Harvesting Random Telegraph Noise for True Random Number Generation

Con en s lis s a ailable a ScienceDi ec
In . J. Elec on. Commun. (AEÜ)
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Regula pape
Ha es ing andom eleg aph noise o ue andom numbe gene a ion
F.J. Rubio-Ba be o ∗, F. de los San os-P ie o, R. Cas o-Lopez , E. Roca, F.V. Fe nandez
Ins i u o de Mic oelec ónica de Se illa, IMSE, CNM (CSIC, Uni e sidad de Se illa), Amé ico Vespucio 28, 41092, Se illa, Spain
A R T I C L E I N F O
Keywo ds:
C yp og aphy
Ha dwa e secu i y
TRNG
CMOS
RTN
A B S T R A C T
A i s glance, Random Teleg aph Noise (RTN) in deeply scaled CMOS ansis o s may seem like a eliabili y
nuisance. Ye , behind he disc e e apping-and-de apping e en s lu ks a po en sou ce o ha dwa e en opy. In
his pape , we ha ness RTN o build a dual-pu pose secu i y module ha se es as bo h a Physical Unclonable
Func ion (PUF) and a T ue Random Numbe Gene a o (TRNG). By measu ing he so-called Maximum Cu en
Fluc ua ion (MCF) a ca e ully chosen obse a ion windows, ou design swi ches e o lessly be ween he s able
ou pu s needed o a PUF and he maximally unp edic able bi s eams demanded by a TRNG. Al hough single-
de ec RTN has long been deemed ideal o andomness, we show ha mul i-de ec RTN scena ios, much mo e
p e alen in eal-wo ld manu ac u ing, can also yield high-quali y andom bi s, especially when aided by
ligh weigh pos -p ocessing. Simple s a is ical me ics guide he ini ial uning, a e which he inal bi s eams
pass he NIST SP 800-22 es sui e o alida e he s a is ical soundness o ou p oposal. In doing so, we add ess
key challenges ha a ise when designing an RTN-based TRNG and compa e ou esul s agains s a e-o - he-a
solu ions, highligh ing ad an ages in ci cui simplici y, bi - a e scalabili y, and dual-use capabili y.
1. In oduc ion
In oday’s hype -connec ed wo ld, sma de ices a e mo e han
ools— hey a e he backbone o mode n communica ion. F om wea -
ables and sma phones o sma implan s, hese echnologies cons an ly
exchange da a, much o i highly sensi i e. This g owing in e connec-
i i y demands no only inno a ion bu also uncomp omising secu i y.
To sa egua d his da a, ad anced c yp og aphic algo i hms such as he
Ad anced Enc yp ion S anda d (AES) [1] and Ri es –Shami –Adleman
(RSA) c yp osys em [2] p o ide obus le els o p i acy and secu i y,
sa egua ding he da a being ansmi ed. These algo i hms ely on
he gene a ion o secu e c yp og aphic keys, a ask ha demands bo h
p ecision and us wo hiness. To mee his need, mode n ha dwa e
p imi i es like Physical Unclonable Func ions (PUFs) and T ue Random
Numbe Gene a o s (TRNGs) ha e eme ged as game-change s. By o -
e ing compac , cos -e ec i e, and ene gy-e icien solu ions, hey pa e
he way o unpa alleled secu i y. Whe he p e en ing coun e ei ing,
enabling de ice au hen ica ion, o p o ec ing sensi i e communica-
ions, hese echnologies ep esen a signi ican leap o wa d in he
ques o secu e and esilien sma sys ems [3].
TRNGs a e designed o gene a e uly andom numbe s by ha -
nessing unp edic able physical phenomena, such as ambien noise,
ji e , chao ic sys em dynamics, o me as abili y. These na u al a i-
a ions ensu e he andomness and unp edic abili y equi ed o secu e
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (F.J. Rubio-Ba be o), [email p o ec ed] (F.d.l. San os-P ie o), [email p o ec ed] (R. Cas o-Lopez),
[email p o ec ed] (E. Roca), [email p o ec ed] (F.V. Fe nandez).
