Co esponding au ho : And is lukss
Copy igh © 2025 Au ho (s) e ain he copy igh o his a icle. This a icle is published unde he e ms o he C ea i e Commons A ibu ion Liscense 4.0.
Quan um-Resis an Key Gene a ion Using QBLH Geome ic S uc u es and
Te ahed al T ina y Encoding: A No el App oach in Pos -Quan um C yp og aphy
And is lukss *
Independen esea che , Canbe a ins i u e echnology, Aus alia.
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(02), 1532-1542
Publica ion his o y: Recei ed on 13 July 2025; e ised on 18 Augus ; accep ed on 21 Augus 2025
A icle DOI: h ps://doi.o g/10.30574/wja .2025.27.2.3017
Abs ac
The dawn o he dis up i e quan um compu ing scena io ma ks a se ious h ea o he exis ence o adi ional
c yp osys ems. Wi h laws such as Sho ’s, capable o ac o ing la ge in ege s in polynomial ime, and G o e ’s, able o
speed up b u e- o ce key sea ches, hese a acks make con en ional public-key in as uc u es inc easingly ulne able,
whe eas e en symme ic ciphe s lose good measu e o hei s eng h. In his a icle, we ocus on an elabo a i e
desc ip ion o a pa en ed me hod o quan um-secu e key gene a ion, whe ein Qabbalah (QBLH) complexi y is u ilized
in he geome ic-symbolic ealm, in conjunc ion wi h magic numbe squa es, phi/pi coo dina e weigh ing, and
e ahed al ina y s a e encoding. The p oposed sys em o T iGa e QBLH Quan um-Sa e Enc yp ion con e s seed
inpu s o mul idimensional keys ha esis linea algeb aic a acks owing o non-linea pe mu a ions, i a ional
cons an weigh ing, and opological complexi y. No mally, pseudo- andom numbe gene a o s spa ialize en opy in
Euclidean geome y, as opposed o he p esen echnique ha places en opy in a comple ely non-Euclidean domain,
whe e classical as well as quan um ad e sa ies ind i ha d o a e se. We desc ibe he me hod in de ail, p esen i s
bene i s o e la ice- and hash-based pos -quan um schemes, and walk h ough an example o i s implemen a ion.
Conside a ion is also gi en o i s po en ial in eg a ion wi h PQC s anda ds, blockchain au hen ica ion, and decen alized
inance applica ions. The sys em uses symbolic ma hema ics, such as he 231 Ga es o QBLH, wi h ina y logic mapped
on o e ahed al s a es o no only c ea e enc yp ion keys bu also e i iable geome ic signa u es. This ep esen s a
pa adigm shi owa d geome ic c yp og aphy, which may be a iable me hod o ealize scalable and us wo hy
digi al in as uc u e in a quan um- h ea ened en i onmen .
Keywo ds: Pos -Quan um C yp og aphy; Quan um-Resis an Key Gene a ion; QBLH Geome ic S uc u es;
Te ahed al T ina y Encoding; La ice-Based C yp og aphy; Secu e Key Exchange
1. In oduc ion
The ad en o quan um compu ing has p esen ed emendous isks o con en ional c yp og aphy, especially public-key
sys ems like RSA and ECC, whose secu i y hinges on he abili y o ac o ize huge numbe s o compu e disc e e
loga i hms— asks ha scalable quan um machines heo e ically can accomplish by use o Sho 's algo i hm. On he o he
hand, G o e 's algo i hm dec eases he e ec i e secu i y le el o symme ic ciphe s, showing how all o ou exis ing
digi al in as uc u e is unde h ea (NIST, 2022; Gee, 2024).
Pos -Quan um C yp og aphy schemes in end o mi iga e hese h ea s ia la ice, code, mul i a ia e, and hash-based
schemes. S ill, hese algo i hms a e on e y shaky g ounds o hey a e algeb aic in na u e and a e basically allied o
pseudo- andom numbe gene a o s ha open up he possibili y o u he exploi s (Gee & Lukss, D a 2025).
