@gmail.com
29.12.2018
Abstract.
Under certain circumstances, determinism of a block cipher can lead
to a disclosure of sensitive information about working mechanism of
underlying machine. Unveiled restrictions of the mechanism can also
give a possibility for an adversary to brute-force the cipher at a
reasonable period of time.
We propose a nondeterministic algorithm operating on variable-length
groups of bits with dynamically varying parts of round ciphertext. We
named it as “Neuron Cipher”. It does not use as public as private key.
In compared with symmetric or asymmetric encryption, it has obvious
practical advantages. Among them is a “Perfect Secrecy” [4].
1. Introduction
In cryptography, there are a number of cryptographic hash functions that
use deterministic algorithms with some restrictions on input data. However,
the most restrictions are realized in the deterministic algorithm, the bigger
range of attacks the adversary is given. The one who designs an
authorization application should always remember that a percent of publicly
presented information about the procedure of authorization is to converge
to zero.
Minimizing predictability of what next step is going to be in an encryption
process, we came up with a neurobiology-based solution. The basis of our
research is in the field of memory formation. The principles of
neurotransmission in the case of influence of dynamically changing
dendrite formation. We also base on Theory of One Synapse [7].
The goal of this article is to present a new innovative approach of
encryption mechanism. The mechanism that will allow a user, first, not to
worry about publicly transmitted ciphertext, second, to get a ciphertext that
makes any brute-force attacks meaningless.
2. Construction details
Neuro-amorphic structure of the algorithm is based on features of dendrites
of a nerve cell and of amorphous substances. As a basis we took a
simplified model of the nerve cell.
The algorithm can have as many rounds as it needs but in most practical
cases, it will require not more than two or three rounds at all. In a first
round, process of generating a hash consists of four stages:
Stage 1: Plaintext is split into two or more pieces, randomly. Length of each
piece of the plaintext can or cannot be equal to other one’s length.
Stage 2: Generating a random number of 256, 512, 1024 or more bit size.
Stage 3: Calculating a synapse value (sv) by XORing the random number
and the piece of the plaintext chosen by Fsp (round function). The chosen
piece is a primary piece (pp).
Stage 4: Calculating hashes for remaining pieces of the plaintext by using
sv. Obtained hashes of all pieces including sv, form a first hash value,
ciphertext, for the whole plaintext.
Figure 1. “Amorphic Construction”.
Fsp – a multi-valued function the main task of which is to take a plaintext as
an argument and split it into two or N pieces of different sizes. A piece of
the plaintext can be a sentence, a phrase, a word, half a word, a letter (‘s),
a number (‘s), special symbol (‘s) or a combination of word and special
symbol, etc.
Hash value of the plaintext is different from round to round and its length as
well.
K1, K2, K3,… KN – key hashes generated by the synapse value.
H1, H2, H3,… HN – round hash values of pieces of the plaintext.
3. Mode of operation
In our case, an algorithm that provides a confidentiality is based on two
components, a unique binary sequence (256, 512, 1024, etc.) and an
initialization vector (iv) that is calculated by Fsp on a random basis.
Figure 2. “Encryption mode”. Cipher Neuron Chaining (CNC).
Figure 3. “Decryption mode”. Cipher Neuron Chaining (CNC).
Formula for CNC encryption mode can have the following expression:
H0 = Si K, ,
Hj = Fsp(sv Pj), ,
Formula for CNC decryption mode can have the following expression:
H0 = Ki sv, ,
Pj = Fsp(Hj Ki), , , i.
In compared with CBC (Cipher Block Chaining) [1], PCBC (Propagating
Cipher Block Chaining) [2] and other modes of operation, CNC has several
advantages. Among them:
1. Encryption and decryption processes can be parallelized. Thus, it can
result in a fast overall performance of the entire hash process.
2. Ki values are being changed in every single round. It helps reach a
good level of Avalanche effect.
