Lattice-Face Key Infrastructure (LFKI) for Quantum Resistant Computing.

Abstract

In this paper we present a new idea by exploring a hybrid system designed to share symmetric and asymmetric properties. LFKI is code named, end-to-end cryptographic system for cloud, mobile, internet of things (IOT) and devices (ECSMID). Until now, there had not been much done on lattice faces as a hybrid cryptographic solution. Here in, we do not owe respect to only randomization reduction or deterministic reduction. We embrace a collective approach to defining the old age question of what problem is hard enough in NP to resist a quantum assailant. These biases, especially non-deterministic reduction is used to show that lattices are interesting hard problems within the set of NP Complete problems. Though the shortest vector problem (SVP) seems promising. It is nearly enough to facilitate and establish lattice basis an exception from the priori art [1]. The many configurations of their vertices seem to dismiss the wonderful properties of the dynamic faces that abound in various lattice constructs. The elements of these faces found in between regions bounded by the vertices and edges are of great interest to cryptography. When represented as numerical values serve as mathematical images of the lattice basis distribution. It is demonstrated that each vector representation has the potential to generate cryptographically secure number of keys. They follow a somewhat rigid rule; deterministic and yet a chaotic arrangement of the lattice vectors represented within a matrix of column (c) and rows (r), where (c=>16 and r=>16). A fitting rule is already available with the necessary mechanism to produce 1: n relationship for a plaintext against many ciphertext. This is found in Open/Knight Tour (O/KT) movements and can easily be modified to absorb larger lattice basis. Lattice face are ready made with properties that are closely related to the regular vectors of Euclidean space.

Full text

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Lattice-Face Key Infrastructure (LFKI) for Quantum
Resistant Computing
**USC Incubator 1225 Laurel St. Columbia, SC 29201
LokDon security research project:
**Josiah Johnson Umezurike | jumezurike@lokdon.com | September 12th 2018
Keywords: OTP, Qubit, SVP, CVP, Lattice-basis, 2048 Bits, AES, Cryptography, QR,
QI, Blockchain
Abstract
In this paper we present a new idea by exploring a hybrid system designed to share symmetric and asymmetric properties. LFKI is
code named, end-to-end cryptographic system for cloud, mobile, internet of things (IOT) and devices (ECSMID). Until now, there had
not been much done on lattice faces as a hybrid cryptographic solution. Here in, we do not owe respect to only randomization reduction
or deterministic reduction. We embrace a collective approach to defining the old age question of what problem is hard enough in NP
to resist a quantum assailant. These biases, especially non-deterministic reduction is used to show that lattices are interesting hard
problems within the set of NP Complete problems. Though the shortest vector problem (SVP) seems promising. It is nearly enough
to facilitate and establish lattice basis an exception from the priori art [1]. The many configurations of their vertices seem to dismiss
the wonderful properties of the dynamic faces that abound in various lattice constructs. The elements of these faces found in between
regions bounded by the vertices and edges are of great interest to cryptography. When represented as numerical values serve as
mathematical images of the lattice basis distribution. It is demonstrated that each vector representation has the potential to generate
cryptographically secure number of keys. They follow a somewhat rigid rule; deterministic and yet a chaotic arrangement of the lattice
vectors represented within a matrix of column (c) and rows (r), where (c=>16 and r=>16). A fitting rule is already available with the
necessary mechanism to produce 1: n relationship for a plaintext against many ciphertext. This is found in Open/Knight Tour (O/KT)
movements and can easily be modified to absorb larger lattice basis. Lattice face are ready made with properties that are closely
related to the regular vectors of Euclidean space.
Introduction
This article is an observation from over
20 years research work. The work is not
done by a mathematician but by a
security professional. The sole intent is to
solve the common problem of our time
from practitioner’ perspective. It is agreed
on all grounds, the havoc quantum
computing will bring to the modern
cryptography. Consequently, it is
sufficiently relevant to be prepared pre
and post quantum. The understanding of
Euler, Hamiltonian cycle and lattice basis
paved the way in drawing the relationship
needed to harmonize the open knight
tours (OKT) in the genre of Hamilton’
path. The similarities under study shows
the pervasiveness of Hamilton’s path in
grid (n x n) formation. In absence of any
back track; It does enumerate all points
in Euclidean space if and only if n=>5.
Hamilton’ cycle, when applied to grid or
chessboard, it clearly proves that it is
indeed a hard NP as the grid become
richly connected. When this exercise
extended to the operation of AES
(Rijndael) which commonly lies on 4 x 8
grid. It is possible to expand the scope of
AES to develop a 2048-Bit AES-hybrid
using the bounded region between the
edges and vertices of a lattice face. The
result is a low-cost, high entropy,
endpoint to endpoint cryptographic
system for cloud, mobile and IoT devices
(ECSMID). The reference specification is
a category of hard NP problems closely
related to numbered faces of a lattice
basis or matrix. This cryptography shows
the properties of both symmetric,
asymmetric cryptography or public key
infrastructure (PKE, KEM and DS).

2
OBJECTIVE:
To show that there is a cryptographic
formation following a lattice basis that fits
into an ideal set of hard NP complete
problems known to be resistant to
quantum computing. A matrix could be
observed as a numerical image of a
lattice basis to bring about a low cost,
pervasive and high entropy cipher which
hybridizes and increases AES capacity to
roughly 10 times. Thereby, resists post-
quantum attacks and cancel the effects
of pre and post-quantum breaches.
An Overview of current cryptography
The Frailty of PKI and AES: There are
numerous talks about PKI. Ponemon
institute, Gartner, IBM and many other
reliable and prolific sources had
mentioned their worries about the future
of PKI as we know it. More so, PKI and
AES are the dominant part of the
mechanism securing the internet
transactions of today. The banks, health,
retail, government and all entities use
these two pieces of technology.
They are supposed to secure and make
private each communication whenever
you access any secure website. It is a
scientific knowledge that PKI is based on
mathematics:
Where in, (N = p * q), φ(N) = (p-1) (q-1)
where (e, N) is the public keys and (d, N)
is the private key.
There is a condition e {integer;
1<e<phi(N); co-primes (sharing no
factors) with N & φ (N)}. Choose d such
that,
{ed mod φ (N) = 1}:
In the wake of these problems are many
proposals for the direction of modern
cryptography. There are:
1. Lattice basis cryptography
2. Code base cryptography
3. Multivariate cryptography.
Some of these are the second runners up
of NIST’s call for paper in cryptography
2019. This means that they are still being
considered in the second round of NIST
standardization for modern
cryptography. It is a scientific fact that
any mathematical problem is there to be
solved. One can clearly the reason for
NIST call. However, considerations were
only given to the use of these schemes to
deliver AES keys following the 50year old
trend or tradition.
This means that our crown jewel cannot
depend on any mathematical function
based on Fermat's theorem or any other.
To achieve the desired goal. A favorable
design will be that which references a
quantum cryptographic model (QCM) as
a relevant strategy for securing the
internet in times to come. Otherwise,
quantum computing will wreak havoc on
modern day cryptography whenever it
finally gets into the hands of consumers.
Let us take a serious look at what a lattice
really means in a mathematical sense of
it.
The Problem:
If anyone can obtain the factors of the
large number N with d (public key) any
message will be decrypted. At the time of
writing it is known that RSA is cracked.
You should also note that Quantum
computing has the potentials to solve the