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Symbolic and geome ic c yp og aphy p o ides a undamen ally di e en ou look, whe eby keys ge embedded in
mul idimensional and opological spaces. This is how he T iGa e QBLH Quan um-Sa e Enc yp ion Sys em wo ks: i
combines Qabbalis ic 231 Ga es pe mu a ions, magic squa es weigh ed by i a ional cons an s (ϕ, π), and e ahed al
ina y encoding; keys a e ealized as bo h c yp og aphic bi s ings and geome ic signa u es, he eby p o iding a dual
laye o en opy subjec o nei he algeb aic no quan um a acks (Se e Ye zi ah, 2004; Gee, 2024).
By c ea ing mul idimensional keys ha canno be educed by any con en ional linea o polynomial app oach, symbolic
elemen s combined wi h ina y-s a e logic c ea e a mechanism o key gene a ion esis an o educ ion. I u he s he
es ablishmen o blockchain walle s, quan um c yp ocu encies, as well as IoT au hen ica ion, and by doing so, i
de ends agains a acks posed by algo i hms o bo h Sho and G o e . P o iding he al e na i e landscapes o pos -
quan um c yp og aphy along wi h a sus ainable way o wa d in a pos -quan um e a, he T iGa e will he e o e wo k
alongside PQC.
1.1. Challenges in Quan um-Resis an C yp og aphy
While quan um echnologies appea p omising, hey b ing abou an u gen equi emen o secu e c yp og aphic
me hods o be esis an o ad e sa ies using quan um compu e s. The classical c yp osys ems, namely RSA and Ellip ic
Cu e C yp og aphy (ECC), bu ess hei secu i y by assuming he unde lying numbe - heo e ic p oblems o be e y
ha d o sol e on con en ional machines. Howe e , wi h Sho 's algo i hm, in ege ac o iza ion and disc e e loga i hms
can be sol ed in polynomial ime on a quan um compu e , lea ing such widely deployed sys ems lacking any semblance
o secu i y (NIST, 2022).
In esponse o such h ea s, Pos -Quan um C yp og aphy (PQC) has sp ung in o exis ence- he name gi en o
c yp og aphy ha will p o ide algo i hms in ac able o bo h classical and quan um ad e sa ies. Among he mos
p ominen candida es a e la ice-based algo i hms like CRYSTALS-Kybe and Dili hium, code-based algo i hms such as
Classic McEliece, mul i a ia e polynomial schemes, and hash-based schemes such as SPHINCS+ (NIST, 2022).
Candida es o hese schemes a e essen ially being s anda dized o ede al and comme cial use.
Ne e heless, and despi e hei p omises, PQC algo i hms d ama is all sho in many espec s. The mos e iden o all
conce ns ega ds he ac ha s ong eliance is placed on pseudo- andom numbe gene a o s (PRNGs) o he
gene a ion o c yp og aphic key ma e ial, esul ing in inciden s o en opy weakness o dis ibu ions skewed in ce ain
cases (Gee & Lukss, D a 2025). Secondly, implemen a ions o PQC s ill emain ulne able o ce ain side-channel
a acks, such as iming, powe , and elec omagne ic leakages, and hese become e en s onge in he eal wo ld, whe e
ad e sa ies capi alize on implemen a ion laws a he han pu e ma hema ical weaknesses. Key gene a ion and en opy
sou ces emain he weakes links in PQC.
Following geome ic and symbolic c yp og aphy, complexi y is no only posed by algeb aic ha dness bu also by
embedding c yp og aphic s uc u es in non-Euclidean spaces and opological ne wo ks. These opological schemes
a ge p oducing key ma e ials which canno be educed ia linea algeb aic me hods, he eby ende ing he
supe posi ion- en anglemen based a ack s a egies ine icien .2.2 Symbolic and Geome ic App oaches
His o ically, c yp og aphy ope a ed on bina y logic and algeb aic compu a ion. On he o he hand, symbolic and
geome ic mechanisms use di e en ma hema ical ields o gene a e some deg ee o complexi y. Ea ly s udies in ina y
logic, spin-based compu a ion, and opological encoding e ealed ha bypassing bina y logic could enhance esis ance
o ce ain classes o a acks (Gee, 2024).
Especially mul i- alued and ina y logic sys ems ins an ia e mo e s a es pe one compu a ional uni , keyspace hus
expands exponen ially in compa ison o bina y sys ems. To ci e, a bina y sys em can e e p oduce s a es 2^n o an n-
leng h sequence, whe eas a ina y sys em can p oduce 3^n, making b u e- o ce sea ch so di icul ha e en G o e 's
quad a ic speedup canno o e big leads, (Gee, 2024). Spin-based models o quan um compu ing, which include qu i
sys ems, essen ially in es iga e simila mul i- alued encoding echniques.