c yp og aphic p ocesses. An example o a silicon-based TRNG is one
ha exploi s me as abili y in lip- lops o ing oscilla o s o ha es
andomness om he mal noise [4]. Con e sely, PUFs se e a unda-
men ally di e en pu pose. Unlike TRNGs, PUFs depend on s abili y
o e ime and ex e nal condi ions. In c yp o-sys ems whe e he key
has o be s o ed in con en ional non ola ile memo ies (NVMs) ha
a e ulne able o coun e ei ing, e e se enginee ing, and ampe ing,
PUFs ha a e mo e adap able and immune o hese h ea s can be used
ins ead o NVMs since he key is no sa ed bu a he p oduced e e y
ime i is demanded. Simila o TRNGs, PUFs also ely on an en opy
sou ce. In silicon PUFs, one common example is he andom a iabili y
in ansis o h eshold ol ages, 𝛥𝑉 h. This en opy can be ha es ed
h ough mechanisms such as he delay a ia ions in ing oscilla o s
caused by 𝛥𝑉 h. The esul is a unique, consis en , and non-p edic able
ou pu , in he o m o a digi al bi s eam. These dis inc i e ou pu s make
PUFs highly aluable o applica ions such as de ice au hen ica ion,
c yp og aphic key gene a ion, and ob usca ion. Toge he , TRNGs and
PUFs p o ide ounda ional ools o building secu e, esilien ha dwa e
sys ems. Simila ly, TRNGs play a pi o al ole in hese applica ions
by p o iding he high-quali y andomness equi ed o secu e c yp o-
g aphic p o ocols, such as gene a ing enc yp ion keys o nonces o
secu e communica ion. Beyond c yp og aphy, TRNGs also ind applica-
ions in a eas like s ochas ic compu ing, whe e hei abili y o p oduce
h ps://doi.o g/10.1016/j.aeue.2025.155801
Recei ed 9 Janua y 2025; Accep ed 7 Ap il 2025
In . J. Elec on. Commun. (AEÜ) 196 (2025) 155801
A ailable online 17 Ap il 2025
1434-8411/© 2025 The Au ho s. Published by Else ie GmbH. This is an open access a icle unde he CC BY-NC-ND license
( h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ ).
F.J. Rubio-Ba be o e al.
uly andom bi s eams suppo s e icien , low-powe p obabilis ic
compu a ion [5].
Mos silicon-based PUFs and TRNGs ely on Time-Ze o Va iabili y
(TZV) as a p ima y en opy sou ce. Once seen as a non-ideali y, TZV
is now alued o i s po en ial in ha dwa e secu i y applica ions. Wi h
con inued CMOS scaling, hough, conce ns ha e shi ed owa d Time-
Dependen Va iabili y (TDV), a ising om mechanisms like aging and
Random Teleg aph Noise (RTN). Add essing TDV has become a collab-
o a i e e o , om i s p ecise cha ac e iza ion [6] o he de elopmen
o TDV- esilien ci cui s [7]. Rema kably, TDV, like TZV, is mo e han
a challenge—i o e s a no el en opy sou ce. In pa icula , RTN has
eme ged as a compelling sou ce o en opy o ha dwa e p imi i es,
wi h applica ions anging om c yp og aphic sys ems o secu e com-
munica ions and secu i y echnologies like PUFs [8]. This po en ial
s ems om i s unique physical cha ac e is ics: RTN is a s ochas ic
phenomenon caused by cha ge apping and de apping a de ec s (also
known as aps) wi hin he ga e oxide o nanoscale ansis o s. These
e en s p oduce ime-dependen luc ua ions in he ansis o ’s d ain
cu en , c ea ing a dis inc i e ‘‘on–o ’’ eleg aph-like signal. The du a-
ion o he ‘‘on’’ and ‘‘o ’’ s a es depends on he iming cha ac e is ics
o each indi idual de ec .