3
math and/or crack these large primes (N)
in a short period of time according to
Shor' Algorithm [2]. The time to perform
the feat is usually said to be in polynomial
time. In that case the RSA math show
earlier, will no longer be a hard problem
of a non-deterministic polynomial (NP).
Qubit Is the stable standard signal state
of a quantum computer: Again, the
development of any quantum resistant
algorithm could not afford to dismiss that
notation typical to a qubit. In fact, one
cannot neglect this idea and it cannot be
over emphasized. This new qubit factor
will also render any form of primitive
cryptography useless. Another problem
arises with the periodicity of lattice
constructs. That begs the question. Is
there a way to infuse the lattice with
enough diffusion that will trigger more
than translational changes in bases to
bring about entropy and complexity so
dynamic that it will be impossible to
decipher the permutation of the basis
transformation?
Another serious problem that we must
consider lies in the diffusion of AES.
1. Maximum distance separable
(MDS) matrix introduced by
Shannon Claude. – This is an
identity matrix multiplication of A
to produce a transformed matrix
A;
Let the matrix A = 
 , by joining
identity matrix I to A. This introduces
an invertible linear transformation.
2. Ciphertext produced after the
transformation of AES will have
just a key to lead to a plaintext.
3. The constant nature of the
produced ciphertext will make it
susceptible to byte-wise brute-
force attacks.
According the substitution box (S-
box) AES. This matrix is based on
Rijndael finite field. An attack byte-
wise will be successful today. One will
have about 256 options or
combinations for each byte in the
32byte arrangement of 256 bits block
of AES. This means that in a 256 bits
AES block size. Each sub block
contains at the most 4 bytes or 32
bits. There are about 256 elements
that could fit in one sub block at some
point. This sub block will have about
(256 s-box elements * 32 sub blocks)
= 8192 combination of s-box
elements or 32bits in it. Here 32 bits
is equivalent to double words.
Attacking bitwise could seem
hopeless at first but going about the
attack from byte perspective is
another smart way achieving the
same goal. A study of the diagram in
fig. 1.0 will quickly show why
attacking in byte-wise will be
successful. It is much better to
integrate the bits in a way to have
more structure or visual patterns.

4
Brute force analysis of AES 256
Fig 1.0
SOLUTION TO THE PROBLEM:
Observation of the weaknesses of AES is
the motivation behind this paper. A
generous diffusion of semantically
unsound messages was presented on
behalf of AES. Going by the facts known
about AES, such entropy cannot be
accomplished in less than 14 rounds. In
fig. 1.0 notice the patterns in cipher key 1
and ciphertext 1. This is clear by a visual
comparison to state, 32byte of structured
messages. After the completion of the
14th round of AES, the resulting output is
ciphertext1. Imagine using the S-Box
directly with 256 elements where each
matrix cell has the capability of 32bits or
more. Lattices in their natural forms
explain this abstraction in a new light.
What then is a lattice?
A lattice is a set of all integral linear
combinations of a given set of linearly
independent points in Zn. For a basis B=
{b1,...,bd} we denote the lattice it
generates by    

 .
Its rank is d, and the lattice is said to be
of full rank if d=n. We identify the basis
{b1,...,bd} with the n×d matrix containing
b1,...,bd as columns, which enables us to
write the shorter L(B) ={Bx|xZd}. We
use both the terms lattice point and lattice
vector to describe the elements of a
lattice basis.
A bit more time will be spent to introduce
a new insight, towards quaint
understanding of the shortest vector
problem (SVP) of graph and path. While
it is generally a consideration for being an
NP-Hard problem. Randomization
reduction without considering the face
holding the basis is not enough to
establish this as a case of NP-Hard
problem [3]. That alone, could have been
insufficient, or not good enough for
quantum resistant encryption. Quantum
computers are created to solve
mathematical problems impossible to
human mental speed. If a lattice basis
must be retained as the frontier of
modern cryptography; it must be an
interesting one with some elegant
properties that could be reduced to
randomized, non-deterministic and
deterministic biases. The intention is not
to be overly critical. There is a need to be
proactive. One’s intent will be to find the
right solution out of many; not to accept a
solution that is not ripe. This is neither to
wait for a solution to present itself. A
potent lattice or ideal lattice and its image
must be dynamic, with certain rigid rules,
yet precise in decision making affirmative
to the output. It must possess a
distribution of probabilistic basis
transformation with respect to the input

