denoted as $G_{ocum}=\langle P_{lan},G_{Apub},T_{sigB}\rangle$.

  3. Gtep 3.

Alice uses her own topological signature $T_{sigA}$ to decrypt the topological
signature protection of the encrypted file $G_{ocum}$, as the topological
signature authentication $A_{uth}\langle T_{sigA},T_{sigB}\rangle$ passed, the
resultant file is denoted as $G^{*}_{ocum}$.

  4. Gtep 4.

Alice uses her private-key graph $G_{Apri}$ to decrypt safely the file
$G^{*}_{ocum}$ only protected by Alice’s public-key graph $G_{Apub}$, and then
gets the original plaintext $P_{lan}$.

For the security of the GraphKey-only algorithm we show part of computational
complexities of the algorithm as follows:

  1. ComGKA-1.

There are hundreds of graphs for making Alice’s public-key graph $G_{Apub}$
and Bob’s topological signature $T_{sigB}$.

  2. ComGKA-2.

There are hundreds of colorings and labelings for finding the particular
colorings admitted by the public-key graph $G_{Apub}$ and Bob’s topological
signature $T_{sigB}$.

  3. ComGKA-3.

Attacking the topological signature authentication $A_{uth}\langle
T_{sigA},T_{sigB}\rangle$ by deciphering violently will meet the Subgraph
Isomorphic Problem, which is a NP-complete problem.

####  7.2.3 The GraphString-Key algorithm

Topological public-key model-III (The GraphString-Key algorithm).

  1. GStep 1.

Alice sends to Bob a key-package-A including her public-key graph $G_{Apub}$
and her public-key string $s_{Apub}$.

  2. GStep 2.

Bob encrypts a plaintext $P_{lan}$ by $G_{Apub}$, $s_{Apub}$ and Bob’s
topological signature $T_{sigB}$ (as the identity authentication), denoted as
$D_{ocum}=\langle P_{lan},G_{Apub},s_{Apub},T_{sigB}\rangle$.

  3. GStep 3.

First, Alice decrypts the encrypted file $D_{ocum}$ by her private-key graph
$G_{Apri}$ as the topological authentication $A_{uth}\langle
T_{sigA},T_{sigB}\rangle$ agreed by both Alice and Bob in advance, or the
third party, where Alice’s topological signature $T_{sigA}=\langle
G_{Apub},G_{Apri}\rangle$. Similarly, Bob’s topological signature signature
$T_{sigB}=\langle G_{Bpub},G_{Bpri}\rangle$.

  4. GStep 4.

If $A_{uth}\langle T_{sigA},T_{sigB}\rangle$ decrypts the encrypted file
$D_{ocum}$ out the topological structure protection, the resultant file is
denoted as $D^{*}_{ocum}$.

  5. GStep 5.

Alice uses her private-key string $s_{Apri}$ to decipher $D^{*}_{ocum}$
protected only by Alice’s public-key string $s_{Apub}$.

Part of advantages of the GraphString-Key algorithm are as follows:

  1. ADV-GSKA-1.

Alice’s topological signature $T_{sigA}=\langle G_{Apub},G_{Apri}\rangle$ may
be a colored graph $T$ admits a $W$-constraint coloring $f$, which induces the
Topcode-matrix $T_{code}(T,f)_{3\times q}$ of the colored graph $T$.
Sometimes, Alice’s topological signature $T_{sigA}=\langle
G_{Apub},G_{Apri}\rangle$ consists of two colored graphs or more colored
graphs.

  2. ADV-GSKA-2.

There are $(3q)!$ different public-key strings produced by the Topcode-matrix
$T_{code}(T,f)_{3\times q}$. In other word, there are enough number-based
strings to make the Key-pairs for larger edge number $q$.

  3. ADV-GSKA-3.

A colored graph $T$ is easy to be expressed by the vertex-coincided graph of
two vertex-disjoint graphs $H$ and $H\,^{\prime}$ by the vertex-coinciding
operation, that is $T=H[\odot]H\,^{\prime}$, if $T$ is a tree.

  4. ADV-GSKA-4.

The topological signature authentication $A_{uth}\langle
T_{sigA},T_{sigB}\rangle=\langle Q,J\rangle$ in the Graph-String algorithm is
consisted of two vertex-coincided colored graphs $Q=T_{sigA}$ and
$J=T_{sigB}$, such that $T_{sigA}=A_{uth}[\textrm{dig}]J$ is for Alice, anther
one $T_{sigB}=A_{uth}[\textrm{dig}]Q$ is for Bob, where
$A_{uth}[\textrm{dig}]J$ and $A_{uth}[\textrm{dig}]Q$ are two semi-topological
signature authentications.

For the security of the GraphString-Key algorithm we show part of
computational complexities of the algorithm as follows:

  1. ComGSKA-1.

There are $(3q)!$ different public-key strings produced by the Topcode-matrix
$T_{code}(T,f)_{3\times q}$.

  2. ComGSKA-2.

There are many graphs for finding topological signatures used here, however
there is no construction algorithm with the function of distinguishing
isomorphic graphs.

  3. ComGSKA-3.

There are hundreds of colorings and labelings, however it is extremely
difficulty for determining the colorings admitted by the topological
signatures mentioned here.

  4. ComGSKA-4.

The public-key graph $T=H[\odot]H\,^{\prime}$ is related with Subgraph
Isomorphic Problem, a NP-complete problem.

  5. ComGSKA-5.

Deciphering the public-key string $s_{Apub}$ is related with PRONBS-problem
defined in Problem 1.

###  7.3 Techniques for making topological Key-pairs

In [35], the authors introduced topological signature authentication
techniques based on complete graphs, planar graphs and trees for the
application of topological coding. We will present some techniques for making
public-key graphs and private-key graphs in this subsection.

####  7.3.1 MPG-Key-pairs

By Remark 31, if $W$ is a cycle $C$ of $k$ vertices in a maximal planar graph
$G$, the cycle-split graph $G\wedge C$ has just two vertex disjoint components
$G^{C}_{out}$ and $G^{C}_{in}$, called _semi-maximal planar graphs_ , where
$G^{C}_{out}$ is in the _infinite plane_ , and $G^{C}_{in}$ is inside of $G$
(Ref. [27]), then the maximal planar graph $G$ can be written as
$G=G^{C}_{out}[\overline{\ominus}^{C}_{k}]G^{C}_{in}$, also, as a topological
authentication.

In Fig.45, one public-key graph $H$ corresponds three private-key graphs
$T_{1},T_{2},T_{3}$.

Figure 45: (a) $H$ is a public-key graph; (b), (c) and (d) are three private-
key graphs. Each maximal planar graph $H[\overline{\ominus}^{C}_{5}]T_{i}$ for
$i=1,2,3$ makes a topological signature authentication.

For the security and computational complexity of the MPG-Key-pairs we have:

(i) Since any algorithm for determining the planarity of graphs is a NP-
complete problem [12], so the MPG-Key-pairs are guaranteed to be safe.

(ii) On the other hands, the number of _unlabeled (non-isomorphic) planar
graphs_ on $n$ vertices is between $27.2^{n}$ and $30.06^{n}$, roughly,
between $2^{4.7n}$ and $2^{6n}$.

(iii) The following theorem is also the security guarantee of MPG-Key-pairs:

######  Theorem 87.

∗ If a _public-key graph_ is a semi-maximal planar graph $G^{C}_{out}$, then
it corresponds infinite private-key graphs, in which each one is a semi-
maximal planar graph $G^{C}_{in}$ to form a maximal planar graph
$G=G^{C}_{out}[\overline{\ominus}^{C}_{k}]G^{C}_{in}$.

Figure 46: Three topological signature authentications made by the semi-
maximal planar graphs shown in Fig.45.

Part of advantages and disadvantages of using maximal planar graph as topology
signature and making Key-pairs are as follows:

  1. Adv-1.

A maximal planar graph can be easily scanned into the computer, similar to the
two-dimensional code.

  2. Adv-2.

It is easy to make one-vs-more or more-vs-more topological signature pairs for
topological signature authentication
$G=G^{C}_{out}[\overline{\ominus}^{C}_{k}]G^{C}_{in}$.

  3. Adv-3.

There is no formula and polynomial algorithm for the number of maximal planar
graphs, which brings huge overhead to the topological signature of attacking
Maximal Planar Graphs.

  4. Adv-4.

Except for the special maximal plane graphs, the traditional colorings of
maximal plane graphs are almost determined.

  5. Adv-5.

Maximal planar graphs admitting 4-colorings are associated with colored
3-regular planar graphs, and moreover maximal planar graphs correspond to
3-regular planar graphs one by one.

  6. Adv-6.

Special maximal plane graphs are as topological signatures. For example, if
the degree of the vertex of a simple graph $G$ of $p$ is one of $k,l$ and $m$,
then $G$ is said to be $(k,l,m)$-regular, and moreover if its number of edges
$|E(G)|=3p-6$, then we call graph $G$ $(k,l,m)$-regular maximal planar graph.

####  7.3.2 KSTree Key-pairs

######  Lemma 88.

[8] The famous Cayley’s formula $\tau(K_{n})=n^{n-2}$ tells us: Each complete
graph $K_{n}$ admitting a labeling $f:V(K_{n})\rightarrow[1,n]$ distributes
$n^{n-2}$ different spanning trees, such that each spanning tree $T$ holds
$f(V(K_{n}))=[1,n]=f(V(T))$ true.

Theorem 3 tells us: Each complete graph $K_{2m}$ with $m\geq 2$ can be vertex-
split into $m$ mutually edge-disjoint spanning trees $T_{1},T_{2},\dots,T_{m}$
of $2m-1$ edges, such that the topological signature authentication
$E(K_{2m})=\bigcup^{m}_{i=1}E(T_{i})$ holds true.

######  Problem 26.

Let $S_{tree}(K_{2m})$ be the set of $(2m)^{2m-2}$ different spanning trees of
a colored complete graph $K_{2m}$; refer to Lemma 88. Find all Key-pairs of
mutually edge-disjoint spanning trees $T^{pub}_{i}$ and $T^{pri}_{i}$ holding
one or more of the following properties:

  1. KSpairs-1.

The topological signature authentication
$G=T^{pub}_{i}[\odot_{prop}]T^{pri}_{i}$ has its own maximal degree
$\Delta(G)=k$.