In c yp og aphy, he geome ic ways y o embed en opy in o s uc u es such ha he algeb aic la ening is di icul .
Fo ins ance, mapping gi en da a in o highe dimensional la ices, poly opes, o essella ions can p o ide secu i y
p ope ies ha a e no a leas alone based on a ha dness assump ion;
1.2. Founda ions in QBLH and Sac ed Geome y
The T iGa e QBLH Quan um-Sa e Enc yp ion Sys em s ems om symbolic ma hema ics and sac ed geome y, ha is,
Qabbalis ic s uc u es as desc ibed in he Se e Ye zi ah (2004). In his ancien ex , 231 "Ga es" co espond o
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pe mu a ions o Heb ew le e s ha make up a web o symbolic ans o ma ions. Lacking a desc ip i e c yp og aphic
model o ga es, hei s uc u e can be modeled as a g aph, which ollows om possibili ies o ans o ma ion among
i s nodes ep esen ing s a es. Thus, i p o ides an unde pinning combina o ial amewo k o pe mu a ion-based
enc yp ion, p esen ing ma hema ically mo e meaning ul al e na i es o linea key de i a ion.
Magic squa es complemen QBLH; hese ancien -ma hema ical concoc ions a e a angemen s in which he sums o
ows, columns, and diagonals all mus equal some cons an alue. In c yp og aphy, he magic squa es p esen he laye
o symme y and balance o lessen any chance o biased key dis ibu ion. These mappings combine wi h i a ional
weigh ings based on he golden a io (ϕ ≈ 1.618) and on pi (π ≈ 3.14159) o yield coo dina e sys ems wi h buil -in
algeb aic esis ance. By spa ializing he seed inpu s wi hin a magic squa e by phi/pi-weigh ed coo dina es, T iGa e
e ec i ely embeds en opy in his s uc u e, which canno be u he immed down by linea algeb a o polynomial
educ ion.
Te ahed al encoding in oduces a u he e inemen by ex ending bina y logic in o bo h i s ina y and o a ional
s a es. A egula e ahed on wi h i s ou iangula aces is used o geome ically ep esen logical s a es:
• Open (O) – analogous o bina y 0
• Closed (C) – ha is, bina y 1
• Righ (R) – clockwise spin s a e
• Le (L) – coun e clockwise spin s a e
These ou s a es may be combined in o o a ional pa hs o e a e ahed al su ace o o m a highly compac and non-
linea key ep esen a ion. In quan um implemen a ions, such s a es could be ealized as qu i s o as pai s o qubi s,
p ese ing hei abili y o in e ac wi h quan um ha dwa e, ye unable o be classically educed (Gee, 2024).
In quan um c yp ocu ency, simila ideas ha e been used in applying e na y ga es h ough e ahed al essella ions
owa ds he p o ec ion o blockchain (Gee, 2024). Building upon ha p emise, he T iGa e sys em, in u n, in e laces
QBLH's symbolic ne wo k, magic squa e weigh ing, and e ahed al ina y encoding o o m a single key gene a ion
mechanism. By impa ing en opy in o sphe ical opologies and spino dynamics, i c ea es opological ba ie s
incapable o being c ossed e icien ly by classical o quan um ad e sa ies.
Table 1 Compa ison o Con en ional PQC Me hods s. T iGa e QBLH Fea u es
Fea u e
La ice-Based
PQC (e.g., Kybe )
Hash-Based PQC (e.g.,
SPHINCS+)
Code-Based PQC
(e.g., McEliece)
T iGa e QBLH
Secu i y
Basis
Algeb aic
ha dness (LWE,
RLWE)
Hash collision
esis ance
Linea code
decoding
ha dness
Symbolic- and geome ic-
opology (QBLH, magic squa es,
e ahed al encoding)
Key
Gene a ion
PRNG + Gaussian
sampling
PRNG + hash expansion
PRNG + e o
ec o s
Phi/Pi weigh -magic squa es;
231 Ga es pe mu a ions
A ack
Su ace
Algeb aic
educ ions, la ice
sie ing
Hash
p eimage/collision
sea ch
Decoding +
s uc u al
leakage
Topological complexi y, ina y
s a es, and non-linea mapping
1.2.1. Founda ions in QBLH and Geome ic C yp og aphy
The T iGa e QBLH sys em desc ibes he hyb id me hodology consis ing o symbolic ma hema ics, sac ed geome y, and
mode n c yp og ahpy o key gene a ion ha is esis an o algeb aic and quan um a acks. En opy is embedded in o
complex mul idimensional symbolic and geome ic s uc u es, passing beyond he o dina y linea app oaches.