5
and output (references are made to
homomorphic encryption). -- The image
of the lattice basis is a bounded matrix of
interest. Let us look, once more into the
relationship between encrypted and
decrypted messages.
Asymmetric:
Encrypted data (c) = msge mod N
Decrypted data (msg) = cd mod N
It is clear from the above that an assailant
only requires (d, N) to decrypt the
message. Although, this does not apply
to AES in mathematically sense. It quietly
applies in byte-wise brute force of AES
cipher key.
Symmetric:
Let Ct = cipher template length; where
the length is the same as the keys used
to perform wholistic encryption of the
message. The message is added to
extended key K of period D which could
be a 64 bits passphrase or more. Note
that, a modulo arithmetic (XOR) is used
herein. It is a common knowledge that
AES is one form of the family of
symmetric key cryptography. The
strength of AES is synonymous to the
irreducibility of polynomials of GF (28) or
8th degree. Symmetric key cryptography
(SKC) uses a secret key: They are
commonly known as passwords or
passphrases and mostly manual driven.
It is interesting to note that the key used
to perform the actual encryption in AES
sometimes are derived from these
passwords via key derivation mechanism
capable of a pseudo random number
generator (PRNG). Password based key
derivation function (PBKDF-2) is a good
example.
The block sizes of AES are defined
(128bits, 192bits and 256bits). These
and many other reasons add to their
weaknesses before quantum computing
brute force attacks. PRNG will generate
AES keys of 16, 24 and 32 bytes to match
the block sizes respectively. If the
message doesn't fit the block. It is then
padded with IV so that it will fit the chosen
block. Grover's algorithm is a quantum
algorithm that finds with high probability
the unique input to a black box function
that produces an output of a defined
value, using just ) evaluations of
the function, where  is the size of the
function's domain. Despite the effort
vested in making AES secure, Grover is
saying that it is probable half the time, to
brute-force AES – 128 in 264 iterations. At
least, one can unravel useful information
that will lead to breaking of such scheme
using quantum computer as a level
playing field [4]. Here in, it is implied that
the time is in quantum domain not
polynomial time.
ECSMID proposes the use of seeds in
social security numbers, driver license
number and phone numbers. It is
recommended to use 10-20 digits
number arranged in one order. These
numbers could be picked off vectors
capable of becoming seeds for
generating 680 digits number from each
position on the matrix of n * n. We will talk
more about this on another paper. These
data are fed into the algorithm just like
traditional classics and/or primitives of
today.
Encrypted data (c) = (msg, D): (msg xor
D) mod Ct
Decrypted data (msg) = (c, D): (c xor D)
mod Ct.

6
AES cipher substitution is directly derived
from the substitution box (S-Box).
LFKI default matrix is 16*AES Fig. 2.0
In Fig. 2.0 transformation of LFKI is
completed via the natural occurrence of
the lattice basis which under observation
was noted analogous to OKT. One can
say there is a similarity between the new
protocol and AES. It is necessary to
perform an exercise; a proof by
visualization. This will help us to clearly
establish without a doubt these claims.
We will attempt to prove this abstract
connection, not only to dispel doubts, but
to deepen understanding of the trail
modern cryptography could blaze.
AES will suffer a similarly if not the same
fate as RSA. If we do not apply this new
mechanism. The future of quantum
computer will certainly vilify it as well as
any other contraption that does not
comply to the dynamic reality of quantum
computing model (QCM). We don't really
have to wait into the future anyway.
People are already saving petabytes of
data in the cloud. In due time these could
be disclosed as soon as quantum
computer becomes available.
AES will suffer a similarly if not the same
fate as RSA. If we do not apply this new
mechanism. The future of quantum
computer will certainly vilify it as well as
any other contraption that does not
comply to the dynamic reality of quantum
computing model (QCM). We don't really
have to wait into the future anyway.
People are already saving petabytes of
data in the cloud. In due time these could
be disclosed as soon as quantum
computer becomes available.
The minimum OKT necessary is 5
modes. Which simply translates to 5
different ciphers because the state is
mapped or substituted to the knight
template (KTn) to produce cipher
templates (CTn). Structured messages
are fed into the CT(n≤5). This method will
accommodate future changes in code
points and or code units. This will cover
any future changes in the value sizes for
types e.g word (WORD), double word
(DWORD) and quantum word
(QWORD).

7
Dynamic LFKI rounds Fig. 3.0
One of the major advantages observed in
this paper is the fact that all other inputs
in the circuit change except for the
original message. This is a superior
mechanism to surpass the challenges of
quantum superposition.
Technical Specification
To solve this problem from a technical
perspective. It is imperative to draw an
analogy from 3D shapes and their
properties: Especially surface area (face)
with breadth. A cuboid and other
favorable dimensions of lattice basis will
suffice for this development. Their
properties like face, edges and vertices
come in handy in unbounded and
bounded space. You can get a flux from
these properties as a result of vectors
forming regular point in Euclidean space
to enhance orientation as seen in lattice
basis. In programmatical (code) terms,
the idea of a matrix transformation:
Translation, transposition, and
substitution serves us well by forming an
algorithm that covers lattice face key
infrastructure and architecture. The face
on lattices are commonly known to have
points or vectors. Same goes to a matrix
which is a quantitative representation of
the lattice following certain strict rules.
Therefore,
Total flux = ∫∫ f.n. dS [where n=1].
For our purpose the vector accent will not
be needed. As scalar and vector
delineation blurs in the region of SVP.
It means that any normal face in a shape
will have a regular arrangement of point
in Euclidean space (lattice). In this sense,
following the elements of Galois' field; a
matrix, mathematically can hold a
lattice's contents: It is then noted that a
lattice is only a form which can be
reflected or translated. It will have points
upon which forces can interact with it.
This means that changes in choosing any
of these points could change the matrix
or the indices they bear. Below is the
explanation of informational entropy.
Mathematically, this is expressed as
H(C) = H (M|C), where H(M) is the
informational entropy of the plaintext and
H(M|C) is the conditional entropy of the
plaintext given the ciphertext C. This
implies that for every message M and
corresponding ciphertext C, there must
be at least one key K that binds them as
a one-time pad. Mathematically
speaking, this means K=>C=>M, where
K, C, M denotes the distinct quantity of
keys, ciphers and messages. In other
words, if you need to be able to go from
any plaintext in message space M to any
cipher in cipher-space C (encryption) and
from any cipher in cipher-space C to a

8
plain text in message space M
(decryption), you need at least |M|=|C|
keys (all keys used with equal probability
of 1/|K|) to ensure perfect secrecy [5].
It is also a standard practice to increase
entropy by introducing seed candidates
capable of deriving PRNs, silent noise
(passphrases or codes) penetration and
other manipulations that permeates
cipher text to remove structure in
plaintext (original message) cause
entropy in the ciphertext. This could be
achieved through modulo arithmetic by
adding (XOR-ing) numerical values of
passphrases (characters, special
characters) of the UTF-8 to the original
messages. This will be fully explained
later in this paper. According Shannon,
the common knowledge of entropy is in
the information content Hx of a value x
that occurs with probability Pr[x] is
  .
The entropy of a random source is the
expected information content of the
symbol it outputs, that is
 