RTN has also been he en opy sou ce o choice o a hand ul
o explo a o y esea ch wo ks, inspi ing a ious p oposals on how o
ha ness and ha es i o build silicon-based TRNGs [9–13]. In [9],
a sampling equency is de ined o pe iodically sample he single-
de ec d ain cu en le els, eco ding he ansi ions be ween he ‘‘on’’
and ‘‘o ’’ s a es. These ansi ions, d i en by he s ochas ic na u e o
apping and de apping e en s, o m he ounda ion o gene a ing
andom bi s: he ‘‘on’’ s a e co esponds o a bi alue o ‘1’, while he
‘‘o ’’ s a e co esponds o ‘0’. Howe e , a signi ican challenge a ises
om he sampling equency, which can in oduce a bias: an unequal
p opo ion o ‘1’s and ‘0’s in he bi s eam. Add essing his bias equi es
co ec ion mechanisms, such as pos -p ocessing algo i hms, which sig-
ni ican ly educe he inal bi h oughpu (i.e., he bi a e). The wo k
in [10] ex ends he seminal idea in [9] o add ess single-de ec RTN
and also p opose a solu ion o bene i om mul i-de ec RTN scena ios
(i.e., ansis o s wi h many aps). In he la e case, hey p oposed
a mo e sophis ica ed app oach in ol ing he digi iza ion o he d ain
cu en le els and a bi unca ion scheme, unde he hypo hesis ha
he leas -signi ican bi s p o ide he bes sou ce o andomness, while
he mos -signi ican bi s a e disca ded. Ano he app oach is p oposed
in [11], whe e only comple e e en s a e conside ed: a cap u e ollowed
by an emission, o ice e sa (equi alen ly, an ‘‘on’’ s a e ollowed
by an ‘‘o ’’ s a e, o ice e sa). This me hod, when pai ed wi h an
app op ia ely chosen sampling equency, has he po en ial o elimina e
bi bias. Howe e , he desc ibed solu ion ails o le e age he addi ional
en opy inhe en in mul i-de ec scena ios, e ec i ely educing he
p oblem o he single-de ec case and yielding li le o no bene i
om he p esence o mul iple de ec s. The RTN-based TRNG in [12]
p oposes an edge- o-pulse scheme mainly ocused o mul i-de ec RTN
scena ios. The digi ized pulses hen sample a high- equency oscilla o ,
elimina ing he need o addi ional pos -p ocessing bu binding he bi
a e di ec ly o he iming p ope ies o he RTN de ice de ec s. Finally,
a ecen TRNG has been p oposed in [13] whe e di e en ypes o noise
( licke and RTN) a e used as en opy sou ces. Howe e , he epo ed
esul s a e limi ed o a hand ul o non-comme cial ansis o samples
only.
Despi e signi ican ad ancemen s in RTN-based TRNGs, none o he
epo ed app oaches u ilize he same en opy sou ce, RTN, o con-
s uc ing bo h PUFs and TRNGs. Addi ionally, RTN is a ela i ely slow
phenomenon, wi h he du a ions o he ‘‘on’’ and ‘‘o ’’ s a es ypically
anging om 10−6 s o se e al seconds [12,13], making high bi a es
a pe sis en challenge. Howe e , he exis ing wo ks lack a de ailed
analysis o he ade-o s ha single- and mul i-de ec RTN in mode n
CMOS ansis o s o e in his con ex . This wo k aims o add ess hese
challenges by p oposing s a egies o e ec i ely manage bo h single-
Fig. 1. Illus a ion o RTN in he ansis o ’s d ain cu en , highligh ing he wo ap
s a es and speci ying he key RTN pa ame e s.
and mul i-de ec scena ios, while p o iding a comp ehensi e pe spec-
i e on he ade-o s be ween imp o ing andomness and enhancing
bi a es o RTN-based andom bi s eams in comme cial echnologies.
Building on p io wo k on RTN-based PUFs [8], i in oduces a e ined
a chi ec u e ha , h ough a sub le modi ica ion, enables he gene a ion
o uly andom bi s eams, ex ending i s applicabili y o bo h PUFs and
TRNGs.
The s uc u e o his pape is as ollows: Sec ion 2 in oduces he
undamen als o RTN and he en opy ex ac ion me hod o cons uc -
ing bo h a PUF and a TRNG. Sec ion 3 de ails he p oposed PUF and
TRNG a chi ec u e, explaining how bi esponses a e gene a ed and he
adjus men s needed o swi ch be ween he wo. Sec ion 4 ou lines he
me hodology o e alua ing TRNG quali y, om basic o indus y-g ade
andomness es s, and p esen s he amewo k used o gene a e RTN-
based andom bi s eams. Sec ion 5 summa izes he esul s and indings
ac oss p og essi ely complex scena ios, while Sec ion 6 concludes he
pape . Auxilia y p oo s and o mulae a e p o ided in Appendices o D.