QBLH S uc u es and he 231 Ga es
Taken om he Qabbalis ic Se e Ye zi ah, 231 Ga es in QBLH ep esen all wo-le e Heb ew combina ions. Gene a ed
as a di ec ed g aph, ga es p oduce high-en opy sequences h ough pe mu a ion pa hs and hus p o ide non-linea
complexi y much beyond ha ecei ed by he con en ional pseudo- andom gene a o s.
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Magic Squa es and Symme y in C yp og aphy
Magic squa es weigh ed by i a ional cons an s- he golden a io ϕ o π-make hese coo dina e sys ems non-
algeb aically educible. Ac ing like spa ial en opy ma ices, hey con ey hose non-linea ela ionships ha a e mean
o c ea e mo e-e icien key con ibu ion and be e p o ec ion in c yp og aphy applica ions.
Te ahed al Encoding and T ina y Logic
The sys em hus implemen s a o m o e ahed al encoding o ep esen he ou s a es (Open, Closed, Righ , and Le )
as i s. Such ina y logic gi es exponen ial g ow h o he s a e space and se es o insc ibe da a in o o a ional pa hs
ac oss he e ahed on, hus p o iding immuni y agains linea algeb a-based a acks.
In eg a ion o QBLH, Magic Squa es, and Te ahed al Logic
The p ide o T iGa e is ha i o e s a laye ed app oach consis ing o pe mu a ion QBLH ga es, magic squa es weigh ed
by i a ional numbe s, and e ahed al ina y encoding. This leads o a mul idimensional en opy engine, which
p oduces digi al keys and geome ic signa u es o quan um-sa e au hen ica ion.
Rela ed Wo ks
In ac , esea ch in geome ic c yp og aphy along wi h ha in quan um blockchain end up suppo ing non-linea ,
mul idimensional sys ems. Thus, ina y swi ch-ga es, e ahed al essella ions, and opological enc yp ion amewo ks
o e a be e esilience agains classical and quan um a acks, he eby legi imizing he T iGa e app oach.
2. Me hodology
2.1. T iGa e QBLH Key Gene a ion
The T iGa e QBLH sys em enc yp s h ough a mul is age pipeline ha akes in an en opy- ich seed and ou pu s a
quan um- esis an key. While o he c yp og aphic sys ems ely on algeb aic ha dness assump ions alone, T iGa e QBLH
combines symbolic s uc u es (QBLH 231 Ga es), geome ic embeddings (magic squa es weigh ed wi h i a ional
cons an s), and ina y opologies ( e ahed al encoding). The ollowing sec ions desc ibe he s eps in ol ed in his
me hodology.
2.1.1. Seed Inpu No maliza ion
The p ocess begins wi h a seed inpu . The seed inpu may come om a ied en opy sou ces, such as:
• Use passph ases (alphanume ic s ings)
• Biome ic hashes ( inge p in , i is scan, oice p in )
• Quan um andom numbe gene a o (QRNG) ou pu s
The objec i e shall be o i s ensu e ha he seed achie es i s high en opy and uni o m dis ibu ion be o e i can be
aken u he o ans o ma ion. Fo ins ance, he passph ase TRIGATE2025 is con e ed o ASCII and nume ical
cha ac e s. Each cha ac e is mapped o some in ege alue and conca ena ed in o a sequence o numbe s. The sequence
may, howe e , be s eng hened by applying, o example, a SHA3-512 hash o ano he NIST-app o ed diges be o e
p oceeding wi h u he s eps [Wo all e al., 2025; Veale, 2024].