 

 

It is submitted in quality from observation:
That it is not a common knowledge to
think of n=ꚙ with respect to the equation
of Galois field GF(2p) which essentially
claims its validity from Euclidean space.
In programmatical (code) terms, the idea
of a matrix translation, transposition,
transformation and substitution serves us
well by forming an algorithm that covers
lattice face key infrastructure and
architecture. Imagine that, this is in
opposition to present day symmetric
cryptography limiting scopes.
In cryptography this means that those
points can represent encryption and
decryption components of data by
satisfying GF (Pn) where n=ꚙ. The flux
analogy herein depends on the surface
area or orientation of the shape and
forces (analysis) on them. The changing
flux will be likened to the changing
entropy at every turn of the algorithm
(operation) owing to noise. The total flux
is the product of the basis surface area,
force and normal vectors. It is therefore
possible to create a system of quantum
immunity or resistance for the
computation by replacing the vectors or
points with characters of written words.
Carefully chosen, are certain Unicode
characters (i.e numbers). These
formulate the standard state (ST):
Subsequent generation of numbers from
these face/s or seeds, following position
P(n=0) - P(n=255) give rise to other sets (680
digits long) which could be used as
cipher templates (CT). These points
become numbers generated from the
chaotic regularity found in faces of sky,
snowflakes and silicon shapes (of course
in 2D and 3D).
The proposed algorithm comes with a
powerful wrapping (Mode1 – Mode5 or
M1-M5) mechanism. That’s what makes it
possible to be used as an exchange
channel in the order of PKI public and
private key. However, the school of
thought defers from the popular opinion
of Shortest Vector Problem (SVP)
associated with the current lattice basis
solution for cryptography. It is deduced
from the research that open knight tour
on a lattice face is a harder NP problem

9
than the notion of SVP [6]. It cannot be
solved by a quantum computer as long
as the matrix is equal and greater than 16
for the columns as well as the rows:
Given a matrix in column major (c,r),
where a full rank is n x d. let n = d. It
follows that {c>=16=<r}.
Note: The point closest to the chosen
vector in SVP is orthogonal to all other
points of interest. Finding the shortest
path is the reason why this problem is of
interest to cryptographers. This can
never be deducted with certainty needed
for integer mathematics. In lattice
diagram ‘A’ fig. 1.0 the periodicity is very
clear more so, all points sought to
determine the shortest vector path are
orthogonal. Now, look very closely at
lattice diagram ‘B’ fig. 2.0 The periodicity
is also clear as the denoted impression in
‘A’. Although the bases are replaced by
numbers just like a matrix would have
within. When the numbers or the lattice
bases are rearranged. A measure of
difficulty arises in a way the problem
becomes harder. The path to finding the
shortest vector is no longer a linear one.
Or is it? By Pythagoras it still is
orthogonal respecting the base
orientation.
Lattice base and matrix mix Fig. 5.0
A quantum Turing-machine with qubits
orientation cannot sniff with certainty the
positions of any legal open knight tour
(OKT) on lattice face if the column (c) and
the row (r) of the matrix are respectively
of c=>16 and r=>16. If the position (Pn)
that generates any set of
cryptographically secure keys is
unknown. If any set of keys generated
from the matrix positions (Pn) follow n!
where n=>256 is unknown. If comparing
any two positions (P1) to (P2) on the
lattice does not sniff out similar 680-digit
long keys. Giving any input, it is said that
the decision is impossible. Else, this is
probably the hardest NP problem and will
not resolve in polynomial-time.
Computational difficulty Fig. 4.0
P ≠ NP and no one is sure of P = NP as
it is not polynomial resolvable as earlier
explained. In corollary, one can find a
common NP-Hard problem which allows
similar inputs as the OKT. In that case
lattice basis are best suited for this
reduction. Let X represent a lattice with
regular point(s) in Euclidean space.
It is agreed on equal footing that
Hamiltonian path and open knight tour
(OKT) are NP Complete [7].
It is also a common knowledge that the
Shortest Vector Problem (SVP) of a
lattice-based cryptography is an NP-
Hard problem. See Ajtai works for details.
We will only try to reduce the hard

10
problem to NP to prove that OKT is
equally a hard problem.
To prove that OKT is a hard NP
problem: We only need to re-state the
theorems. We will follow these steps:
1) We deduce that X ꞒNP
This could be done in (i) or (ii)
i) Polynomial time algorithm
ii) Certificate and verifiers
2) Reduce from known NP to the
problem Y to X.
i. If Y ꞒP the X ꞒP
ii. If YꞒNP, then X ꞒNP
X not in P unless P = NP
X is NP Complete if X ꞒNP & X is NP-
Hard.
X is NP-Hard if every problem Y ꞒNP
reduced to X
In this case inputs for X and Y are the
same e.g coordinates, . There
will be no polynomial time algorithm for
this proof. There is still a known problem
3DM (Ș) that is NP-Hard. If we can fit this
problem into Y, then Y too is NP-Hard.
Proof: Y is NP-Hard
Given: 3D matching (variable gadget).
Disjoint set x, y, z each size n given
triples T  x *y*z.
Is there a subset   such that every
element, Ꞓx y  z is in exactly one,
Ꞓ Following a legal knight move OKT
could only be on black dot (Y) or white
square (N) at once?
Method: Reduction of X to Y.
Three-dimensional matching (3DM) is
NP Complete (Theorem). It is going to be
a graphical prove. To make this easy, we
set up an 8 by 8 matrix of black dots and
white squares. See diagram C Fig 4.0.
Let X represent the lattice basis (SVP)
and Y represent OKT. Lattice basis
(SVP) had been reduced to NP-Hard
problem earlier[8]. Other precedence, 3-
SAT was reduced to 3DM [7].
To Prove:
Ș ≤p Y (If we could solve Ș we could
solve Y).
3DM reduction to OKT Fig. 4.0
It is noted that in a deterministic Turing
machine the answer is in affirmative for
all inputs following the algorithm. You
have the graph and the path to trace.
This is quite analogous to the knight on a
standard chase board. This same
analogy is akin to non-deterministic
mechanism given any input for decision
of Y (black) or N (white). In this is more
like a black dot or white square.
Following a certain strict rule which
compels the knight or the input to touch
on one of two (2) nodes if at the vertex
(corner); four (4) nodes if on the edges
and eight (8) nodes if at the middle of the
board. It will trace the path to the nearest