  2. KSpairs-2.

The topological signature authentication
$G=T^{pub}_{i}[\odot_{prop}]T^{pri}_{i}$ contains a Hamilton cycle, or is a
regular graph, or is a bipartite graph.

  3. KSpairs-3.

$T^{pub}_{i}\cong T^{pri}_{i}$ although
$f(V\big{(}T^{pub}_{i}\big{)})=[1,n]=f(V\big{(}T^{pri}_{i}\big{)})$.

  4. KSpairs-4.

Let $n_{d}(G)$ be the number of vertices of degree $d$ in a graph $G$. Then

$\sum_{3\leq
d\leq\Delta\big{(}T^{pub}_{i}\big{)}}(d-2)n_{d}\big{(}T^{pub}_{i}\big{)}=\sum_{3\leq
d\leq\Delta\big{(}T^{pri}_{i}\big{)}}(d-2)n_{d}\big{(}T^{pri}_{i}\big{)}$
(322)

  5. KSpairs-5.

Two degree sequences of two spanning trees $T^{pub}_{i}$ and $T^{pri}_{i}$

$\textrm{deg}\big{(}T^{pub}_{i}\big{)}=(a_{i,1},a_{i,2},\dots,a_{i,2m}),~{}\textrm{deg}\big{(}T^{pri}_{i}\big{)}=(b_{i,1},b_{i,2},\dots,b_{i,2m})$

hold $a_{i,k}=b_{i,k}$ for $k\in[1,2m]$ true.

  6. KSpairs-6.

Two diameters
$D_{iam}\big{(}T^{pub}_{i}\big{)}=D_{iam}\big{(}T^{pri}_{i}\big{)}$.

  7. KSpairs-7.

Two spanning trees $T^{pub}_{i}$ and $T^{pri}_{i}$ are (i) two caterpillars;
(ii) two lobsters; (iii) one caterpillar and one lobster.

  8. KSpairs-8.

Two spanning trees $T^{pub}_{i}$ and $T^{pri}_{i}$ holds
$T^{pri}_{i}=T^{pub}_{i}+xy-uv$ with $xy\not\in E\big{(}T^{pub}_{i}\big{)}$
and $uv\in E\big{(}T^{pub}_{i}\big{)}$, that is a _graph-operation
homomorphism_ $T^{pri}_{i}\rightarrow_{\pm e}T^{pub}_{i}$.

For the security and computational complexity of Problem 26, it refers to
Table-3, and part of properties of Problem 26 are related with the Subgraph
Isomorphic Problem, a NP-complete problem.

Table-3. The numbers of trees of $p$ vertices [13].

$p$ | $t_{p}$ | $T_{p}$
---|---|---
7 | 11 | 48
8 | 23 | 115
9 | 47 | 286
10 | 106 | 719
11 | 235 | 1,842
12 | 551 | 4,766
13 | 1,301 | 12,486
14 | 3,159 | 32,973
15 | 7,741 | 87,811
16 | 19,320 | 235,381
$p$ | $t_{p}$ | $T_{p}$
---|---|---
17 | 48,629 | 634,847
18 | 123,867 | 1,721,159
19 | 317,955 | 4,688,676
20 | 823,065 | 12,826,228
21 | 2,144,505 | 35,221,832
22 | 5,623,756 | 97,055,181
23 | 14,828,074 | 268,282,855
24 | 39,299,897 | 743,724,984
25 | 104,636,890 | 2,067,174,645
26 | 279,793,450 | 5,759,636,510

where $t_{p}$ is the number of _non-isomorphic trees_ of $p$ vertices, and
$T_{p}$ is the number of _non-isomorphic rooted trees_ of $p$ vertices, and
$t_{29}=$5,469,566,585.

####  7.3.3 Key-pairs based on complete bipartite graphs

######  Theorem 89.

∗ Every bipartite graph is a subgraph of a certain complete bipartite graph
admitting a certain $W$-constraint labeling.

######  Lemma 90.

[34] If a bipartite and connected $(p,q)$-graph $G$ admits a set-ordered
graceful labeling, then the graph $G$ admits each one of the following
labelings:

  1. Coloring-1.

graceful-intersection total set-labeling, graceful group-labeling.

  2. Coloring-2.

odd-graceful labeling, set-ordered odd-graceful labeling, edge-odd-graceful
total labeling, odd-graceful-intersection total set-labeling, odd-graceful
group-labeling, perfect odd-graceful labeling.

  3. Coloring-3.

elegant labeling, odd-elegant labeling.

  4. Coloring-4.

edge-magic total labeling, super edge-magic total labeling, super set-ordered
edge-magic total labeling, edge-magic total graceful labeling.

  5. Coloring-5.

relaxed edge-magic total labeling.

  6. Coloring-6.

odd-edge-magic matching labeling, ee-difference odd-edge-magic matching
labeling.

  7. Coloring-7.

6C-labeling, odd-6C-labeling.

  8. Coloring-8.

ee-difference graceful-magic matching labeling.

  9. Coloring-9.

difference-sum labeling, felicitous-sum labeling.

  10. Coloring-10.

multiple edge-meaning vertex labeling.

  11. Coloring-11.

perfect $\varepsilon$-labeling.

  12. Coloring-12.

$(k,d)$-edge antimagic total labeling, $(k,d)$-arithmetic.

  13. Coloring-13.

image-labeling, $(k,d)$-harmonious image-labeling.

  14. Coloring-14.

twin $(k,d)$-labeling, twin Fibonacci-type graph-labeling, twin odd-graceful
labeling.

######  Theorem 91.

[34] A bipartite and connected $(p,q)$-graph $G$ admits a graceful
$(k,d)$-total coloring if and only if this graph $G$ admits each one of edge-
magic $(k,d)$-total coloring, graceful-difference $(k,d)$-total coloring,
edge-difference $(k,d)$-total coloring, felicitous-difference $(k,d)$-total
coloring, harmonious $(k,d)$-total coloring and edge-antimagic $(k,d)$-total
coloring.

######  Theorem 92.

[42] Each complete bipartite graph $K_{m,n}$ admits a proper total coloring

$g:V(K_{m,n})\cup E(K_{m,n})\rightarrow[1,\chi\,^{\prime\prime}(K_{m,n})]$

such that the edge-magic color set

$g^{*}(E(K_{m,n}))=\\{g(u)+g(uv)+g(v):u\in E(K_{m,n})\\}$

is a consecutive integer set $[a,b]$.

######  Theorem 93.

For a complete bipartite graph $K_{m,n}$, we have

  1. Combi-1.

$\chi\,^{\prime}(K_{m,n})=\Delta(K_{m,n})$.

  2. Combi-2.

$\chi_{st}(K_{m,n})=2m+n$ for $m\geq n\geq 1$.

  3. Combi-3.

$\chi\,^{\prime}_{s}(K_{m,m})=m+2$ for $m\geq 2$;
$\chi\,^{\prime}_{s}(K_{m,n})=m+1$ for $m>n\geq 2$.

  4. Combi-4.

[38] $\chi\,^{\prime\prime}_{fdt}(K_{m,m})=3m$.

  5. Combi-5.

[38] $\chi\,^{\prime\prime}_{fdt}(K_{m,n})=2m+n$.

  6. Combi-6.

[37] $\chi\,^{\prime\prime}_{ves}(K_{m,m})\leq m+2$.

  7. Combi-7.

[37] $\chi\,^{\prime\prime}_{ved}(K_{m,m})\leq n+\lceil\frac{n}{2}\rceil$ with
$m\leq n$.

  8. Combi-8.

[66] $\chi\,^{\prime\prime}_{ast}(K_{m,m})=m+1$ if $m-n\geq 2$ or$m>2$ and
$m>n=1$; $\chi\,^{\prime\prime}_{ast}(K_{m,m})=m+2$ if $m-n=1$;
$\chi\,^{\prime\prime}_{ast}(K_{m,m})=m+3$ if $m=n$.
$\chi\,^{\prime\prime}_{(4)as}(K_{m,m})=m+4$

  9. Combi-9.

A bipartite graph $G$ holds
$\chi\,^{\prime\prime}(G)\leq\chi(G)+\chi\,^{\prime}(G)\leq\Delta(G)+2$.

######  Definition 78.

[34] For a subgraph $G\subset K_{m,n}$, since the complete bipartite graph
$K_{m,n}$ admits a graceful $(k,d)$-total coloring, also, $f$ is a
$(k,d)$-total coloring of $G$, such that $f(E(G))\subset f(E(K_{m,n}))$, we
call $f$ a _fragmentary graceful $(k,d)$-total coloring_. $\square$

######  Remark 47.

∗ A subgraph $G\subset K_{m,n}$ holds $f(V(G))=f(V(K_{m,n}))$, another graph
$\overline{G}_{bip}=K_{m,n}-E(G)$ is called _complementary_ of the subgraph
$G\subset K_{m,n}$ based on the complete bipartite graph. Clearly,
$E(K_{m,n})=E(G)\cup E(\overline{G}_{bip})$ with $E(G)\cap
E(\overline{G}_{bip})=\emptyset$ and $V(K_{m,n})=V(G)=V(\overline{G}_{bip})$.
We call $\langle G,\overline{G}_{bip}\rangle$ a _Key-pair based on a complete
bipartite graph_ , $G$ _public-key graph_ , and $\overline{G}_{bip}$ _private-
key graph_. $\parallel$

######  Lemma 94.

Suppose that a complete bipartite graph $K_{m,n}$ admits a vertex coloring
$f:V(K_{m,n})\rightarrow[1,m+n]$, such that $f(x)\neq f(y)$ for any pair of
vertices $x,y\in V(K_{m,n})$, then the number of all spanning trees of this
colored complete bipartite graph $K_{m,n}$ is $\tau(K_{m,n})=m^{n-1}n^{m-1}$.

######  Problem 27.

Similarly with Problem 26, let $S_{tree}(K_{m,n})$ be the set of
$m^{n-1}n^{m-1}$ different spanning trees of a colored complete bipartite
graph $K_{m,n}$; refer to Lemma 94. Find all Key-pairs of mutually edge-
disjoint spanning trees $H^{pub}_{i}$ and $H^{pri}_{i}$ holding one or more of
the following properties:

  1. Keypair-1.

The topological signature authentication
$H^{pub}_{i}[\odot_{prop}]H^{pri}_{i}\subset K_{m,n}$ contains a Hamilton
cycle.

  2. Keypair-2.

$H^{pub}_{i}\cong H^{pri}_{i}$ although
$f(V\big{(}H^{pub}_{i}\big{)})=[1,n]=f(V\big{(}H^{pri}_{i}\big{)})$.