En opy becomes signi ican because weak o easily guessed seeds a e suscep ible o b u e- o ce o dic iona y a acks—
e en in quan um esis ance en i onmen s [Soyege e al., 2024]. T iGa e achie es such unp edic abili y a ha basic le el
wi h he QRNGs, ne e inhe en ly depending upon pa en ed de e minis ic pseudo- andom numbe gene a o s ha ha e
been p o ed o weaken classical c yp og aphy [Wa anabe, 2020].
2.2. Magic Squa e Mapping Using Phi/Pi Weigh ing
The no malized nume ic seed hen inds i s place in o one o he sizes o magic squa es chosen as pe he equi ed key
leng h and en opy densi y. Typically 3×3, 5×5, o 7×7 g ids may be made use o . The special p ope y o magic squa es
is ha sums ac oss all ows, columns, and main diagonals a e equal o he same numbe . In pe ad en u e one changes
e minology: he p ope y symme ically and edundan ly sha es alues, p e en ing localized en opy collapse[Palka,
2020].
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To add u he complexi y, he magic squa e's cells a e weigh ed by means o i a ional ma hema ical cons an s:
Phi= ac{1 + sq {5}}{2} app ox 1.61803 qquad pi app ox 3.14159
I a seed alue S is loca ed a coo dina es (i, j), he weigh ed alue is de ined as:
V(i, j) = S imes ( Phi^i imes pi^j).
This weigh ing scheme embeds he seed inside an i a ional coo dina e sys em, gene a ing a geome ical key space no
amenable o algeb aic educ ion. In a manne o speaking, hese c ea e non- epea ing dis ibu ions akin o hose ound
in quasi-c ys al s uc u es, which esis ac o iza ion and canno be easily mapped by polynomial- ime algo i hms
[Fa hana e al., 2025].
Figu e 2. Flowcha o T iGa e QBLH key gene a ion pipeline (seed → no maliza ion → magic squa e embedding → QBLH
pe mu a ion → ina y encoding → key ex ac ion).
Figu e 1 Flowcha o T iGa e QBLH key gene a ion pipeline
3. QBLH 231 Ga e Pe mu a ion
The weigh ed magic squa e is he basic en ance in o he QBLH 231 Ga es ne wo k. This symbolic amewo k, de i ed
om he ancien Se e Ye zi ah, ep esen s 231 pe mu a ions o Heb ew le e s and o ms he di ec ed g aph o s a es
and ans o ma ions. In T iGa e, e e y magic squa e cell is u ned in o a node in such a g aph, and he a e sal p oceeds
acco ding o de e minis ic o pseudo- andom ules seeded by he inpu sequence.
As each s ep in he a e sal is gene a ed, a ina y s a e is c ea ed:
Open (O) ≡ bina y 0
Closed (C) ≡ bina y 1
Righ (R) ≡ clockwise spin
Le (L) ≡ coun e clockwise spin
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Con a y o bina y, he ina y encodings hold di ec ional and opological in o ma ion, hus collision- esis i ely and
opologically embedding o a ion and o ien a ion in o he sequence. Each a e sal pa h leads o a combina o ial
explosion o possible ou pu s. G o e 's algo i hm, wi h i s quan um sea ch applica ions, p o ides only quad a ic speed-
up o uns uc u ed sea ches, whe eas his opological expansion ou uns quan um sea ch e iciencies [Mekdad e al.,
2025].
Fu he mo e, he symbolic esonance o he 231 Ga es ensu es he ou pu s a e no only non-linea bu also non-
e e sible wi hou ull knowledge o he seed and pa h o a e sal. This assu es ha he sys ems ollow mode n
c yp og aphic p inciples o o wa d sec ecy and un o geabili y [Dohe y e al., 2024].
4. Te ahed al T ina y Encoding
Upon pe mu a ion, he ina y s a es a e mapped in o a e ahed al opology. The Te ahed on, a Pla onic solid wi h
ou aces, comp ises a na u al geome ical con aine o ina y logic. Each plane was assigned one o he s a es:
• Plane 1 → Open (O)
• Plane 2 → Closed (C)
• Face 3 → Righ (R)
• Face 4 → Le (L)
Sequences o s a es become o a ional pa hways on he e ahed on. Fo ins ance, he 4- uple (O → R → L → C)
ep esen s a unique spin ajec o y on he poly ope. These ajec o ies hus ac as sho e encodings o in o ma ion, in
a manne akin o qu i -based ep esen a ions in quan um compu ing [Romanelli e al., 2015].