11
node no backtrack is allowed. This
solution could go in a loop within a
changing or expanding bases.
Open knight path traced from any corner
of n x n graph will have 2n nodes of
connection for 3 moves at the most. This
is counted from n=0 position (Initial point)
where n=0 is not really a move.
1. The assumed position (Pn) on the
corner is not counted as the first
move such that no move is
considered for initial position n=0.
This means that the number of
nodal connections at any chosen
path will have 2n nodes; where
0=<n=<3. Only one node will be
activated to move on to the next
point of decision in the path. This
is how the numbers are
generated.
End of proof:
The open knight tours satisfy the
condition of 3DM where in, a response of
true (Y) or false (N) is entered to satisfy
that only one element of the triplets could
the held in T. If the path found for the
legal knight is correct. The clause must
be black dot else white square. The path
of a legal move, is a certificate which the
machine must verify (one can also say
that it is polynomial algorithm satisfied by
the input and output of instruction sets)
by counting black as a YES or white
square as a NO. This method does not
need to worry about garbage collection in
the circuit for fear of tautology [9].
Notice a clear demonstration that there is
no need to perform the garbage
collection technique in this
implementation reference. The said
technique is performed as an extra layer
in the reduction to show that the set Ș will
connect to every other dot in Euclidean
space. Relying on the above claims,
premises and theorems we submit this
reference specification of an algorithm
that combines symmetric and
asymmetric cryptography using zero
knowledge triangle flow and
homomorphic encryption, standing
strong enough to resist attacks from
quantum computing. - Lattice-Face Key
Infrastructure (LFKI)-- It recognizes and
applies:
a) public key encryption - 2048 bits AES-
hybrid is used for encryption in wraps or
modes
b) key encapsulation - positions of key
sets are encrypted with msg and
separated
c) digital signature - attributes are formed
and stored as encrypts (HE properties
are used)
d) Hashes are not used in the classical
sense for authentication: They only
suffice for initial plain text integrity
(digest) check
e) CRC or checksum is not pushed here
because of HE: If the hashes match, the
original plaintext is the same as the
current one.
The minimum modes for any encryption
done is usually 5 or M5 for this system.
However, you can encrypt anything (a
message etc.,) from M1 to Mnth. This
security could be applied in
telecommunications, cyber physical
systems (CPS), IoT, information
technology (IT), aeronautics, lithography,
medicine and health, retail, finance and

12
education. This will be the hybrid of all
times.
Infrastructure of LFK
It has an elegant, simple and easy to
implement approach. Our social mode of
interaction on the media had made
possible for us to easily figure out what
works. Many profiles today are
comprised of attributes. Therefore, we
reduce data into certain groups for
seemingly public key implementation.
Digital Nucleus Aggregator (DnA): These
are attributes that can be converted to
encrypted strings for various
intermediate representation in the digital
space. e.g Name, SS#, eFRI, DOB, PIN,
Address, password Gender, Driver
license# etc. It could be anything of your
choosing. Profiles rely on DnA as their
building blocks for intermediate
representation in this reference. DnA are
derived from profiles attributes as we will
demonstrate later.
Digital Data Nucleus Authority (DDnA):
These are integration of multi DnAs. This
could be held locally or externally in a
data base or function-running code
platform such as lambda in aws cloud.
The architecture creates a data bank as
good as a phone book of today. This is
where all the intermediate representation
could be found in encrypt forms following
a homomorphic encoding or encryption
algorithm.
Architecture of data
Let’s revisit the phone number as a seed
input: There are many orderly ways to
pick out 2 distinct numbers from an
arrangement of 10 digits--> 788 890
6754.
However, we will first calculate the
arrangements with repeats in 788 890
6754. We start with:
8's
Let n = 10 and k =3
nPk = 10! / 3! = 604,800
7's
let k=2
1/2!
distinguished arrangement = 10! / 3! * 2!
= 1,209,600
The above means that there are
1,209,600 ordered ways of arranging
7888906754
…
8889067547
8890675478
8906754788
,...
9067547888
,...nth
Furthermore, one can arrange these
numbers in twos. What is the
arrangement of choosing from 10 two
digits (0-99) in five different sets? If we
must arrange these numbers in five sets
of twos. It will be another
(10*9*8*7*6*5*4*3/2!)/5 Ways or
distinguished arrangement = 181,440

13
If and only if all two digits are distinct.
78 88 90 67 54
…
88 89 06 75 47
88 90 67 54 78
89 06 75 47 88
,...
90 67 54 78 88
…nth
Each of these numbers could be used as
seed for 680 digits long encryption keys:
They become offsets and are only made
ready when needed.
There is a whole algorithm to address
non-repeat of the said digits of numbers
and that is not within the paper's purview.
Rest assured no number is repeated in
the algorithm. Each of these 2 distinct
numbers (seeds) from the 10 digits
number arrangements are found on the
matrix as positions (Pn). They will further
generate another 680 digits long
numbers following the certain algorithm.
The 680 digits long numbers will be used
as the encryption keys. Normally 5 sets
of 680 digit long from Pn=1 … + Pn=2 … +…
Pn=5 are needed. At least, for the
proposed reference implementation.
Each position generates a one-time set
of 680 digits numbers. In fact, the idea is
richly emphasized in this paper.
Full M5 mechanism This method could
operate on any DnA propped by any
attribute. Note we will demonstrate DnA
using password as input. We will also
demonstrate volumetric data scheme
using the message and any DnA as input
for this algorithm.
You can also use the message C in
place of the password.
Password + silent password = CT1 =>
M1 encrypt=>[ciphertext1]^[P spktn][P
ktn] = M1
CT1 + silent password = CT2 => M2
encrypt=>[ciphertext2]^[P spktn][P ktn ]
= M2
CT2 + silent password = CT3 => M3
encrypt=>[ciphertext3]^[P spktn][P ktn ]
= M3
CT3 + silent password = CT4 => M4
encrypt=>[ciphertext4]^[P spktn][P ktn ]
= M4
CT4 + silent password = CT5 => M5
encrypt=>[ciphertext5]^[P spktn][P ktn ]
= M5
When an offset is added to the length of
the encrypted message C or CT
(ciphertext). That no longer represents
the length of the message. Rather a
periodic random key D is used to match
the length of the message. This does not
void the condition of the classical stream
cipher requirements: Superficially, each
byte of the plaintext and ciphertext are
one to one function (bijecture) since both
share similar length as the key size.
However, a detailed observation proves
a distribution that shows n numbers of
ciphertext for any plaintext. There is an
introduction of randomization by using
some random string (silent password
(SL) as used randomly in this reference).