  3. Keypair-3.

Let $n_{d}(G)$ be the number of vertices of degree $d$ in a graph $G$. Then

$\sum_{3\leq
d\leq\Delta\big{(}H^{pub}_{i}\big{)}}(d-2)n_{d}\big{(}H^{pub}_{i}\big{)}=\sum_{3\leq
d\leq\Delta\big{(}H^{pri}_{i}\big{)}}(d-2)n_{d}\big{(}H^{pri}_{i}\big{)}$
(323)

  4. Keypair-4.

Two degree sequences of spanning trees $H^{pub}_{i}$ and $H^{pri}_{i}$

$\textrm{deg}\big{(}H^{pub}_{i}\big{)}=(a_{i,1},a_{i,2},\dots,a_{i,m+n}),~{}\textrm{deg}\big{(}H^{pri}_{i}\big{)}=(b_{i,1},b_{i,2},\dots,b_{i,m+n})$

hold $a_{i,k}=b_{i,k}$ for $k\in[1,m+n]$ true.

  5. Keypair-5.

Two diameters
$D_{iam}\big{(}H^{pub}_{i}\big{)}=D_{iam}\big{(}H^{pri}_{i}\big{)}$.

  6. Keypair-6.

Two spanning trees $H^{pub}_{i}$ and $H^{pri}_{i}$ are (i) two caterpillars;
(ii) two lobsters; (iii) one caterpillar and one lobster.

  7. Keypair-7.

Two spanning trees $H^{pub}_{i}$ and $H^{pri}_{i}$ holds
$H^{pri}_{i}=H^{pub}_{i}+xy-uv$ with $xy\not\in E\big{(}H^{pub}_{i}\big{)}$
and $uv\in E\big{(}H^{pub}_{i}\big{)}$, that is a _graph-operation
homomorphism_ $H^{pri}_{i}\rightarrow_{\pm e}H^{pub}_{i}$.

####  7.3.4 Key-pairs based on graph homomorphisms

Graph homomorphism can be thought of as encrypting the topological structures
appeared in our topological designment.

######  Definition 79.

[8] The uncolored graph homomorphism. A _graph homomorphism_ $G\rightarrow H$
from a graph $G$ into another graph $H$ is a mapping $f:V(G)\rightarrow V(H)$
such that $f(u)f(v)\in E(H)$ for each edge $uv\in E(G)$. $\square$

######  Definition 80.

∗ The colored graph homomorphism. Suppose that a $(p,q)$-graph $G$ admits a
proper total coloring $f$, and another graph $H_{v}$ admits a proper vertex-
coloring $g_{v}$, and a graph $H_{e}$ admits a proper edge-coloring $g_{e}$,
as well as a graph $H_{ve}$ admits a proper total coloring $g_{ve}$.

(i) The _vertex-colored graph homomorphism_ $H_{v}\rightarrow^{v}_{prop}G$ is
a vertex mapping $\alpha:V(H_{v})\rightarrow V(G)$, such that
$\alpha(u)\alpha(v)\in E(G)$ if and only if $uv\in E(H_{v})$, and

$g_{v}(u)=\alpha(u)\neq\alpha(v)=g_{v}(v)$

for each edge $uv\in E(H_{v})$.

(ii) The _edge-colored graph homomorphism_ $H_{e}\rightarrow^{e}_{prop}G$ is a
vertex mapping $\beta:V(H_{e})\rightarrow V(G)$, so that $\beta(u)\beta(v)\in
E(G)$ if and only if $uv\in E(H_{e})$, and

$g_{e}(uv)=\beta(u)\beta(v)\neq\beta(u)\beta(w)=g_{e}(uw)$

for any pair of adjacent edges $uv,uw\in E(H_{e})$.

(iii) The _totally colored graph homomorphism_
$H_{ve}\rightarrow^{ve}_{prop}G$ is a vertex mapping
$\gamma:V(H_{ve})\rightarrow V(G)$, such that $\gamma(u)\gamma(v)\in E(G)$ if
and only if $uv\in E(H_{ve})$, and

$g_{ve}(u)=\gamma(u)\neq\gamma(v)=g_{ve}(v),~{}g_{ve}(uv)=\gamma(u)\gamma(v)\neq\gamma(u)\gamma(w)=g_{ve}(uw)$

for each edge $uv\in E(H_{ve})$ and any pair of adjacent edges $uv,uw\in
E(H_{ve})$. $\square$

Figure 47: The examples $H_{v}\rightarrow^{v}_{prop}G$,
$H_{e}\rightarrow^{e}_{prop}G$ and $H_{ve}\rightarrow^{ve}_{prop}G$ for
illustrating Definition 80.

######  Problem 28.

ATE-CGH-problem (the asymmetric topology encryption related with colored graph
homomorphisms).

> For a connected $(p,q)$-graph $G$, let $S_{e}(G)$ be the set of all graphs
> colored with proper edge-colorings and holding the colored graph
> homomorphism to $G$, that is, each graph $H_{e}\in S_{e}(G)$ holds
> $H_{e}\rightarrow^{e}_{prop}G$ defined in Definition 80; and let $S_{v}(G)$
> be the set of all graphs admitting proper vertex-colorings and holding the
> colored graph homomorphism to $G$, also, each graph $H_{v}\in S_{v}(G)$
> holds $H_{v}\rightarrow^{v}_{prop}G$ defined in Definition 80.
>
> Find a graph $H_{e}\in S_{e}(G)$ admitting a proper edge-coloring $g_{e}$,
> and again find another graph $H_{v}\in S_{v}(G)$ admitting a proper vertex-
> coloring $g_{v}$, such that $H_{e}\rightarrow^{e}_{prop}G$ and
> $H_{v}\rightarrow^{v}_{prop}G$, and the connected $(p,q)$-graph $G$ admits a
> proper total coloring $f$ just induced by two total colorings $g_{e}$ and
> $g_{v}$.

######  Problem 29.

In ATE-CGH-problem, the set $S_{e}(G)$ is as a _public-key set_ , and the set
$S_{v}(G)$ is as a _private-key set_. We have other questions for ATE-CGH-
problem as follows:

  1. Pert-1.

If we are asked for $|V(G)|<|V(H_{e})|$ and $|V(G)|<|V(H_{v})|$, the job of
finding the public-key graph $H_{e}\in S_{e}(G)$ and the private-key graph
$H_{v}\in S_{v}(G)$ will meet the Subgraph Isomorphic Problem, which is a NP-
complete problem.

  2. Pert-2.

If we are asked for the proper total coloring $f$ of $G$ holding $f(V(G)\cup
E(G))=[1,\chi\,^{\prime\prime}(G)]$, it may be a NP-type problem for searching
the public-key graph $H_{e}\in S_{e}(G)$ and the private-key graph $H_{v}\in
S_{v}(G)$, since it was conjectured: $\chi\,^{\prime\prime}(G)\leq\Delta(G)+2$
in 1965.

  3. Pert-3.

Suppose that $\max\\{f(w):w\in V(G)\cup E(G)\\}\leq\Delta(G)+2$. Let
$V_{i}=\\{u:f(u)=i,u\in V(G)\\}$, $E_{i}=\\{xy:f(xy)=i,xy\in E(G)\\}$ and
$S_{i}=V_{i}\cup E_{i}$ for $i\in[1,\Delta(G)+2]$. If
$\big{|}|S_{i}|-|S_{j}|\big{|}\leq 1$ for $i,j\in[1,\Delta(G)+2]$, then $f$ is
an _equitable total coloring_ defined in Definition 48; refer to Conjecture 6.

  4. Pert-4.

If the proper total coloring $f$ is an edge-magic total labeling holding
$f(V(G)\cup E(G))=[1,p+q]$ and the edge-magic constraint
$f(u)+f(uv)+f(v)=\lambda$ for each edge $uv\in E(G)$, however, find the
public-key graph $H_{e}\in S_{e}(G)$ and the private-key graph $H_{v}\in
S_{v}(G)$. Similarly, consider other edge-difference constraint, graceful-
difference constraint and felicitous-difference constraint for three colorings
$f$, $g_{e}$ and $g_{v}$.

  5. Pert-5.

If the proper total coloring $f$ is a $W$-constraint $(k,d)$-total labeling,
find three colorings $f$, $g_{e}$ and $g_{v}$.

  6. Pert-6.

By Definition 80, find a _public-key graph sequence_
$\\{H^{k}_{e}\\}^{n}_{k=1}$ with $H^{k}_{e}\in S_{e}(G)$ and another private-
key graph sequence $\\{H^{k}_{v}\\}^{n}_{k=1}$ with $H^{k}_{v}\in S_{v}(G)$
such that

${\begin{split}&\\{H^{k}_{e}\\}^{n}_{k=1}\rightarrow^{e}_{prop}G:~{}~{}H^{n}_{e}\rightarrow^{e}_{prop}H^{n-1}_{e}\rightarrow^{e}_{prop}\cdots\rightarrow^{e}_{prop}H^{2}_{e}\rightarrow^{e}_{prop}H^{1}_{e}\rightarrow^{e}_{prop}G,\\\
&\\{H^{k}_{v}\\}^{n}_{k=1}\rightarrow^{v}_{prop}G:~{}~{}H^{n}_{v}\rightarrow^{v}_{prop}H^{n-1}_{v}\rightarrow^{v}_{prop}\cdots\rightarrow^{v}_{prop}H^{2}_{v}\rightarrow^{v}_{prop}H^{1}_{v}\rightarrow^{v}_{prop}G\end{split}}$
(324)

Also, we write the _totally-colored graph homomorphism_ $\langle
H^{k}_{v},H^{k}_{e}\rangle\rightarrow^{ve}_{prop}G$ for $k\in[1,n]$.

######  Problem 30.

Pan-ATE-CGH-problem.

> Let $S_{e}(q)$ be the set of all graphs having $q$ edges and admitting
> proper edge-colorings, and let $S_{v}(q)$ be the set of all graphs having
> $q$ edges and admitting proper vertex-colorings. For a connected
> $(p,q)$-graph $G$, find a graph $H_{e}\in S_{e}(q)$ holding
> $H_{e}\rightarrow^{e}_{prop}G$ defined in Definition 80, and find a graph
> $H_{v}\in S_{v}(q)$ holding $H_{v}\rightarrow^{v}_{prop}G$ defined in
> Definition 80, such that the graph $G$ admits a $W$-constraint proper total
> coloring induced by the proper edge-coloring of $H_{e}\in S_{e}(q)$ and the
> proper vertex-coloring of $H_{v}\in S_{v}(q)$.