2. Exploi ing RTN o PUF & TRNG
2.1. RTN undamen als
As b ie ly ou lined ea lie , RTN mani es s in he d ain cu en o
CMOS ansis o s as as , andom, and disc e e shi s. F om a de ec -
cen ic pe spec i e, hese shi s a ise due o cha ge ca ie s unde going
apping and de- apping e en s a de ec s wi hin he ansis o ’s ga e
oxide. These e en s lead o luc ua ions in he h eshold ol age (𝑉 h)
and consequen ly o a ia ions in he d ain cu en [14], as illus a ed
in Fig. 1. RTN can be cha ac e ized by se e al key pa ame e s: he
numbe o de ec s in he de ice, he ampli ude o he d ain cu en
shi s (𝛥𝐼D) induced by each de ec , and he s a is ical ime cons an s
go e ning he emission (𝑡𝑒) and cap u e (𝑡𝑐) o cha ge ca ie s. Emission
e e s o he ime equi ed o a de ec o elease a cha ge ca ie when
occupied, while cap u e ep esen s he ime lapse equi ed o a de ec
o ap a cha ge ca ie when unoccupied. These dynamics, as depic ed
in Fig. 1, mani es as luc ua ions in he ansis o ’s d ain cu en du ing
RTN e en s.
Ca ie emission and cap u e imes a e ypically desc ibed by a
Ma ko p ocess, whe e he pa ame e s a e s ochas ic and exhibi ex-
ponen ial dis ibu ions (1). The dis ibu ions unde conside a ion ha e
means deno ed by 𝜏𝑒 and 𝜏𝑐, espec i ely. These mean alues a e also
commonly e e ed o as dwell imes. Fo simplici y, om his poin
onwa d, and unless explici ly s a ed o he wise, we will use he e ms
cap u e imes, emission imes, o dwell imes o e e speci ically o
he mean alues 𝜏𝑒 and 𝜏𝑐, a he han o indi idual ins ances (𝑡𝑐 o 𝑡𝑒).
Recu si ely, he mean alues a e also andomly dis ibu ed, ollowing
he bi a ia e logno mal dis ibu ion in (2) [15].
𝑃𝑒∕𝑐(𝑡) = 1
𝜏𝑒∕𝑐
⋅exp (−𝑡
𝜏𝑒∕𝑐)(1)
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
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F.J. Rubio-Ba be o e al.
𝐷de (𝑥, 𝑦) = 1
2𝜋𝜎𝜏𝑒𝜎𝜏𝑐√1 − 𝜌2
⋅exp (−1
2√1 − 𝜌2[(𝑥−𝜇𝜏𝑒)2
𝜎2
𝜏𝑒
+(𝑦−𝜇𝜏𝑐)2
𝜎2
𝜏𝑐
+2𝜌(𝑥−𝜇𝜏𝑒)(𝑦−𝜇𝜏𝑐)
𝜎𝜏𝑒𝜎𝜏𝑐]) (2)
whe e 𝜇𝜏𝑒
and 𝜇𝜏𝑐
ep esen he mean cap u e and emission imes o
he log-no mal dis ibu ion, espec i ely, while 𝜎𝜏𝑒
and 𝜎𝜏𝑐
deno e he
s anda d de ia ions, and 𝜌 is he co ela ion coe icien . Addi ionally,
he numbe o de ec s in a de ice is ypically modeled as a Poisson-
dis ibu ed andom a iable [16]. In his con ex , he pa ame e 𝜆 o
he Poisson dis ibu ion ep esen s he expec ed numbe o de ec s,
such ha he p obabili y o a ansis o ha ing exac ly 𝑘 de ec s is
de e mined by 𝜆. Fo a speci ic echnology, wi h an a e age de ec
densi y 𝜆 (e.g., numbe o de ec s pe 𝑛𝑚2) and a de ice a ea 𝐴, he
expec ed numbe o de ec s can be exp essed as:
P (𝑘) = (𝜆𝐴)𝑘𝑒−𝜆𝐴
𝑘!(3)
The numbe o de ec s can be ex ac ed by di ec obse a ion as he
coun o disc e e cu en le els in he d ain cu en [17]. In pa icula ,
he cu en ace in Fig. 1 exhibi s a single-de ec RTN wi h shi
𝛥𝐼D. As hese pa ame e s a e andom in na u e, wo equally designed
de ices p esen di e en RTN-induced shi s. As an example, in a 65-nm
comme cial echnology, imes and shi ampli udes can be as low as a
ew mic oseconds and nanoamps, espec i ely.