3. Confusion property is totally realized in part of dependency of Hj on
sv.
4. Diffusion property is realized completely. Changes in Si values will
drastically change bits in the ciphertext (over 50%).
Also, it should be noted that, first, Si value (iv) is chosen randomly by Fsp
and second, the length of the ciphertext for the same plaintext is different
for a single encryption process.
4. Perfect Secrecy
As [4] claims, “Perfect Secrecy is defined by requiring of a system that after
a cryptogram is intercepted by the enemy the a posteriori probabilities of
this cryptogram representing various messages be identically the same as
the a priori probabilities of the same message before the interception”. In
other words, the chances to decrypt a ciphertext for an attacker must be
the same in both situations, when the attacker gets known about the
ciphertext and when he or she gets known nothing about it. That is, the
ciphertext gives absolutely no additional information about the plaintext.
According to Shannon’s proof, a one-time pad has the perfect secrecy
property. But a practical realization of the one-time pad has serious
drawbacks. Among them:
1. “Security place”. A place where the one-time pad is stored must be
as secure as a military territory.
2. “Limit of users”. A number of people which have an access to the
one-time pad as minimum as possible.
3. “Transport efficiency”. It becomes practically impossible if there is an
urgent need for transportation of the one-time pad from one place in
planet to another without using Internet.
As we see, non-deterministic property of NACA can eliminate above-
mentioned drawbacks.
Let’s consider the following practical example (Internet version).
Suppose, agent A has to transmit to agent B some secret message “Meet
me at 8 o’clock, October 31, 7799 Broadway, New York”. In case of a one-
time pad usage, agent A must share his or her one-time pad with agent B
before any message transmission.
Moreover, keeping the perfect secrecy property agent A will always need to
generate and transmit a new one-time pad each time when he or she
needs to send a secret message. This requirement is time-consuming and
costly for both agents.
Now, imagine that a generation of the one-time pad is executed on the side
of agent A. Then, in order to send a secret message agent A will not need
to share his or her generated pad with agent B. In this situation, only one
secret agent B has to know is a sv.
Figure 4. “Secret message”.
As seen on Figure above, snippet “October 31” (S7) is used as a iv value in
Round 1. sv is generated by XORing a 256 (512, 1024, etc.) bits key and
iv. Then, sv is used for generating a key hash K7 by XORing sv and a
randomly chosen new iv. For each round, there is a unique iv. Thus, the
length as well as hash value of the plaintext will change from round to
round. In this case, the ciphertext can be as longer as shorter than the
plaintext. This property of NACA is crucial as it allows the ciphertext, first,
not to be strictly tied to the following inequality , where is a
ciphertext, is a plaintext and is a key hash.
[4] claims, “if a secrecy system with a finite key is used, and letters of
cryptogram intercepted, there will be, for the enemy, a certain set of
messages with certain probabilities that this cryptogram could represent.
As increases the field usually narrows down until eventually there is a
unique solution to the cryptogram”. In case of NACA, there is no need to
worry about interception at all as a ciphertext of any plaintext can be
presented publicly with no any disclosure of plaintext information.
Another crucial property of NACA is a set of different ciphertexts for a
single plaintext. It became possible because of randomness of iv value.
5. Neural entropy
The degree of uncertainty is a crucial property of any cryptographic
algorithm in part of outcome value. Ideally, an outcome (ciphertext) of the
cryptographic algorithm should, first, be presented publicly without giving a
possibility to hack it, second, have a unique value compared with another
outcomes of the same input value (plaintext).
We believe that this two features of an outcome are self-sufficient and let
the degree of uncertainty reach its maximum.
The first feature implies an absolute identity of both, priori and posteriori
probabilities of the outcome. It is different from the interpretation of “Perfect
Secrecy” formulated by Shannon [4]. In our case, there is no need to
separate a priori probability from a posteriori one as the outcome of NACA
can be unveiled as much public as a public key in asymmetric
cryptography.