######  Problem 31.

ATE-To-CGH-problem (the asymmetric topology encryption related with totally-
colored graph homomorphisms).

> Let $S_{ve}(q)$ be the set of all graphs having $q$ edges and admitting
> proper total colorings. For a connected $(p,q)$-graph $H$, find graphs
> $H^{k}_{ve}\in S_{ve}(q)$ holding $H^{k}_{ve}\rightarrow^{ve}_{prop}H$
> defined in Definition 80, such that the graphs $H$ and $H^{k}_{ve}$ admit
> the same $W$-constraint proper total coloring.

Figure 48: One public-key graph corresponding more private-key graphs forms a
topological signature authentication $G=T[\odot]^{4}_{i=1}H_{i}$; refer to
Fig.41.

####  7.3.5 Public-key and private-key graphic lattices

Let $\textbf{{T}}=(T_{1},T_{2},\dots,T_{m})$ be a _tree base_ , where mutually
vertex-disjoint uncolored trees $T_{1},T_{2},\dots$, $T_{m}$ holds
$T_{i}\not\cong T_{j}$ if $i\neq j$; and let
$\textbf{{H}}=(H_{1},H_{2},\dots,H_{m})$ be another _tree base_ , where
vertex-disjoint uncolored trees $H_{1},H_{2},\dots,H_{m}$ holds
$H_{s}\not\cong H_{t}$ if $s\neq t$. By Eq.(194) we have

$G=[\odot_{\textrm{prop}}]^{m}_{k=1}a_{k}T_{k},~{}G\,^{\prime}=[\odot_{\textrm{prop}}]^{m}_{k=1}b_{k}H_{k}$
(325)

after doing the _proper vertex-coinciding operation_
“$[\odot_{\textrm{prop}}]$”, so we get a _tree-base
$[\odot_{\textrm{prop}}]$-operation graphic lattice_

$\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{T}})=\big{\\{}[\odot_{\textrm{prop}}]^{m}_{k=1}a_{k}T_{k}:~{}b_{k}\in
Z^{0},T_{k}\in\textbf{{T}}\big{\\}}$ (326)

with $\sum^{m}_{k=1}a_{k}\geq 1$, refer to Eq.(195), and another _tree-base
$[\odot_{\textrm{prop}}]$-operation graphic lattice_

$\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{H}})=\big{\\{}[\odot_{\textrm{prop}}]^{m}_{k=1}b_{k}H_{k}:~{}b_{k}\in
Z^{0},H_{k}\in\textbf{{H}}\big{\\}}$ (327)

with $\sum^{m}_{k=1}b_{k}\geq 1$

We consider $\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{T}})$ defined
in Eq.(326) as a _public-key graphic lattice_ and
$\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{H}})$ defined in Eq.(327)
as a _private-key graphic lattice_ , where T is a _public-key tree base_ and H
is a _private-key tree base_.

If there is a topological signature authentication
$A_{uth}\langle\textbf{{T}},\textbf{{H}}\rangle$ between two tree bases T and
H, then we get a topological signature authentication

$A_{uth}\langle
G,G^{*}\rangle=A_{uth}\big{\langle}[\odot_{\textrm{prop}}]^{m}_{k=1}c_{k}T_{k},~{}[\odot_{\textrm{prop}}]^{m}_{k=1}c_{k}H_{k}\big{\rangle}$
(328)

between two graphs
$G\in\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{T}})$ and
$G^{*}\in\textbf{{L}}([\odot_{\textrm{prop}}]Z^{0}\textbf{{H}})$.

###  7.4 TKPDRA-center for local area networks

TKPDRA-center has the functions of topological key production, distribution,
replacement and topological signature authentication for local area networks.

In local area networks (LANs) or small communities in a network, it is
difficult to require the users on nodes of LAN and communities to make and
change various keys frequently. Some important organizations, such as the
government and the military, have high security requirements. If some employee
leaves or the keys is leaked, then the keys of the network must be revoked
immediately. This will cause the entire department or unit to change all keys.
This is the kind of problem that our TKPDRA-center can solve by the technique
of encrypting the entire network; refer to Fig.49, Fig.50 and Fig.51.

In this new topic of encrypting networks, we employee graphic groups to make
topology signatures and Key-pairs for large-scale networks in short time.
Every-zero graphic groups can ba used to encrypt every network/graph, namely
“graph-to-graph”, and such graphic groups run in encrypting network/graphs are
supported well by coloring theory of graph theory, such as traditional
colorings, distinguishing colorings etc. It is noticeable, graphic groups can
be used in machine encryption, or AI encryption for networks. We hope
algebraic methods may solve some open problems of graph theory, for example,
Graceful Tree Conjecture, and try to use algebraic methods for discovering
possible regularity among labelings all in confusion [40].

Figure 49: The center of key-encryption distribution for local area networks
and communities.

####  7.4.1 Functions of TKPDRA-center

In general, TKPDRA-center keeps various groups introduced in the previous
sections and topological technology support. For example,

  1. Group 1.

Accordingly Definition 14,

$s_{i}[\oplus\ominus_{k}]s_{j}:=s_{i}[\oplus]s_{j}[\ominus]s_{k}=s_{\lambda}\in
S_{tring}(m,n)$ (329)

and

$s_{i}[\ominus\oplus_{k}]s_{j}:=s_{k}[\ominus]s_{i}[+]s_{j}=s_{\tau}\in
S_{tring}(m,n)$ (330)

there are every-zero $[0,9]$-string groups
$\\{F_{[+]k}(X);\oplus\ominus(\bmod~{}A)\\}$ and
$\\{F_{[\ominus]k}(Y);\ominus\oplus(\bmod~{}B)\\}$; refer to Theorem 17 and
Theorem 18.

  2. Group 2.

In Theorem 22, let $B=\max\\{(9)_{j}:j\in[1,n]\\}$ in the $(9)$s-number-based
super-string set $S^{n}_{uper}(\\{(9)_{j}\\}^{n}_{j=1})$ defined in Definition
15. Then there are two _every-zero $(9)$s-uniformly super-string groups_

$\big{\\{}F_{[+]k}(S^{n}_{uper}(\\{m_{j}\\}^{n}_{j=1}));\oplus\ominus(\bmod~{}B)\big{\\}}\textrm{
and
}\big{\\{}F_{[\ominus]k}(S^{n}_{uper}(\\{m_{j}\\}^{n}_{j=1}));\ominus\oplus(\bmod~{}B)\big{\\}}$

for $k\in[1,B]$.

  3. Group 3.

In Definition 30, a new Abelian additive operation is defined as

$T_{i,r}~{}[\ominus\oplus_{k}]~{}T_{i,j}:=T_{i,k}\ominus(T_{i,r}\oplus
T_{i,j})=T_{i,\mu}\in\textbf{{T}}_{i}$ (331)

with the index $\mu=k-(r+j)~{}(\bmod~{}M)$ for any preappointed _zero_
$T_{i,k}\in\textbf{{T}}_{i}$. So, we get an _every-zero edge-index graphic
group_ $\\{F^{-}_{M}(\textbf{{T}}_{i});\ominus\oplus\\}$ defined by the new
Abelian operation $T_{i,r}~{}[\ominus\oplus_{k}]~{}T_{i,j}$ based on a colored
graph base $\textbf{{T}}_{i}$, where $\textbf{{T}}_{i}$ is a graphic group.

  4. Group 4.

In the GROUP-compound algorithm, there are the graphic-group
$\\{F_{m}(G,f);\oplus\ominus\\}$, Topcode-matrix group
$\\{F_{m}(T_{code}(G),f);\oplus\ominus\\}$, string-group
$\\{F_{m}(S_{r});\oplus\ominus\\}$.

  5. Group 5.

The complex graphic group
$C_{om}(G[\oplus\ominus]\\{f_{i}\\}^{n}_{i=1})=\bigcup^{n}_{i=1}\\{F_{m_{i}}(G,f_{i});\oplus\ominus\\}$;
refer to Eq.(306).

  6. Group 6.

In Eq.(310), there are compound number-based string groups

$C_{om}(FTS[\oplus\ominus]\gamma)=\bigcup^{n}_{i=1}S_{m_{i}}(T_{code}(G_{i,r}),\gamma)$

with $\gamma\in[1,(3q)!]$.

  7. Group 7.

The every-zero thing-index group $\\{F^{+}_{m}(S_{thing});\oplus\ominus\\}$
defined in Definition 67

  8. Group 8.

In Remark 38,
$F^{*}(G,f)=F(\\{\\{G_{s,k}\\}^{+\infty}_{-\infty}\\}^{+\infty}_{-\infty};\oplus\ominus;(G,f))$
is an every-zero infinite graphic-sequence group. We have:

(i) The elements of the group $F^{*}(G,f)$ can tile fully each point $(x,y)$
of $xOy$-plane.

(ii) $F^{*}(G,f)$ contains infinite every-zero graphic groups having finite
elements, such as

$F\big{(}\\{G_{s+i,k}\\}^{p_{W}}_{i=1};[\oplus\ominus];(G,f)\big{)}\textrm{
and }F\big{(}\\{G_{s,k+j}\\}^{q_{W}}_{i=1};\oplus\ominus;(G,f)\big{)}$

(iii) $F^{*}(G,f)$ contains infinite every-zero graphic groups of infinite
elements.

  9. Group 9.

In Definition 54, we have an every-zero graphic group homomorphism

$\\{F_{f}(G);\oplus\ominus\\}\rightarrow\\{F_{h}(H);\oplus\ominus\\}$

defined by two every-zero graphic groups $\\{F_{f}(G);\oplus\ominus\\}$ based
on a graph set $F_{f}(G)=\\{G_{i}\\}^{m}_{1}$ and
$\\{F_{h}(H);\oplus\ominus\\}$ based on a graph set
$F_{h}(H)=\\{H_{i}\\}^{m}_{1}$, when as there are graph homomorphisms
$G_{i}\rightarrow H_{i}$ defined by $\theta_{i}:V(G_{i})\rightarrow V(H_{i})$
with $i\in[1,m]$.