2.2. The maximum cu en luc ua ion: a uni ied me hod o RTN en opy
ha es ing
An e ec i e app oach o ha ness en opy om RTN is h ough he
concep o Maximum Cu en Fluc ua ion (MCF) [18]. The MCF ep-
esen s he cumula i e di e ence be ween he maximum and minimum
alues o a d ain cu en ace o e a gi en ime in e al. The key
ad an age o using MCF lies in i s abili y o encapsula e all ele an
RTN in o ma ion, including emission and cap u e imes, cu en shi s,
and he numbe o de ec s. Ma hema ically, MCF is a nonnega i e
and e e -inc easing unc ion because i accumula es he di e ences
be ween he maximum and minimum alues o he d ain cu en ,
which a e di ec ly in luenced by he la ges and smalles a ia ions
in 𝑉 h o e a gi en pe iod. De ices wi h mo e de ec s, and hus mo e
p onounced RTN luc ua ions, esul in highe MCF alues. Ano he
no able ea u e o he MCF is i s inhe en insensi i i y o a ia ions
caused by misma ches in he mean alues o 𝑉 h. Ins ead, i ocuses
solely on he absolu e di e ences be ween he minimum and maximum
alues, ensu ing obus en opy ha es ing.
In ac , ecen s udies ha e u ilized his me ic o de elop an aging-
esis an PUF [8]. In his wo k, he MCF is calcula ed as he cumula i e
di e ence, o e a speci ic obse a ion window (𝑡MCF), be ween he
maximum and minimum d ain cu en a ia ions o a p ope ly bi-
ased de ice. As an example, he MCF o a single de ec ansis o is
depic ed in Fig. 2. In his app oach, he RTN-based PUF desc ibed
in [8] calcula es he MCF du ing 𝑡MCF o wo iden ically designed
ansis o s (selec ed om an a ay o 𝑀 de ices) using Peak-De ec -and-
Hold ci cui s (PDHmax and PDHmin) [19]. The esul ing MCF alues a e
hen compa ed o a ain a 1-bi PUF esponse.
The 𝑡MCF alue holds signi ican ele ance o a eliable PUF e-
sponse: he la ge his alue, he mo e RTN can be cap u ed in o he
MCF me ic, and he mo e eliable he esponse bi will be om one
challenge o he nex (i.e., he MCF compa ison will ideally yield he
same alue e e y ime a ce ain challenge is gi en). This a ionale
yields he ounda ional ideal behind his pape : I educing 𝑡MCF beyond
a ce ain alue causes un eliable dispe se bi s, po en ially his un elia-
bili y is wha can be used o gene a e a ue andom numbe . In ac ,
as will be demons a ed, by me hodically selec ing he alue o 𝑡MCF, i
would be possible o a ain PUF-like esponse bi s o p oduce andom
bi s eams, laying he ounda ion o a dual PUF-TRNG a chi ec u e
ha uses RTN.
Fig. 2. A ansis o ’s d ain cu en showing RTN ( op plo , op ace), i s compu ed
MCF o each 𝑡MCF ( op plo , bo om ace) and i s digi ized coun e pa , DMCF (bo om
plo ), compu ed by compa ing he MCF wi h a e e ence cu en 𝐼 e .
3. Ha dwa e secu i y p imi i es based on RTN & MCF
3.1. PUF a chi ec u e
This wo k p ima ily le e ages he wo co e componen s o he
p e iously p oposed [8] and implemen ed RTN-based PUF [19]: he
ansis o a ay, which ac s as he en opy sou ce wi h app op ia ely bi-
ased MOS ansis o s, and he sensing ci cui y, esponsible o ex ac ing
and measu ing RTN o p oduce he ou pu esponse.