F om he pe spec i e o ha dwa e implemen a ion, hese s a es may be encoded wi h e na y logic ga es o wi h pai ed
qubi s. This e na y mapping hus p esen s wo majo ad an ages:
• Highe in o ma ion densi y: a single i ep esen s log₂3 ≈ 1.585 bi s, gi ing a sligh ly be e e iciency han a
bi .
• Resis ance agains a acks ocusing on bina ies: mos c yp analysis me hods a e gene ally concei ed o bina y
inpu s, leading o ina y s a es causing a misma ch wi h he a acke 's model [Chan e al., 2023].
Table 2 Mapping be ween T ina y S a es (O, C, R, L), Bina y Equi alen s, and Faces o he Te ahed on
T ina y S a e
Symbol
Bina y Equi alen
Te ahed on Face Rep esen a ion
Open
O
0
Face 1 (Top - Open e ex)
Closed
C
1
Face 2 (Base - S able/Closed su ace)
5. Key ex ac ion and dual ou pu s
The las s ep yields wo simul aneous ou pu s:
• P ima y bi s ing key - P oduced by collapsing ina y ajec o ies in o bina y ep esen a ions and,
subsequen ly, conca ena ed in o a ixed-leng h bi s ing (e.g., 256 o 512 bi s). Such key can be used as a di ec
inpu o s anda d c yp og aphic algo i hms, like AES, o la ice-based enc yp ion, o hyb id PQC schemes
[Za ale a-Mones el e al., 2023].
• Geome ic Signa u e - Seconda ily ou pu ed in he shape o a geome ic pa h o ma ix eco ding he
p og ession h ough e ahed a. This signa u e will be enowned as a e i ica ion ool ha gua an ees keys
canno be ep oduced absen he exac ina y encoding p ocess. I also allows o mul i- ac o e i ica ion,
which b ings oge he symbolic, geome ic, and bi s ing ep esen a ions [Rebolledo e al., 2022].
• The dual-ou pu sys em o i ies agains ad e sa ial a acks: e en i he bi s ing is somehow exposed, wi hou
he geome ic signa u e, he ad e sa y canno ully alida e he key. This model is eminiscen o mul i-laye
au hen ica ion in cybe secu i y bu is embedded a he le el o c yp og aphic gene a ion [Schul z e al., 2021].
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6. In eg a ion Wi h PQC F amewo ks
Al hough in oducing a no el symbolic–geome ic pa adigm, he T iGa e QBLH sys em was designed o con o m o
exis ing PQC s anda ds. The ex ac ed bi s ing can be used as:
• A key exchange pa ame e o la ice-based p o ocols,
• A hash inpu o Me kle signa u e schemes, o
• A session key in hyb id schemes o classical and PQC ciphe s.
Hence, T iGa e a oids being an isola ed, expe imen al sys em and has applica ions along cu en lines o NIST PQC
s anda diza ion [Tonin e al., 2021].
7. A Summa y o Me hodological Ad an ages
The me hodology comes o se e al key ad an ages:
Assu ing majo esis an quali ies in T iGa e QBLH may be ou lined as a unc ion o ea u es:
• En opy in an ampli ied manne by i a ional weigh ing (phi, pi).
• Combina o ial explosion by a e sing h ough QBLH 231 Ga e.
• Topological encoding by e ahed al ina y s a es.
• Dual ou pu s (bi s ing + geome ic signa u e) o a mo e ho ough e i ica ion.
• Seamless PQC in eg a ion o ensu e o wa d compa ibili y.
Bold in combining symbolic ma hema ics, geome y, and c yp og aphic heo y, T iGa e QBLH ci cum en s o he
amewo ks dimissed by adi ional algeb aic ha dness assump ions. The e o e i is a possible candida e o quan um-
sa e key gene a ion wi h espec o o he ci ilian and de ense cybe secu i y in as uc u es.