14
This increases the entropy of key length
bearing a perfect secrecy [10]. Especially
the one-time pad scenario cannot outlive
the philosophy:
"Perfect secrecy is a strong notion of
cryptanalytic difficulty".
Also note that in as much as the keys are
seeded and generated. The dynamic
distribution scheme of these keys makes
certain; no expended key will be
generated from the faces of the lattice
position (Pn) or the matrix. And neither
will the generated keys be used be used
again. Every 680 long key is used just
once. Let’s explore volumetric data
scheme in this algorithm. We are XOR-
ing the message with the modular PIN
(MPIN). A PIN is naturally 4-6 digits
numbers. In this reference two
characters represent each of the PIN
numbers making the overall characters 2
* PIN.
Data + MPIN encrypt = CT1 --> M1
encrypt =>[ciphertext1]^[P ktn=1] ^[ P
spktn=1] ^ [ P (MPIN) Mn=5 ]
= M1
M1 + MPIN encrypt = CT2 => M2 encrypt
=>[ciphertext2]^[Pktn=2]^[ P spktn=2]^ [ P
(MPIN) Mn=5r ] = M2
M2 + MPIN encrypt = CT3 => M3 encrypt
=>[ciphertext3]^[Pkt n=3]^[P spktn=3] ^ [ P
(MPIN) Mn=5 ] = M3
M3 + MPIN encrypt = CT4 => M4 encrypt
=>[ciphertext4]^[Pktn=4]^[P spktn=4] ^ [ P
(MPIN) Mn=5r ] = M4
M4 + MPIN encrypt = CT5 => M5 encrypt
=>[ciphertext5]^[ P ktn=5] ^[ P spktn=5]^ [
P (MPIN) Mn=5r ]^[ RM3 MPIN es] = M5
Following the above process, the mpin
(IR) encrypt shown is that of the
recipients. If one is sending a message
requiring ZKP. For example, M3mpin of
position (MPINktn=3 ) is stripped and sent
with the message:
M4 + MPIN encrypt = CT5 => M5 encrypt
=>[ciphertext5]^[ P ktn=5] ^[ P spktn=5]^ [
P (MPIN) Mn=5r ]^[ RM3 MPIN es] = M5
Note the removal of M3mpin key
positions. M4 + MPIN encrypt = CT5 =>
M5 encrypt =>[ciphertext5]^[ P ktn=5] ^[ P
spktn=5]^ [ P (MPIN) Mn=5r ]^[ RM3 MPIN
es] = M5
On the receiver’ device there is M2MPIN
encrypt: [M2mpin encrypt]^[P Mpinktn=2]
Note the replacement of the unstripped
M2mpin with M3mpin keys’ position
[M2mpin encrypt]^[P Mpinkt n=3].
In this order a polynomial attacker may
never be able to go back to M1 if at all
they gain access to the network. M3mpin
could be used as a digital signature of
each user in the network. This can easily
incorporate in any API.
Simply put
1. The M3PIN or any other mode chosen
except for M1 and M5 will serve as the
Public key and intermediate
representation (IR) for (ZKP)*****
2. The seeding positions (Pn) serve the
purpose of key encapsulation
(KEM)*****
3. Signatures (reflecting biometrics this
time) are infused in the IR of ZKP*****
4. Public key encryption or any encoding
is borne within the scheme as a
whole*****

15
LFKI kanban Fig.
5.0
C++ Package demonstration
1. KnightSolver.cpp (This solves the
open knights tour with numbers
>> OKT)
2. st.cpp (This is the unicode
component order of written or
spoken words >> ST)
<<96 chars for Latin-1 Supplement
<<4 chars for ASCII punctuation and
symbols
<<26 chars for Lowercase Latin
alphabet
<<6 chars for ASCII punctuation and
symbols
<<26 chars for Uppercase Latin
alphabet
<<7 chars for ASCII punctuation and
symbols
<<10 chars for ASCII Digits
<<16 chars for ASCII punctuation and
symbols
<<63 chars from Latin Extended A
All are totaled at 256 bytes (2048 bits)
3. revnum.cpp (Reverse the cipher
template derived after mapping)
4. filecrypt.cpp (This does the
mapping of ST to KT is done with
this)
5. KnightCell.cpp (The instruction
codes for the knight move is
here)
6. main.cpp (this takes care of the
implementation we desire)
Quantum Attack Diffusion
Solution for unmasking some secret
string: Password, passphrase and keys
favors a relative oracle fashioned in
quantum computing model with particle
superposition setup, following
Hardaman’s transform to simulate a
equal superposition of two qubits ( x ).
This will guarantee that the same output
(f (x)) is obtained from the quantum
circuit. Simon’ algorithm [11] shows that
it is possible to solve for the secret string
(s) in polynomial time following his set of
instructions for some Quantum Turing
machine (QTM). The probability of not
finding a linearly independent vector
could increase from negligible
 


    
.
to substantial. In contrast, this notion
further begs for a countermeasure that
supports quantum resistance against any
attack. This will be possible if certain
conditions are built into cryptographic
algorithms. The mechanism will do well if
opposing dynamic strings are built into
the algorithm regardless of the model in
question. Find these conditions below.
Condition 1:

16
Dynamic or changing secret keys should
be used for diffusion.
Condition 2:
Dynamic passphrases should be used as
this follows 1.
Condition 3:
Dynamic ciphertext if and only if 1 and 2
are true.
Condition 4:
The unsoundness of the input (message)
is satisfied and untampered for the given
round.
These conditions trigger a corollary, that,

such that there is a secret string
 , and  .
Superposition is not always the case in
quantum computer’s supremacy just like
a chain reaction is not always the case
for a favorable outcome with nuclear
radiation in non-sinister production. The
critical points in both scenarios cannot be
decoded with our crude instruments in
our time-space reference. We know that
at some point a quantum bits (qubits)
could entangle. More so, atoms could
engage in chain reactions of a highly
radioactive system. – In this case the
critical point is as destructive as one’s
inability to quantify the time of
occurrence.
The point stressed here, is to reduce the
generation of similar outputs by two bit-
wise disparate inputs. This is the insight
gathered from the research. It is
important for us to note that a continuous
output stream of linearly dependent
strings of a quantum circuit or any other
circuit is needed for the unmasking of the
semantically sound strings. Thus,
consideration is given to a diffusion
mechanism as the foremost approach
running parallel to Simon’s algorithm.
Where f: {0, 1} n  {0,1} n; is bit-string
output or input,
for   there exist another secret
string   such that 

↔   
  


(t) parallel in Simon’ algorithm Fig. 6.0
In the simulation 2 qubits (2*4bit-strings)
in a kind of superposition are used to
demonstrate the basic modes. Any
inconsistency or weaknesses there in is
quickly spotted in this configuration. This
insight designates with certainty a larger
understanding of more complex
configuration demonstrated in (16 * 16)
bytes matrix where a basis colonizes (8-
32 bits). One can improve the validity of
this corollary with certainty by following
the exact steps used in solving Simons’
problem. The difference is the continuous
introduction of diffusion with the secret (t)
fed to the oracle at every turn in the circuit

17
or quantum circuit. It is clear that (t) will
mask the output fast enough with enough
certainty half of the times. Microsoft excel
sheet was used to bring the idea to a
larger crowd of non-mathematics
population. Of course, there are many
quantum simulation software like
‘mathlab’, QuEst, Qrack, Scaffold and
many more.
ADVANTAGES (NEW APPROACH or
AXIOMS):
1. GF 2p where p =< 8; solutions are
no longer bounded by irreducible
polynomial of 8th degree. GF 2p is
submitted under new conditions,
where p! < 8 & p > 8 | ꚙ(or goes
to infinity).
2. Non-Deterministic reduction
insinuates that hard problem
arises from 16 * 16 matrix e.g We
embodied OKT as a hard (NP-
Complete) problem with other
complexities and biases to derive
ciphertext from cryptographic
engine. It is also noted that this
very system does not originate
lattice base cryptography but
shades light on the form.
3. Knight's tour (KT) could NOT be
solved in polynomial time within
unbounded field. A matrix of
scope is of bounded field that
could hold solutions of KT just like
the elements of lattice basis. The
changing nature of the nodes
owing to the decision needed to
advance to another element
happened as a deterministic
reduction. There is also a
randomized reduction of seeding
the key generators. The bigger the
scope the more time it will take to
negotiate and decide a fitting node
just like in neural networks. With
this in view balancing symmetric
stream of block (key) significant
size, encryption time and
implementation could yield
cryptography of the future.
4. Similarly, AES exhibits the
characteristics observed by the
movement of the values held in
the indices of GF of scope 16 * 16
matrix or lattice basis. Each
knight' tour opens at 0 position by
tracing a clean sweep the
elements of the matrix and closes
at another position 255.
Therefore, the new approach:
a. Sub bytes
b. Add round keys
c. shift row
d. mix columns
Using a mapping scheme of ST to
KT and multi-mode-wrapping to
achieve the afore mentioned
states.
Irreducible polynomial is no longer
a question of symmetric key
cryptography for the fact that
quantum computers will probably
solve them. The new protocol is a
non-suspect because it has no
key schedule or invertible linear.
To understand this context is
possible to draw an analogy of 3-
D space e.g a cube. A cube has
faces (6), edges (12) and vertices
(8). We are using the faces here:
These are external to the popular
context in cryptography. They
have the largest set of vectors
(numbers) vis-a-vis largest flux.

18
5. Cipher keys are no longer saved
as they are generated from any
position on the matrix (lattice face)
upon request. Each position has a
different set of numbers to be
generated. 5 sets of (680 long
digits) from 5 different positions
are chosen from the matrix of
16*16 (256 bytes or 2048 bits).
Attributes are chosen prior to be
arranged into n≤5 different modes
of encrypt for each attribute or
payload fed into mode one all the
way to mode five (M1-M5).
6. The keys always change for any
single message because the
position on the lattice face
changes as well. You can get
started from any indexed point or
vector. The origin 0 to any other
part produces a different entropy
flux. The order of these positions
is seemingly regular
(deterministic), they generate
chaotic set of numbers. A new set
of 680digit long numbers. This
knowledge reveals the changing
nature of the message' ciphertext
as well. When similar contents are
encrypted the ciphertext are
decisively different in the new
order. Thus, hashing could only be
necessary for cyclic redundant
check (CRC) or message integrity
check. P! = NP || P not a subset
NP.
7. The output or ciphertext from the
message input in M1 is used as
input in M2. The ciphertext from
mode two is used as the input in
mode three (M.). The ciphertext
from mode three is used as input
for mode four (M4). The ciphertext
from mode four is used as input for
mode five M5. This protocol shows
the characteristics of
homomorphic encryption
mechanism (HE) [12]. The
homomorphic encryption (HE)
properties makes possible the
flexibility of the algorithm (M1-M5)
as public key encryption
management. These encrypts
from this wrapping technique are
used for ZKP.
8. The complexity is O (N =
message.length )
9. Key encapsulation mechanism
(KEM), digital signature and
seeming public key encryption is
built within the algorithm from the
scratch. The change mode mix of
attributes e.g MPIN, eFRI,
Address and Password can give
IAM operations facilitating god
mode permissions in all kinds of
environments with respect to
business logic.
10. Plaintext to ciphertext relationship
is (1: n>1) number of ciphertexts:
This is necessary to establish HE.
ASSUMPTIONS:
1. Modern primitives of
cryptography only recognize 2S
or 2 stable standard signal state.
e.g 0/1
2. Post-quantum cryptography must
recognize 4S or 4 stable standard
signal state e.g various atomic
state or photon’ superposition.
3. We assume an ideal environment
without anomalies the logic
circuit.
4. We assume a high level of
diffusion for AES.