The co e componen o he RTN-PUF is an a ay o 𝑀 iden ically
designed ansis o s wi h minimal dimensions. These small dimensions
enhance he isibili y o RTN, as la ge , o e -scaled de ices ypically
exhibi Flicke noise (1∕𝑓) a he han he Lo en zian noise (1∕𝑓2)
cha ac e is ic o RTN [10]. Each ime a esponse bi is eques ed, a
challenge is p esen ed o he PUF, speci ying which wo ansis o s a e
pai ed and in wha o de . The pai ing o de is c i ical, as e e sing
i changes he sign o he di e ence be ween he MCFs, hus al e ing
he esponse bi . The d ain cu en s (𝐼𝐷𝑆 ) o he selec ed ansis o s
a e con inuously moni o ed, and hei MCFs a e compu ed o e a
speci ied ime in e al (𝑡MCF, e.g., 100 μs). A e his ime, he MCFs
a e compa ed: i one ansis o ’s MCF is la ge , he ou pu esponse bi
is assigned a alue o ‘1’; o he wise, i is ‘0’. In his PUF opology, he
challenge selec s he speci ic pai o ansis o s om he a ay. The
MCFs a e de e mined by measu ing and sub ac ing he cumula i e
maximum and minimum d ain cu en le els o each ansis o . The
esponse ime o he sys em depends solely on 𝑡MCF, in oducing a ade-
o : a longe 𝑡MCF collec s mo e RTN in o ma ion, inc easing en opy
and eliabili y, bu educes he bi a e ha can be ex ac ed om he
PUF.
3.2. P oposed TRNG a chi ec u e
The RTN-based PUF a chi ec u e p oposed in [19] can be seamlessly
adap ed o unc ion as a TRNG. While he PUF design equi es wo an-
sis o s o gene a e a eliable esponse, a TRNG, as will be shown, needs
only a single ansis o . This adap a ion e ains he o iginal a chi ec u e
bu eplaces he compa ison be ween wo ansis o s wi h a compa ison
o he MCF alue o one ansis o agains a e e ence alue, 𝐼 e , as
illus a ed in Fig. 3. To p oduce he bi s eam, a digi al ep esen a ion
o he MCF (DMCF) is de i ed by compa ing he compu ed MCF wi h
𝐼 e , and s o ing he esul ing bi in a egis e a he end o each 𝑡MCF
pe iod. The e e ence alue is ca e ully chosen o lie jus abo e he
noise le el, ensu ing ha noise is e ec i ely excluded om he bi
gene a ion p ocess. The ansis o is moni o ed con inuously, wi h i s
DMCF compu ed pe iodically, gene a ing a new bi a a equency o
1∕𝑡MCF.
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
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F.J. Rubio-Ba be o e al.
Fig. 3. Diag am o he p oposed RTN-based TRNG opology di ec ly de i ed om [8]
ha u ilizes only a single de ice as he en opy sou ce.
In con as o PUFs, whe e a longe 𝑡MCF enhances en opy and
eliabili y by collec ing mo e RTN in o ma ion bu educes he bi
a e, he design o a TRNG ocuses on iden i ying he obse a ion
window (𝑡MCF) ha maximizes andomness. Fo a PUF, he gene a ed
bi s mus no only be andom o ensu e unp edic abili y bu also s able,
p oducing he same esponse each ime he same challenge is applied.
This s abili y is c i ical o eliable au hen ica ion. On he o he hand,
andomness is pa amoun o TRNGs, no only wi hin each que y bu
also be ween successi e que ies. A TRNG’s bi s eam should exhibi
high andomness and di e e e y ime i is eques ed. The p ope
adjus men o he obse a ion window, 𝑡MCF, is he key o enabling he
dual unc ionali y o he MCF-based a chi ec u e, allowing i o ope a e
as bo h a PUF and a TRNG. By uning 𝑡MCF, he same a chi ec u e can
achie e he s able esponses equi ed o PUFs o he high andomness
demanded by TRNGs.
4. RTN-based TRNG quali y assessmen
To alida e he easibili y o he RTN-based TRNG concep , we
de eloped a physics-based RTN simula o and analysis amewo k.