7.1. Example Implemen a ion o T iGa e QBLH (~800 wo ds)
Fo p ac ical exposi ion in o he T iGa e QBLH me hodology, a wo ked example wi h he seed `TRIGATE2025` is being
e e ed o. The p ocess illus a es he human- eadable passph ase mo phing in o a quan um- esis an enc yp ion key
by way o hyb idiza ion in ol ing ma hema ical cons an s, geome ic mappings, and ina y encodings.
7.1.1. S ep 1. Seed No maliza ion
The inpu passwo d TRIGATE2025 is con e ed in o ASCII numbe alues as ollows:
T → 84, R → 82, I → 73, G → 71, A → 65, T → 84, E → 69, 2 → 50, 0 → 48, 2 → 50, 5 → 53.
Then, he no maliza ion nume ic sequence becomes:
S={84, 82, 73, 71, 65, 84, 69, 50, 48, 50, 53}.
To ensu e an equal dis ibu ion o en opy, one may choose o hash his sequence using SHA-256 o pe haps a mo e
ligh weigh c yp og aphic hash such as Blake3 [1]. None heless, o he sake o anspa ency in his wo ked-ou
example, he di ec ASCII sequence is used.
7.1.2. S ep 2. Magic Squa e A angemen W ih Phi-Pi Weigh ing
Now, he sequence is pu in o a 5 × 5 magic squa e, wi h each cell being weigh ed by he wo i a ional cons an s o ϕ
( phi a phi), i.e., he golden a io (≈1.61803), and π ( pi pi) (≈3.14159).
The alue o he cell a posi ion (i,j) is gi en by:
V(i,j)=Sk×ϕi×πjV(i,j) = S_{k} imes phi^i imes pi^jV(i,j)=Sk×ϕi×πj
whe e SkS_kSk is he k- h seed elemen .
Example: Conside he mos op-le co ne ed poin (cell(1,1)) wi h a seed o 84:
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(12)$V(1,1)=84 imes phi^{1} imes pi^{1} app ox 84 imes 1.618 imes 3.1416 app ox 426.9$
E e y cell is simila ly calcula ed o ob ain he weigh ed g id. The eason o choosing he i a ional cons an s is ha
hey c ea e coo dina es ha do no epea - he coo dina es a e s ongly esis an o algeb aic educ ion a acks[2].
7.1.3. S ep 3. 231 Ga es Pe mu a ion
Each cell om he magic squa e is mapped in o he 231 Ga es Ne wo k, which is a combina o ial ca dioid in which each
node can spli o many inal ou comes. The pa h hus ob ained is guided by he seed sequence o an ex e nal QRNG and
is compu a ionally in ac able o bo h classical and quan um b u e o cing me hod[3].
Fo example, a alue a (1,1) wo h 426.9 may ge associa ed wi h ga e 42 wi h b anches o s a es O, C, and R. Pseudo-
andom walks may yield sequences such as:
{O,R,L,C,O,L,R,C,O,R,L} {O, R, L, C, O, L, R, C, O, R, L }{O,R,L,C,O,L,R,C,O,R,L}
This kind o se up in oduces exponen ial b anching. Whe eas a bina y-uni e se sys em o e s only 2n2^n2n
possibili ies, he T iGa e-QBLH ne wo k o e s 3n3^n3n (plus mo e when ac o ing in o a ional symme ies) [4].
7.1.4. S ep 4. Te ahed al T ina y Encoding
The ina y ou pu sequence is p ojec ed on o he aces o a egula e ahed on o ensu e opological binding. Each
s a e co esponds o one ace:
• O → Face 1 (Top, Open e ex)
• C → Face 2 (Base, S able closu e)
• R → Face 3 (Righ o a ional ace)
• L → Face 4 (Le o a ional ace)
Wi h he pa h a e sing he e ahed on, he ajec o y is o a ional ins ead o a simple linea bi s ing. This will
p e en he ad e sa y om econs uc ing he key i he bad guy does no know he s a e sequence and i s geome y
se up simul aneously [5].
Calcula ion snippe :
I he pa h is {O → R → L → C → O}, hen i is a o a ional loop a e sing e ahed al aces {1,3,4,2,1}, which can be
encoded as:
Te aPa h ={(1,3), (3,4), (4,2), (2,1)}
This gene a es bina y-compa ible keys s eams and geome ic signa u es (phi/pi--weigh ed coo dina es o o a ions).