The second feature implies that the one-way cryptographic NACA-based
function is multivariable. In other words, for any given plaintext there is an
infinite set of different ciphertexts with different length.
Thus, we are coming to such a definition as “Neural Content” (NC):
“A continuous random unit (text, number, symbol) with probability density
function [5] and function X() =
, where .”
Continuity of is caused by existence of infinitely-large set of possible
hash values generated by Fsp and .
Probability of falling into a given interval is defined by the following
formula:
According to the properties of NC we can formulate the definition of “Neural
Entropy” (NE):
“A frequency expectation expressed by the following formula:
ii,
where
iXii
,
”
The frequency expectation has a series of important properties.
Among them:
1. iii.
2. i i.
3. .
4. ii.
5. .
6. ().
7. 0
In compared with Shannon entropy that calculates an expected value of
information content, “Neural Entropy” primarily focuses on the frequency of
appearance of a random-unit-pattern. This feature of NE has a lot of
practical applications.
For more technical details of the frequency expectation () and
probability density function (), see [5]. For a detailed description of NE,
see [6].
6. Provable security
If an adversary is known about faces of both, agent A and agent B, he or
she will only need to know “Where?”.
If an adversary is familiar with the place where agent A and agent B are
used to or can be, then he or she will be satisfied with only decrypted part
of the plaintext “Meet”.
Neuro-amorphic function or NAF or N-function is a cryptographic
nondeterministic multivalued function , where is a set of bits
and 1,2 1 2 12.
NAF has a series of important properties. Among them:
1. There are no two identical values of codomain for a single value of
domain .
2. It is nondeterministic (kiei).
3. It is irreversible (-1).
In the context of Neuron Cipher and NAF, we can formulate the strict
conditions for the “security” of a cryptographic algorithm:
“An algorithm is secure if and only if the following two conditions are met:
1. The same message always results in a different hash.
2. The brute-force attack is meaningless in both directions.”
First condition can be realized through a direct usage of NACA for any
cryptographic needs.
Second condition implies a lack of information for choosing the right
template or pattern for an iterative algorithm. It is about a situation when an
attacker does not have any information about the nature of data that is
used in an encryption internal process. For example, in case of user
passwords, an attacker systematically (iteratively) checks all possible
passwords until the correct one is found. In other words, the attacker knows
possible variants of letters, numbers or symbols the password might be
consisted of. Nondeterministic property of NACA and Fsp allows a user to
be as much secure as possible from any brute-force attack at all.
7. Conclusion
We presented the new concept for an encryption procedure. The concept
introduces a series of innovative mechanism and definitions that confront
traditional deterministic concept of a cryptographic hash function. Among
the main practical advantages of Neuro-Amorphic Construction Algorithm
(NACA), two of them should be noted separately, “public storage of
ciphertext” and “brute-force attack resistance”.
We hope that our decent work will help researchers, engineers and other
users in their professional endeavors.
References
[1] C. Rackoff, S. Gorbunov, “On the Security of Cipher Block Chaining
Message Authentication Code”, University of Toronto,
http://people.csail.mit.edu/sergeyg/publications/securityOfCBC.pdf
[2] A. Z uquete, P. Guede, “Efficient Error-Propagating Block Chaining”,
http://www.inesc-id.pt/pt/indicadores/Ficheiros/1215.pdf
[3] C. Shannon, “A Mathematical Theory of Cryptography”, 1945,
https://www.iacr.org/museum/shannon/shannon45.pdf
[4] C. Shannon, “Mathematical Theory of Cryptography”, 1949,
http://pages.cs.wisc.edu/~rist/642-spring-2014/shannon-secrecy.pdf
[5] E. Mielberg, "Probability Density M-function”, to be republished, 2019
[6] E. Mielberg, "Neural Entropy", to be published, 2019
[7] E. Mielberg, "Theory of One Synapse", to be published, 2019