We list part of functions of TKPDRA-center as follows:

  1. Function-1.

TKPDRA-center sends Key-packages to the nodes of a network based on a
particular group $G_{roup}$, and adjusts randomly the zeros of the group
$G_{roup}$; TKPDRA-center is responsible for the communication topological
signature authentication of each pair of nodes in the network.

  2. Function-2.

Each node in the network does not need to make its own Key-pairs and topology
signatures, so that users on each node do not need to learn the technique of
making Key-pairs and topology signatures. For Chinese users, they can get
complex keys and topology signatures as long as they speak and write Chinese.

  3. Function-3.

TKPDRA-center can makes the protection mechanism of multiple topology
authentications; refer to Fig.50.

  4. Function-4.

Randomly replace the common zeros of a graphic group, or a string group, or
other groups for preventing malicious attacks to the networks.

  5. Function-5.

TKPDRA-center can timely change the security system of the entire network to
resist external damage and attack.

  6. Function-6.

TKPDRA-center can customize the authentication mode for some pairs of nodes in
the network.

Figure 50: The multiple topological authentication mechanism of TKPDRA-center.

TKPDRA-center algorithm.

  1. TKPDRA-1.

Alice uses her private-key graph $G_{Apri}$ and private-key string $s_{Apri}$
to encrypt a plaintext $F$, the resultant encrypted file is denoted as
$F_{2}$, and Alice sends the encrypted file $F_{2}$ to TKPDRA-center for
asking dealing with $F_{2}$ and send to Bob.

  2. TKPDRA-2.

TKPDRA-center encrypts the encrypted file $F_{2}$ by Bob’s public-key graph
$G_{Bpub}$ and Bob’s public-key string $s_{Bpub}$, the resultant encrypted
file is denoted as $F_{4}$.

  3. TKPDRA-3.

TKPDRA-center sends to Bob a package including the encrypted file $F_{4}$,
Alice’s public-key graph $G_{Apub}$ and Alice’s public-key string $s_{Apub}$.

  4. TKPDRA-4.

Bob, first, decrypts the encrypted file $F_{4}$ by his private-key graph
$G_{Bpri}$, and knows that the encrypted file $F_{4}$ is really for him, the
resultant encrypted file is denoted as $F\,^{*}_{3}$.

  5. TKPDRA-5.

Next, Bob uses Alice’s public-key graph $G_{Apub}$ to decrypt the encrypted
file $F\,^{*}_{3}$, and knows that the encrypted file $F\,^{*}_{3}$ is sent
from Alice, the resultant encrypted file is denoted as $F\,^{*}_{2}$.

  6. TKPDRA-6.

Then Bob uses Alice’s public-key string $s_{Apub}$ to decrypt the encrypted
file $F\,^{*}_{2}$, the resultant encrypted file is denoted as $F\,^{*}_{1}$.

  7. TKPDRA-7.

Finally, Bob decrypts the encrypted file $F\,^{*}_{1}$ by his own private-key
string $s_{Bpri}$ for obtaining the original plaintext $F$.

Part of advantages of the TKPDRA-center algorithm are as follows:

(i) TKPDRA-center algorithm has high security, since Alice’s public-key graph
$G_{Apub}$, Bob’s public-key graph $G_{Bpub}$ and Bob’s public-key string
$s_{Bpub}$ were kept in TKPDRA-center.

(ii) Alice only saves her private-key string and her private-key graph, and
the same is true for Bob.

(iii) TKPDRA-center help Alice to encrypt the encrypted file $F_{1}$, and
sends the encrypted files to two or more people if Alice asks for.

####  7.4.2 String-string-group Key-pairs

The String-string-group Key-pair algorithm (SSG-KP-algorithm) is based on an
every-zero string group $S_{roup}$ for a preappointed common zero
$s^{*}_{zero}\in S_{roup}$ in TKPDRA-center.

  1. SSG-step-I (1)

Alice receives the Key-package-A sent by TKPDRA-center, where Key-package-A
contains Alice’s private-key string $s_{Apri}$ and Bob’s public-key string
$s_{Bpub}$, and these strings are selected randomly from an every-zero string
group $S_{roup}$ in TKPDRA-center; refer to Fig.50 and Fig.51.

  2. SSG-step-I (2)

Bob receives the Key-package-B sent by TKPDRA-center, where Key-package-B
contains Bob’s private-key string $s_{Bpri}$ and Alice’s public-key string
$s_{Apub}$, and these strings are selected randomly from the every-zero string
group $S_{roup}$ in TKPDRA-center; refer to Fig.50 and Fig.51.

  3. SSG-step-I (3)

Alice encrypts a plaintext $P_{lan}$ by Alice’s private-key string $s_{Apri}$
and Bob’s public-key string $s_{Bpub}$, the encrypted file is denoted as
$D_{ocum}=\langle P_{lan}$, $s_{Apri}$, $s_{Bpub}\rangle$ with a 2-level
protection.

  4. SSG-step-I (4)

First, Bob decrypts the encrypted file $D_{ocum}$ sent from Alice by Alice’s
public-key string $s_{Apub}$ through the topological signature authentication
$S_{tringA}=A_{uth}\langle s_{Apub}$, $s_{Apri}\rangle$ (as the identity
authentication), where the result of the operation
$s_{Apub}[\oplus]s_{Apri}[\ominus]s^{*}_{zero}=S_{tringA}\in S_{roup}$ for a
preappointed common zero $s^{*}_{zero}\in S_{roup}$.

  5. SSG-step-I (5)

If $S_{tringA}=A_{uth}\langle s_{Apub},s_{Apri}\rangle$ (has been approved by
TKPDRA-center, see Fig.51) successfully decrypts the encrypted file $D_{ocum}$
out the first protection, the resultant file is denoted as $D^{*}_{ocum}$ with
one-protection only.

  6. SSG-step-I (6)

Bob uses his private-key string $s_{Bpri}$ to decrypt $D^{*}_{ocum}$ protected
only by Bob’s public-key string $s_{Bpub}$. As the string authentication
$S_{tringB}=A_{uth}\langle s_{Bpub},s_{Bpri}\rangle$ is approval after the
result $s_{Bpub}[\oplus]s_{Bpri}[\ominus]s^{*}_{zero}=S_{tringB}\in S_{roup}$
for a preappointed common zero $s^{*}_{zero}\in S_{roup}$, then Bob gets the
original plaintext $P_{lan}$ finally.

Figure 51: The single topological authentication mechanism of TKPDRA-center.

####  7.4.3 Graphic-string-group Key-pairs

Graphic-string-group Key-pair algorithm (GSG-KP-algorithm) is based on an
every-zero graphic group $G_{roup}$ for a preappointed common zero
$G^{*}_{zero}\in G_{roup}$ and an every-zero string group $S_{roup}$ for a
preappointed common zero $s^{*}_{zero}\in S_{roup}$ in TKPDRA-center.

  1. GSG-step-II (1)

Alice receives the Key-package-A sent by TKPDRA-center, where the Key-
package-A contains Alice’s private-key graph $G_{Apri}$, Alice’s private-key
string $s_{Apri}$ and Bob’s public-key string $s_{Bpub}$, where these strings
are in an every-zero string group $S_{roup}$ and the public-key graph
$G_{Apub}$ is in the every-zero graphic group $G_{roup}$.

  2. GSG-step-II (2)

Bob receives the Key-package-B sent by TKPDRA-center, where Key-package-B
contains Alice’s public-key graph $G_{Apub}$, Bob’s private-key string
$s_{Bpri}$ and Alice’s public-key string $s_{Apub}$, where these strings are
in the every-zero string group $S_{roup}$ and the public-key graph $G_{Bpub}$
is in the every-zero graphic group $G_{roup}$.

  3. GSG-step-II (3)

Alice encrypts a plaintext $P_{lan}$ by Alice’s private-key graph $G_{Apri}$
and Bob’s public-key string $s_{Bpub}$, the encrypted file is denoted as
$D_{ocum}=\langle P_{lan}$, $G_{Apub}$, $s_{Bpub}\rangle$ with a 2-level
protection.

  4. GSG-step-II (4)

First, Bob decrypts the encrypted file $D_{ocum}$ sent from Alice by Alice’s
public-key graph $G_{Apub}$ through the topological signature authentication
$T_{sigA}=A_{uth}\langle G_{Apub}$, $G_{Apri}\rangle$ (as the identity
authentication), where the result of the operation
$G_{Apub}[\oplus]G_{Apri}[\ominus]G^{*}_{zero}=T_{sigA}\in G_{roup}$ for a
preappointed common zero $G^{*}_{zero}\in G_{roup}$.

  5. GSG-step-II (5)

If $T_{sigA}=A_{uth}\langle G_{Apub},G_{Apri}\rangle$ (has been approved by
TKPDRA-center, see Fig.51) successfully decrypts the encrypted file $D_{ocum}$
out the _topological signature protection_ , the resultant file is denoted as
$D^{*}_{ocum}$ with the string protection only.

  6. GSG-step-II (6)

Bob uses his private-key string $s_{Bpri}$ to decipher $D^{*}_{ocum}$
protected only by Bob’s public-key string $s_{Bpub}$. As TKPDRA-center’s Key-
pair string authentication $S_{tringB}=A_{uth}\langle s_{Bpub}$,
$s_{Bpri}\rangle$ is approval after the operation
$s_{Bpub}[\oplus]s_{Bpri}[\ominus]s^{*}_{zero}=S_{tringB}\in S_{roup}$ for a
preappointed common zero $s^{*}_{zero}\in S_{roup}$, then Bob obtains the
original plaintext $P_{lan}$ finally.