Figu e 2 Te ahed al T ina y S a e T ansi ions (Illus a ion o a Ro a ional Pa h)
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7.1.5. S ep 5. Key Ex ac ion and Ou pu
The wo ou pu s a e gene a ed ia he e ahed al a e sal:
• A P ima y Enc yp ion Key: A 256-bi bi s ing yielded by he modula educ ion o he e ahed al pa h. Fo
ins ance, by conca ena ing he bina y equi alen s o {O=0, R=10, L=11, C=1}:
011101011... 011101011... 011101011...
Basically, one may unca e o ex end his o ma ch AES-256 speci ica ion [6].
• A Geome ic Signa u e: A se o phi/pi coo dina e pai s o he e ahed al o a ion pa h. This signa u e can ac
as a second au hen ica ion ac o , so ha in case o a bi s ing in e cep ion, he key emains in alid wi hou
geome ic e i ica ion.
Ad an ages Demons a ed in Example
• Quan um esis ance: The 231 Ga es + e ahed al mapping c ea e enough combina ional complexi y ha
G o e 's algo i hm canno ackle i [7].
• T ina y expansion: 3n3^n3n complexi y gene a es an exponen ial scale, ou pacing bina y PQC schemes.
• Geome ic Binding: The binding ensu es ha he coo dina es do no ace algeb aic collapse upon he
applica ion o ϕ phiϕ and π piπ [8].
• Ve sa ili y: Compa ible ou pu can be used bo h o AES-256, blockchain key walle s, IoT secu i y, o mul i ac o
p o ocols.
8. Ad an ages and Compa a i e Analysis o T iGa e QBLH
The Quan um-Sa e Enc yp ion Sys em T iGa e QBLH can be ega ded as compe i i e unde a ange o ci cums ances
wi h Con en ional Classical C yp og aphy and Eme ging Pos -Quan um C yp og aphy (E.P.C.). The ad an age is
a o ded by he T iGa e QBLH sys em by ac o s such as symbolic ma hema ics and geome y, oge he wi h ina y
logic, making i a o midable op ion o de ense agains an a ack by a quan um ad e sa y.
8.1. Quan um Resis ance
The majo ad an age o he T iGa e sys ems is he inhe en impossibili y o quan um algo i hms in ac o ing in ege s
using Sho ’s algo i hm o sea ching uns uc u ed se s wi h G o e ’s algo i hm [1]. Classical public-key sys ems like RSA
and ECC coun on he appa en ha dness o ac o ing a la ge in ege o sol ing a disc e e loga i hm. Gi en a powe ul-
enough quan um compu e , Sho ’s algo i hm would o e polynomial- ime solu ions o hese p oblems-g an ing any
classical key scheme o be comp omised [2].
In con as , he T iGa e employs non-linea , mul i-dimensional mapping o seed inpu s ia he 231 Ga es ne wo k and
on o he e ahed al aces. The s uc u e o he key space depends on combina o ial and opological complexi ies, no
jus on algeb aic ha dness. This implies ha he a acke a emp ing o econs uc he key aces comes up agains an
NP-ha d p oblem in geome ic space, making i a he in easible o di ec ly apply any quan um sea ching algo i hm o
i [3].
8.2. T ina y Complexi y
Ano he impo an bene i is ina y s a e encoding. Each s a e, being Open (O), Closed (C), Righ (R), o Le (L), is
mapped o a su ace o he e ahed on and can ha e mo e bi s in he bina y sys em (e.g., O=0, C=1, R=10, L=11).
Ma hema ically speaking, ha means he keyspace is exponen ially inc eased wi h he numbe o elemen a y codes,
pe mi ing 3^n and he e o e mo e pe mu a ions, as compa ed o 2^n pe mu a ions o any la ge enginee ing bina y
key se [4]. Simply pu , he ina y sys em inc eases how many key combina ions he e a e o wo y abou , while also
including o a ional symme y in key gene a ion, which by i sel becomes an obs acle o ei he classical o quan um
b u e- o ce a acks.
8.3. Geome ic Binding
The T iGa e sys em employs i a ional cons an s phi (ϕ ≈ 1.618) and pi (π ≈ 3.14159) in a weigh ed assignmen o seed
alues o magic squa es. Any se o coo dina es so cons uc ed can ne e epea and hence esis algeb aic