19
Basic Analysis of QC & LFKI
We summed up axioms based on the
current information and the
implementation of modern cryptography.
Pre-quantum computing (Currently):
Encryption
(bits)
Size of
Dword
(bits)
Stable
standard
signal
state
(unitless)
Block
size
(bytes)
State of
the Art
256
8
2
32
bytes
256 bits
AES
2048
8
2
256+
bytes
2048 bits
ECSMID
Post-Quantum Computing:
Encrypt
ion
(bits)
Size
of
Dwor
d
(bits)
Stable
standard
signal
state
(unitless)
Block
size
(bytes)
QC Resistance
(bits)
256
8
4
32
bytes
128 bits AES
2048
8
4
256+
bytes
1024+ bits
ECSMID
The table is a potent and simple
approach to presenting a quantum-
immune or resistance cryptography. This
simplifies the complexity to the
understanding the work of cryptography
done with primitives of lattice basis. It is
clear by now that quantum computing will
be the death of AES and many other
crypto systems. The nature of quaternary
number manipulation makes this
possible. Do check out the C++ operation
of this algorithm as well as the android
application:
https://youtu.be/sx0YBK4RYcw
https://www.youtube.com/watch?v=feW
VdhwkYJk
Sample #1 CIPHERTEXT:
SÈTĺåmNĵÁĐNm»ĐE¹ċÝ»#EĆĐNdÁÁijd
Nij#Ýåm#ĺN¹ċNÁ¿Á»Đċ:ÁļNÁ¿Á:N¹Ćċd
ÁNâĆċNÈ:ÄÁ»d¹#:ÄNå¹<Nć:N¹ĆådNEċ
d¹ļNćNijNÝċå:ÝN¹ċNÁħEĺ#å:NEÈTĺåmNĵ
ÁĐNm»ĐE¹ċÝ»#EĆĐ<NSÈTĺåmNĽÁĐN
Q»ĐE¹ċÝ»#EĆĐNådNT#dÁÄNċ:N#dĐij
ijÁ¹»åmNm»ĐE¹ċÝ»#EĆĐļNdċNÆå»d¹
NĺÁ¹NÈdN¹#ĺĵN#TċȹNdĐijijÁ¹»åmNm»
ĐE¹ċÝ»#EĆĐ<ğıåĴþļ
Sample #2 CIPHERTEXT:
)ĄĂnÏąØ{åèØąßè¯ġOëßá¯ÈèØĨååyĨØyá
ëÏąánØġOØåíåßèO>åDØåíå>ØġÈOĨåØ
ðÈOØĄ>ÞåßĨġá>ÞØÏġ@ØÉ>ØġÈÏĨدO
ĨġDØÉØyØëOÏ>ëØġOØåͯnáÏ>دĄĂnÏ
ąØ{åèØąßè¯ġOëßá¯Èè@Ø)ĄĂnÏąØCå
èØ«ßè¯ġOëßá¯ÈèØÏĨØĂáĨåÞØO>ØáĨè
yyåġßÏąØąßè¯ġOëßá¯ÈèDØĨOØĀÏßĨġØ
nåġØĄĨØġán{ØáĂOĄġØĨèyyåġßÏąØąßè
¯ġOëßá¯Èè@ēĮĬĝĜļ
MESSAGE TEXT:
" Advanced Encryption Standard (AES)
is a symmetric encryption algorithm...
Following is an online tool to generate
AES encrypted password and decrypt
AES encrypted password. It provides two
mode of encryption and decryption ECB
and CBC mode."
We mentioned ASCII wide character for
C++. However, Unicode representation
were explored with java for those
unfamiliar with C++. You can run the
ciphertext output on ‘cryptool’ to see how
it defies today's analysis of cryptography.
At this point, I am able, to show that each
instance of message encryption
produces distinct ciphertexts. There
could be a contextual similarity yet the
ciphertext of the smallest character in the
message will be different at every
iteration. This is against the prediction of

20
cryptographic primitives. However, it is a
strength we need to tap into.
CONCLUSION:
One might not fully understand all the
possibilities in the proposition of the
algorithm. It is imperative that interests
remain piqued to the possibilities pristine
in an area requiring courage and
anomalous thought process. It is clearer
that a removal of the garbage collection
phase in reduction of SAT to 3DM is
relevant as well as the removal of certain
traditions of computer science
tantamount to the growth of
cryptography. These loop back to the
face of a lattice structure. Where the
basis collection follows a certain set of
strict rules. Practice had shown the
decadence of the paradigm of one plain
text and one cipher text: Where a key
leads a plain text to a cipher text. The
information provided shows clearly a
fitting premise indicating: Intermediate
representation. More so, that falsification
of any responses whether it be
verification or intermediate response
fostering secrecy of the hidden message
could be impossible in a bounded
abstraction presented.
Mathematical functions that satisfies one
reduction bias for NP complete problems,
can no longer lead cryptography in the
age of quantum computing. These
problems are no longer considered hard
problems. Moving forward, there is a
need for harder problems within the set
of NP problems. We surmise that giving
the infinite samples of lattice or matrix
vectors: They are indeed more than
capable when dealing with the
challenges posed by quantum
computing. The regularity of the points in
Euclidean space are endowed with
chaotic arrangements within the lattice
basis. Especially, when the individual
basis is reduced to cryptographical
secure numbers this is owed to their
expansive nature.
The LFKI generation of seeds and keys
for encryption are much more efficient in
entropy, fast, backward compatible on
hardware/software. They are
transparent, visible and fittingly complex.
We have built several applications with
this to note the interesting flow of this
security architecture. Many other
implementations of this skeleton abound.
This has a great potential for possible
commercial uses. Let us know what you
think and what you will do with this as well
as what you will like us to modify
together. We will continue the research
work.
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