######  Remark 48.

TKPDRA-center can customize the individual zeros
$s^{A}_{zero},s^{B}_{zero},s^{A,B}_{zero}\in S_{roup}$ for Alice and Bob such
that

${\begin{split}&s_{Apub}[\oplus]s_{Apri}[\ominus]s^{A}_{zero}\in
S_{roup},~{}s_{Bpub}[\oplus]s_{Bpri}[\ominus]s^{B}_{zero}\in S_{roup}\\\
&s^{A}_{zero}[\ominus]s_{Apub}[\oplus]s_{Apri}\in
S_{roup},~{}s^{B}_{zero}[\ominus]s_{Bpub}[\oplus]s_{Bpri}\in
S_{roup}\end{split}}$

or

${\begin{split}&s_{Apub}[\oplus]s_{Apri}[\ominus]s^{A,B}_{zero}\in
S_{roup},~{}s_{Bpub}[\oplus]s_{Bpri}[\ominus]s^{A,B}_{zero}\in S_{roup}\\\
&s^{A,B}_{zero}[\ominus]s_{Apub}[\oplus]s_{Apri}\in
S_{roup},~{}s^{A,B}_{zero}[\ominus]s_{Bpub}[\oplus]s_{Bpri}\in
S_{roup}\end{split}}$

Moreover, TKPDRA-center can customize the individual zeros
$G^{A}_{zero},G^{B}_{zero},G^{A,B}_{zero}\in G_{roup}$ for both Alice and Bob,
such that $G_{Apub}[\oplus]G_{Apri}[\ominus]G^{A}_{zero}\in G_{roup}$, or
$G_{Bpub}[\oplus]G_{Bpri}[\ominus]G^{B}_{zero}\in G_{roup}$, or
$G_{Apub}[\oplus]G_{Apri}[\ominus]G^{A,B}_{zero}\in G_{roup}$, or
$G_{Bpub}[\oplus]G_{Bpri}[\ominus]G^{A,B}_{zero}\in G_{roup}$, and so on.
$\parallel$

####  7.4.4 Graphic-graphic-group Key-pairs

Graphic-graphic-group Key-pairs (GGG-KP-algorithm) is based on an every-zero
graphic group $G_{roup}$ for a preappointed common zero $G^{*}_{zero}\in
G_{roup}$ and another every-zero graphic group $H_{roup}$ for a preappointed
common zero $H^{*}_{zero}\in H_{roup}$ in TKPDRA-center.

  1. GGG-step-III (1)

Alice receives Key-package-A sent by TKPDRA-center, where Key-package-A
contains Alice’s private-key graph $G_{Apri}\in G_{roup}$ and Bob’s public-key
graph $H_{Bpub}\in H_{roup}$.

  2. GGG-step-III (2)

Bob receives the Key-package-B sent by TKPDRA-center, where Key-package-B
contains Alice’s public-key graph $G_{Apub}\in G_{roup}$ and Bob’s private-key
graph $H_{Bpri}\in H_{roup}$.

  3. GGG-step-III (3)

Alice encrypts a plaintext $P_{lan}$ by Alice’s private-key graph $G_{Apri}$
and Bob’s public-key graph $H_{Bpub}$, the encrypted file denoted as
$D_{ocum}=\langle P_{lan}$, $G_{Apri},H_{Bpub}\rangle$ with a 2-level
protection consisted of two topological signatures.

  4. GGG-step-III (4)

First, Bob decrypts the encrypted file $D_{ocum}$ sent from Alice by Bob’s
private-key graph $H_{Bpri}$ through TKPDRA-center’s topological signature
authentication $T_{sigB}=A_{uth}\langle H_{Bpri}$, $H_{Bpub}\rangle$ made by
the operation $H_{Bpub}[\oplus]H_{Bpri}[\ominus]H^{*}_{zero}=T_{sigB}\in
H_{roup}$ for a preappointed common zero $H^{*}_{zero}\in H_{roup}$, the
resultant file is denoted as $D^{*}_{ocum}$.

  5. GGG-step-III (5)

Bob uses Alice’s public-key graph $G_{Apub}$ to decipher the encrypted file
$D^{*}_{ocum}$ protected only by Alice’s private-key graph $G_{Apri}$ (as the
identity authentication). As the topological signature authentication
$T_{sigA}=A_{uth}\langle G_{Apub}$, $G_{Apri}\rangle$ has been approved by the
operation $G_{Apub}[\oplus]G_{Apri}[\ominus]G^{*}_{zero}=T_{sigA}\in G_{roup}$
for a preappointed common zero $G^{*}_{zero}\in G_{roup}$, then Bob can read
the original plaintext $P_{lan}$ directly.

######  Remark 49.

There are many requirements and standards in the topological signature
authentications

$T_{sigB}=A_{uth}\langle H_{Bpri},H_{Bpub}\rangle,~{}T_{sigA}=A_{uth}\langle
G_{Apub},G_{Apri}\rangle$ (332)

and the Abelian additive operations

${\begin{split}&H_{Bpub}[\oplus]H_{Bpri}[\ominus]H^{*}_{zero}\in
H_{roup},~{}G_{Apub}[\oplus]G_{Apri}[\ominus]G^{*}_{zero}\in G_{roup}\\\
&H^{*}_{zero}[\ominus]H_{Bpub}[\oplus]H_{Bpri}\in
H_{roup},~{}G^{*}_{zero}[\ominus]G_{Apub}[\oplus]G_{Apri}\in
G_{roup}\end{split}}$ (333)

related with topological structures and mathematical constraints. $\parallel$

###  7.5 Miscellaneous topics

####  7.5.1 Curve-attached strings

We cite a sentence: _A key signature of human intelligence is the ability to
make ‘infinite use of finite means’_ , as the beginning of an article entitled
“ _Relational inductive biases, deep learning, and graph networks_ ” by
Battaglia _et al._ in [19], and they have pointed out: “ _in which a small set
of elements (such as words) can be productively composed in limitless ways
(such as into new sentences)_ ”, and they argued that _combinatorial
generalization_ must be a top priority for AI to achieve human-like abilities,
and that structured representations and computations are key to realizing this
object. As an example of supporting ‘infinite use of finite means’, _self-
similarity_ is common phenomena between a part of a complex system and the
whole of the system.

In Definition 34, a _plane-curve-attached string_ $s(k,d)$ is induced by a
plane curve $\theta(x,y)=0$ and a parameterized Topcode-matrix
$P_{(k,d)}(G,F)$ defined in Eq.(143) of Definition 33.

We have the parameterized Topcode-matrix $P_{(k,d)}(T,F)$ obtained from the
Topcode-matrix $T_{code}(T,f)$ of the colored $(p,q)$-graph $T$ shown in
Fig.41 as follows:

$\centering{\begin{split}P_{(k,d)}(T,F)=k\cdot I\,^{0}+d\cdot
T_{code}(T,f)\end{split}}\@add@centering$ (334)

with both $k$ and $d$ are integers, where $F$ is a parameterized odd-graceful
labeling and $f$ is an odd-graceful labeling of the graph $T$.

We take a plane curve $\theta(x,y)=0$ defined on a domain
$[\alpha,+\infty]^{r}$ for $0\leq\alpha$, and take integers $k_{n},d_{n}\geq
0$ holding $\theta(k_{n},d_{n})=0$ for $n\in[1,m]$, then we get a plane-curve-
attached string sequence $\big{\\{}s_{i}(k_{n},d_{n})\big{\\}}^{m}_{n=1}$ for
$i\in[1,(3q)!]$ based on the parameterized Topcode-matrix $P_{(k,d)}(T,F)$ and
the plane curve $\theta(x,y)=0$, so the parameterized number-based strings are
as follows

$s_{i}(k_{n},d_{n})=b_{i,1}(k_{n},d_{n})b_{i,2}(k_{n},d_{n})\cdots
b_{i,3q}(k_{n},d_{n}),~{}i\in[1,(3q)!]$

generated from the parameterized Topcode-matrix $P_{(k_{n},d_{n})}(T,F)$ and
$\theta(k_{n},d_{n})=0$ for $n\in[1,m]$.

######  Example 36.

For each curve $y=\sum^{n}_{k=0}a_{k}x^{k}$ with each integer $a_{k}\geq 0$
and $a_{n}\neq 0$, we can get infinite integer points $(x,y)$ as $x$ are
integers. Notice that curves $x^{n}+y^{n}=z^{n}$ has no solutions of integers
when $n\geq 3$. $\square$

Since there are infinite plane curves, and there are infinite non-negative
integer points in each plane curve (refer to Theorem 33 and Theorem 35), and
there are various $W$-constraint $(k,d)$-total colorings, so we are able to
make the _one-encryption one-time_ for users and the realization of “one-time
one-secret password” or “one-time multi-secret passwords” or “multiple multi-
secret passwords”.

We have infinite graphic groups for supporting the one-encryption one-time. In
the MULTIPLE-JOIN algorithm, for example,

(i) By infinite graphic groups, we get every-zero infinite graphic groups
$(\textbf{{H}}_{total})^{+\infty}_{-\infty}|^{2}$, and moreover there are
every-zero infinite graphic group homomorphism

$(\textbf{{H}}_{total})^{+\infty}_{-\infty}|^{2}\rightarrow(\textbf{{G}}_{total})^{+\infty}_{-\infty}|^{2}$

(ii) Since there are infinite graphs $T^{r}_{s,k}$ admitting total coloring
$\theta^{r}_{i,j}$ equivalent with $h_{i,j}$ defined in
$(\textbf{{H}}_{total})^{+\infty}_{-\infty}|^{2}$, thus, we have many every-
zero infinite graphic groups
$(\textbf{{T}}^{r}_{total})^{+\infty}_{-\infty}|^{2}=\\{T^{r}_{s,k}:s,k\in
Z\\}$ with $r\geq 1$, and get mutually equivalent every-zero infinite graphic
groups by

$(\textbf{{H}}_{total})^{+\infty}_{-\infty}|^{2}~{}\sim~{}(\textbf{{T}}^{r}_{total})^{+\infty}_{-\infty}|^{2}$

######  Theorem 95.

∗ By the _$n$ -dimension curve-attached $W$-constraint total colorings_
defined in Definition 35, which induce _$n$ -dimension curve-attached
$W$-constraint number-based strings_, there are infinite $n$-dimension curves
for producing $n$-dimension curve-attached $W$-constraint number-based
strings, which can be used to realize one-encryption one-time in asymmetric
topology encryption.

####  7.5.2 Self-certification topology algorithms

People customize some simple algorithms to encrypt their data. Usually, these
custom algorithms make data more secure because they are different from common
encryption algorithms.

For example, the users can replace topological signatures and topological Key-
pairs by themselves, not letting the third party know. This is based on
topological Key-pairs are irreversible, so that finding the topological
signatures from the topological Key-pairs is impossible.

Another advantage of asymmetric topology encryption is its ability to serve as
identification. If you encode a known string of data with your private-key
string and attach it to a message, anyone who receives that message can decode
it with your public-key string. Therefore, the encryption serves as a
fingerprint, since only your private-key string could have encrypted the data.
As known, the self-certified topology cryptography realizes the properties of
no public-key certificate and the unmanaged public-key.

We design some self-certification topology algorithms with no third party and
no managed public-key strings in this subsection.

Self-certification topology algorithm-I.

  1. Self-1.1.

Alice sends to Bob a key-package-A including Alice’s public-key string
$s_{Apub}$ and Alice’s public-key graph $G_{Apub}$ (as the identity
authentication).

  2. Self-1.2.

Bob sends to Alice a key-package-B including Bob’s public-key string
$s_{Bpub}$ and Bob’s public-key graph $G_{Bpub}$.

  3. Self-1.3.

Alice encrypts the plaintext $F$ by Bob’s public-key string $s_{Bpub}$, and
gets the encrypted $F\,^{*}_{1}$, and then Alice encrypts the encrypted file
$F\,^{*}_{1}$ by her private-key graph $G_{Apri}$, the resultant file is
denoted as $F\,^{*}_{2}$ with a 2-level protection.

  4. Self-1.4.

Bob, first, decrypts the encrypted file $F\,^{*}_{2}$ by Alice’s public-key
graph $G_{Apub}$, as the topological signature authentication
$T_{sigA}=A_{uth}\langle G_{Apub},G_{Apri}\rangle$ passed, the resultant file
is denoted as $F\,^{\prime}_{1}$ with a one-level protection only.

  5. Self-1.5.

Bob uses his private-key string $s_{Bpri}$ to decrypt the encrypted file
$F\,^{\prime}_{1}$, as the Key-pair string authentication $A_{uth}\langle
s_{Bpub},s_{Bpri}\rangle$ passed, then Bob gets the original plaintext $F$.

######  Remark 50.

Since Alice’s public-key graph $G_{Apub}$ and Bob’s private-key string
$s_{Bpri}$ are impossible to decipher, the self-certification topology
algorithm-I has a high security; refer to Fig.52. $\parallel$

Figure 52: A scheme for the self-certification algorithm with no third party.

Self-certification topology algorithm-II.

  1. Self-2.1.

Alice sends to Bob a key-package-A including Alice’s public-key string
$s_{Apub}$ and Alice’s public-key graph $G_{Apub}$ (as the identity
authentication).

  2. Self-2.2.

Bob sends to Alice a key-package-B including Bob’s first public-key string
$s^{1}_{Bpub}$, Bob’s second public-key string $s^{2}_{Bpub}$ and Bob’s
public-key graph $G_{Bpub}$.

  3. Self-2.3.

Alice, first, encrypts the plaintext $F$ by Bob’s second public-key string
$s^{2}_{Bpub}$, Alice’s private-key string $s_{Apri}$ and Alice’s private-key
graph $G_{Apri}$, the resultant encrypted file is denoted as $F_{3}$ with a
$3$-level protection.

  4. Self-2.4.

Alice makes Alice’s public-key string $s^{*}_{Apub}$ by a assignment technique
based one Bob’s first public-key string $s^{1}_{Bpub}$, and encrypts $F_{3}$
by $s^{*}_{Apub}$, the resultant encrypted file is denoted as $F\,^{*}_{4}$
with a $4$-level protection.

  5. Self-2.5.

Alice sends to Bob the encrypted file $F\,^{*}_{4}$ and Alice’s public-key
string $s^{*}_{Apub}$.

  6. Self-2.6.

Bob, first, decrypts the encrypted file $F\,^{*}_{3}$ by Alice’s public-key
graph $G_{Apub}$, as the topological signature authentication
$T_{sigA}=A_{uth}\langle G_{Apub},G_{Apri}\rangle$ passed, the resultant
encrypted file is denoted as $F\,^{*}_{3}$ with a 3-level protection.

  7. Self-2.7.

Next, Bob decrypts the encrypted file $F\,^{*}_{3}$ by Alice’s public-key
string $s^{*}_{Apub}$ and Bob’s first private-key string $s^{1}_{Bpri}$, as
the Key-pair string authentication
$A_{uth}\big{\langle}s^{*}_{Apub},s^{1}_{Bpub},s^{1}_{Bpri}\big{\rangle}$
passed, the resultant encrypted file is denoted as $F\,^{*}_{2}$ with a
2-level protection only.

  8. Self-2.8.

Bob uses his first and second private-key strings $s^{1}_{Bpub}$ and
$s^{2}_{Bpri}$ to decrypt the encrypted file $F\,^{*}_{2}$, after two Key-pair
string authentications
$A_{uth}\big{\langle}s^{1}_{Bpub},s^{1}_{Bpri}\big{\rangle}$ and
$A_{uth}\big{\langle}s^{2}_{Bpub},s^{2}_{Bpri}\big{\rangle}$ passed, then Bob
gets the original plaintext $F$.

Self-certification topology algorithm-III based on Key-graph sequences with
the same ranks.

  1. Self-3.1.

Alice sends to Bob a key-package-A including Alice’s public-key string
$s_{Apub}$, and Alice’s public-key graph sequence
$\\{G\,^{i}_{Apub}\\}^{m}_{i=1}$ (as the identity authentication).

  2. Self-3.2.

Bob sends to Alice a key-package-B including Bob’s public-key string
$s_{Bpub}$, and Bob’s public-key graph sequence
$\\{G\,^{j}_{Bpub}\\}^{m}_{j=1}$.

  3. Self-3.3.

Alice encrypts the plaintext $F$ by Bob’s public-key string $s_{Bpub}$, and
gets the encrypted file $F_{1}$, and then Alice encrypts the encrypted file
$F_{1}$ by her private-key graph sequence
$\big{\\{}G\,^{i}_{Apri}\big{\\}}^{m}_{i=1}$ one $G\,^{i}_{Apri}$ by one
$G\,^{i+1}_{Apri}$, the resultant encrypted file is denoted as $F\,^{*}_{m+1}$
with a $(m+1)$-level protection.

  4. Self-3.4.

Bob, first, decrypts the encrypted file $F\,^{*}_{m+1}$ by Alice’s public-key
graph sequence $\\{G\,^{i}_{Apub}\\}^{m}_{i=1}$ one-by-one, as the topological
signature authentication
$A_{uth}\big{\langle}\\{G\,^{i}_{Apub}\\}^{m}_{i=1},\\{G\,^{i}_{Apri}\\}^{m}_{i=1}\big{\rangle}$
passed, the resultant encrypted file is denoted as $F\,^{*}_{1}$ with a one-
level protection only.

  5. Self-3.5.

Bob uses his private-key string $s_{Bpri}$ to decrypt the encrypted file
$F\,^{*}_{1}$, as the Key-pair string authentication
$S_{tringB}=A_{uth}\langle s_{Bpub},s_{Bpri}\rangle$ passed, then Bob gets the
original plaintext $F$.

For the security and computational complexity of the self-certification
topology algorithm-III we have:

  1. Scta-3.1.

Attacking Alice’s public-key graph sequence $\\{G\,^{i}_{Apub}\\}^{m}_{i=1}$
and Bob’s public-key graph sequence
$\big{\\{}G\,^{j}_{Bpub}\big{\\}}^{m}_{j=1}$ will meet the problems of
topological structures and colorings at $m$ times.

  2. Scta-3.2.

There is an order in encrypting the encrypted file $F_{1}$ by Alice’s private-
key graph sequence $\big{\\{}G\,^{i}_{Apri}\big{\\}}^{m}_{i=1}$, so the
topological signature authentication
$T\,^{i}_{sigA}=A_{uth}\big{\langle}G\,^{i}_{Apub},G\,^{i}_{Apri}\big{\rangle}$
follows the topological signature authentication
$T\,^{i+1}_{sigA}=A_{uth}\big{\langle}G\,^{i+1}_{Apub},G\,^{i+1}_{Apri}\big{\rangle}$,
one-by-one.

Self-certification topology algorithm-IV based on Key-graph sequences with
different ranks.

  1. Self-4.1.

Alice sends to Bob a key-package-A including Alice’s public-key string
$s_{Apub}$, and Alice’s public-key graph sequence
$\big{\\{}G\,^{i}_{Apub}\big{\\}}^{m}_{i=1}$ with $m\geq 1$ (as the identity
authentication).

  2. Self-4.2.

Bob sends to Alice a key-package-B including Bob’s public-key string
$s_{Bpub}$, and Bob’s public-key graph sequence
$\big{\\{}G\,^{j}_{Bpub}\big{\\}}^{n}_{j=1}$ with $n\geq 1$.

  3. Self-4.3.

Alice encrypts the plaintext $F$ by Bob’s public-key string $s_{Bpub}$, and
gets the encrypted file $F_{1}$, and then Alice encrypts the encrypted file
$F_{1}$ by her private-key graph sequence
$\big{\\{}G\,^{i}_{Apri}\big{\\}}^{m}_{i=1}$ one-by-one, the resultant
encrypted file is denoted as $F\,^{*}_{m+1}$ with a $(m+1)$-level protection.

  4. Self-4.4.

Alice encrypts the file $F\,^{*}_{m+1}$ by Bob’s public-key graph sequence
$\big{\\{}G\,^{j}_{Bpub}\big{\\}}^{n}_{j=1}$ one-by-one, the resultant
encrypted file is denoted as $F^{*}_{m+n+1}$ with a $(m+n+1)$-level
protection.

  5. Self-4.5.

Bob, first, decrypts the encrypted file $F^{*}_{m+n+1}$ by Alice’s public-key
graph sequence $\big{\\{}G\,^{i}_{Apub}\big{\\}}^{m}_{i=1}$ one-by-one, as the
topological signature authentication

$T_{sigA}=A_{uth}\langle\\{G\,^{i}_{Apub}\\}^{m}_{i=1},\\{G\,^{i}_{Apri}\\}^{m}_{i=1}\rangle$

passed, the resultant encrypted file is denoted as $F\,^{*}_{n+1}$ with a
$(n+1)$-level protection.

  6. Self-4.6.

Bob uses his private-key string $s_{Bpri}$ and private-key graph sequence
$\big{\\{}G\,^{j}_{Bpri}\big{\\}}^{n}_{j=1}$ to decrypt the encrypted file
$F\,^{*}_{n+1}$. As the following two authentications

$S_{tringB}=A_{uth}\langle s_{Bpub},s_{Bpri}\rangle\textrm{ and
}T_{sigB}=A_{uth}\big{\langle}\\{G\,^{j}_{Bpub}\\}^{n}_{j=1},\\{G\,^{j}_{Bpri}\\}^{n}_{j=1}\big{\rangle}$

passed, finally, Bob obtains the original plaintext $F$.

Figure 53: A scheme for the self-certification algorithm based on the
topological signature sequence.

Self-certification topology algorithm-V based on Key-string sequence and Key-
graph sequence.

  1. Self-5.1.

Alice sends to Bob a key-package-A including Alice’s public-key string
sequence $\big{\\{}s\,^{i}_{Apub}\big{\\}}^{n_{a}}_{i=1}$ with $n_{a}\geq 1$,
and Alice’s public-key graph sequence
$\big{\\{}G\,^{i}_{Apub}\big{\\}}^{m_{a}}_{i=1}$ with $m_{a}\geq 1$ (as the
identity authentication).

  2. Self-5.2.

Bob sends to Alice a key-package-B, which contains Bob’s public-key string
sequence $\big{\\{}s\,^{i}_{Bpub}\big{\\}}^{n_{b}}_{i=1}$ with $n_{b}\geq 1$,
and Bob’s public-key graph sequence
$\big{\\{}G\,^{j}_{Bpub}\big{\\}}^{m_{b}}_{j=1}$ with $m_{b}\geq 1$.

  3. Self-5.3.

Alice encrypts the plaintext $F$ by Bob’s public-key string sequence
$\big{\\{}s\,^{i}_{Bpub}\big{\\}}^{n_{b}}_{i=1}$, and gets the encrypted file
$F_{n_{b}}$, and then Alice encrypts the encrypted file $F_{n_{b}}$ by her
private-key graph sequence $\big{\\{}G\,^{i}_{Apri}\big{\\}}^{m_{a}}_{i=1}$
one-by-one, the resultant encrypted file is denoted as $F\,^{*}_{n_{b}+m_{a}}$
with a $(n_{b}+m_{a})$-level protection.

  4. Self-5.4.

Alice, again, encrypts the encrypted file $F\,^{*}_{n_{b}+m_{a}}$ by Bob’s
public-key graph sequence $\big{\\{}G\,^{j}_{Bpub}\big{\\}}^{m_{b}}_{j=1}$
one-by-one, the resultant encrypted file is denoted as
$F^{*}_{n_{b}+m_{a}+m_{b}}$ with a $(n_{b}+m_{a}+m_{b})$-level protection.

  5. Self-5.5.

Bob, first, decrypts the encrypted file $F^{*}_{n_{b}+m_{a}+m_{b}}$ by Alice’s
public-key graph sequence $\big{\\{}G\,^{i}_{Apub}\big{\\}}^{m_{a}}_{i=1}$
one-by-one, as the topological signature authentication

$T_{sigA}=A_{uth}\big{\langle}\\{G\,^{i}_{Apub}\\}^{m_{a}}_{i=1},\\{G\,^{i}_{Apri}\\}^{m_{a}}_{i=1}\big{\rangle}$

passed, the resultant encrypted file is denoted as $F\,^{*}_{n_{b}+m_{b}}$
with a $(n_{b}+m_{b})$-level protection.

  6. Self-5.6.

Bob uses his private-key string sequence
$\big{\\{}s\,^{i}_{Bpri}\big{\\}}^{n_{b}}_{i=1}$ and private-key graph
sequence $\big{\\{}G\,^{j}_{Bpri}\big{\\}}^{m_{b}}_{j=1}$ to decrypt the
encrypted file $F\,^{*}_{n_{b}+m_{b}}$. As the following two authentications

$S_{tringB}=A_{uth}\big{\langle}\big{\\{}s\,^{i}_{Bpub}\big{\\}}^{n_{b}}_{i=1},\big{\\{}s\,^{i}_{Bpri}\big{\\}}^{n_{b}}_{i=1}\big{\rangle}\textrm{
and
}T_{sigB}=A_{uth}\big{\langle}\\{G\,^{j}_{Bpub}\\}^{m_{b}}_{j=1},\\{G\,^{j}_{Bpri}\\}^{m_{b}}_{j=1}\big{\rangle}$

passed, finally, Bob obtains the original plaintext $F$.

##  8 Conclusion

In order to approach practical applications, resist the attack of quantum
computing, and develop asymmetric topology cryptography, we have introduced a
number of ways to induce various string as follows:

  1. Meth-1.

Number-based strings made by topological coding are as public-key strings.

  2. Meth-2.

Topcode-matrices produce number-based strings for public-key strings.

  3. Meth-3.

Colored graphs have Topcode-matrices, which induce number-based strings.

  4. Meth-4.

Vectors and sets from topological coding make number-based strings.

  5. Meth-5.

Compound number-based strings made by topological coding are as public-key
graphs.

  6. Meth-6.

Number-based strings as Key-pairs are made by the integer partition and the
integer decomposition.

  7. Meth-7.

$[0,9]$-strings, and algebraic operations on $[0,9]$-strings, also, _string
algebra_.

  8. Meth-8.

Number-based super-strings.

  9. Meth-9.

Self-breeding number-based strings.

  10. Meth-10.

Graph-based strings including total graph-colorings; TOTAL-graph-coloring
algorithm-I; TOTAL-graph-coloring algorithm-II; complex number-based strings.

  11. Meth-11.

Number-based strings with parameters,plane-curve-attached strings, space-
coordinate strings, fractional strings.

  12. Meth-12.

A _public-key graph_ $G$ admitting a $W$-constraint parameterized coloring $F$
can be used to make a _private-key graph_ $H=\langle G,F,f(x)\rangle$ based on
a real-valued function $y=f(x)$.

  13. Meth-13.

Assignment Topcode-matrices.

  14. Meth-14.

(abc)-linear colorings.

  15. Meth-15.

flawed $W$-constraint colorings and labelings.

  16. Meth-16.

Plane-coordinate string sequences.

  17. Meth-17.

Graphic lattices for making number-based strings including tree-base graphic
lattices; tree-base $[0,9]$-operation graphic lattices.

  18. Meth-18.

Set-based strings.

  19. Meth-19.

Number-based strings from indexed-colorings including maximal planar graphic
lattices; indexed-colorings; planar graphs and indexed-colorings.

  20. Meth-20.

Hanzi-based strings including planar graphs and Hanzi-based strings; Hanzi-
Lattices; maximal planar graphs and Hanzi-based strings; the
$[\odot_{\textrm{plan}}]$-Hanzi-graphic lattices.

  21. Meth-21.

Every-zero infinite graphic groups, every-zero infinite graphic group
homomorphism, mutually equivalent infinite every-zero graphic groups.

We try to apply various strings to the actual scene:

  1. Appl-1.

Homogeneous number-based string-colorings, non-homogeneous number-based
string-colorings, constructing number-based string-colorings.

  2. Appl-2.

Multi-level multi-rank strings.

  3. Appl-3.

Vector-colorings made by strings.

  4. Appl-4.

Set-colorings made by strings.

  5. Appl-5.

Topcode-matrices with non-number elements: Topcode-matrices with the elements
of strings, vectors and sets; non-number-type Topcode-matrices with
parameters.

  6. Appl-6.

For the overall encryption of networks, or the community encryption of
different communities in a network, we have studied very-zero number-based
string groups including GROUP-compound algorithm; Number-based sub-string
groups; compound number-based string groups; Partial every-zero compound
number-based string groups; and every-zero Topcode-matrix group.

  7. Appl-7.

Topcode-matrix algebra.

There are problems related with strings and string-colorings as follows:

  1. Ques-1.

An _$[a,b]$ -string_ $s=c_{1}c_{2}\cdots c_{n}$ holds $c_{j}\in[a,b]$ with two
integers $a,b$ subject to $0<a<b$. It is may be interesting to generalize the
properties of $[0,9]$-strings to $[a,b]$-strings, and find some connections
between $[a_{k},b_{k}]$-strings with integers $0<a_{k}<b_{k}$.

  2. Ques-2.

Given a number-based string $s=c_{1}c_{2}\cdots c_{n}$ with $c_{i}\in[0,9]$
and $c_{1}\geq 1$, find a positive integer $m$, such that $m$ can be expressed
as a sum $m_{i,1}+m_{i,2}+\cdots+m_{i,a_{i}}$ defined in Eq.(169) of
Definition 42, and there is $s=s^{*}_{i}$ defined in Eq.(170), or
$s=s^{j}_{i}$ defined in Eq.(172), or $s=s_{i,j}$ defined in Eq.(173).

  3. Ques-3.

Given a number-based string $s^{*}=a_{1}a_{2}\cdots a_{m}$ with
$a_{i}\in[0,9]$ and $a_{1}\geq 1$, find a positive integer $n$, such that $n$
can be expressed as a product $p_{i,1}p_{i,2}\cdots p_{i,b_{i}}$ defined in
Eq.(174) of Definition 43, and there is $s^{*}=r^{*}_{i}$ defined in Eq.(175),
or $s^{*}=r^{j}_{i}$ defined in Eq.(177), or $s^{*}=r_{i,j}$ defined in
Eq.(178).

  4. Ques-4.

Let $G_{raph}(p)$ be the set of graphs of $p$ edges. Find a mapping
$\theta:V(G)\rightarrow V(H)$ for two graphs $G,H\in G_{raph}(p)$, such that
there is a vertex $u\in V(G)$ holding $G\wedge u\cong H\wedge\theta(u)$.

  5. Ques-5.

PRONBS-problem defined in Problem 1 is the parametric reconstitution problem
of number-based strings.

  6. Ques-6.

ATE-CGH-problem defined in Problem 28 and Pan-ATE-CGH-problem defined in
Problem 30 and ATE-To-CGH-problem defined in Problem 31 are the problems of
asymmetric topology encryption related with colored graph homomorphisms.

The definition of “ _zero knowledge proof_ ” in the blockchain is that the
verifier can make the verifier believe that a certain assertion is correct
without providing any useful information to the verifier. _Zero knowledge
proof_ is a more secure information verification or authentication mechanism.
Security and privacy are the values of zero knowledge proof. We hope the
techniques of asymmetric topology encryption, such as self-certification
topology algorithms, can be applied into blockchains.

Assuming that attackers are very familiar with the technology of topology
coding, how to generate Key-pairs and how to distribute Key-pairs without
being cracked is one of the focuses of future research, although there is
still a hard and challenging way to go from theory to practice.

## Acknowledgment

The author, _Bing Yao_ , was supported by the National Natural Science
Foundation of China under grants No. 61163054, No. 61363060 and No. 61662066.

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Figure 54: A Chinese sentence
$H_{3907}H_{4582}H_{4043}H_{5414}H_{4147}H_{2083}H_{5341}H_{2382}$ provides
$8!=40320$ different Chinese sentences in total.