This ool se es wo main pu poses: i s , i o e s a lexible pla o m
o gene a ing andom bi s eams unde a ying condi ions, allowing
he explo a ion o di e en a iable combina ions. Second, i e alu-
a es he andomness o he gene a ed bi s eams h ough a wo-s ep
es ing p ocess, beginning wi h basic es s and ad ancing o igo ous
indus ial-g ade assessmen s, as will be demons a ed la e . The sim-
ula o is buil upon ex ensi e expe imen al cha ac e iza ion o CMOS
ansis o s. This includes ex ac ing key pa ame e s such as dwell imes,
he numbe o de ec s, and he impac o h eshold ol age a ia ions.
I also inco po a es he e ec s o ex e nal ac o s such as bias ol ages,
a ea scaling, and empe a u e. These insigh s ensu e ha he simu-
la o accu a ely ep esen s he physical beha io o RTN in p ac ical
scena ios. Using his amewo k ex ensi ely, as will be de ailed la e ,
has allowed us o analyze how ue andomness can be e ec i ely
ex ac ed om RTN. We ha e in es iga ed he condi ions unde which
andomness is op imized, he mos e icien me hods o achie e i ,
and how a ious pa ame e s, such as RTN dynamics and he use o
MCF, impac he quali y o he gene a ed bi s eams. This comp ehen-
si e app oach p o ides he ounda ion o unde s anding and e ining
RTN-based TRNG design.
4.1. RTN-based TRNG amewo k
Fig. 4 depic s a high-le el low diag am o he de eloped Py hon-
coded RTN-based T ue Random Numbe Gene a o TRNG amewo k.
A he hea o his amewo k lies he RTN bi s eam gene a o , whose
pseudocode is p o ided in Algo i hm 1. This algo i hm le e ages he
physics-based RTN model discussed in Sec ion 2 and d aws inspi a ion
om he simula ion me hodology desc ibed in [15]. Bu ins ead o
Fig. 4. Simpli ied low diag am o he RTN-based TRNG emula o .
gene a ing he en i e ansis o ’s d ain cu en ace o cons uc an
𝑛-bi bi s eam (las ing 𝑛⋅𝑡MCF seconds) and subsequen ly calcula e
he alue o DMCF o each obse a ion window o du a ion 𝑡MCF,
his algo i hm employs a mo e e icien me hodology. Speci ically, i
di ec ly iden i ies ansi ions – om cap u e o emission and ice e sa
– wi hin he obse a ion window. These ansi ions esul in a non-ze o
DMCF, p oducing a ‘1’ in he bi s eam. This app oach signi ican ly
educes compu a ional complexi y while main aining he in eg i y o
he andom numbe gene a ion p ocess.
The amewo k enables use s o gene a e RTN-based bi s eams and
explo e a ious scena ios by uning pa ame e s such as he numbe
o de ec s, he speci ic 𝑡MCF alues, he cap u e and emission imes
(𝜏𝑐, 𝜏𝑒), and he leng h o he bi s eams. Many o hese pa ame e s,
including he numbe o de ec s, 𝜏𝑐, and 𝜏𝑒, can ei he be speci ied
manually by he use o sampled om echnology-speci ic dis ibu ions.
Fo ins ance, he numbe o de ec s can be d awn om a Poisson
dis ibu ion (3), while he cap u e and emission imes can ollow
he logno mal dis ibu ion de ined in (2), o any o he dwell ime
dis ibu ion. To enhance e sa ili y, he amewo k allows use s o
gene a e bi s eams o a ixed obse a ion window o sweep ac oss
mul iple 𝑡MCF alues. I can also simul aneously gene a e bi s eams
o mul iple ansis o s, each wi h i s own con igu able numbe o
de ec s, i equi ed. In addi ion o bi s eam gene a ion, he amewo k
p o ides pos -p ocessing capabili ies aimed a imp o ing andomness
in he bi s eams. This lexibili y is u he ex ended by enabling use s
o de ine speci ic es se ings o di e en scena ios h ough a con igu-
a ion ile. The amewo k suppo s ully au oma ed da ase gene a ion,
allowing a ho ough e icien e alua ion o he p oposed RTN-based
TRNG. The gene a ed bi s eams can be seamlessly passed o pos -
p ocessing modules o u he e alua ion, enabling use s o es and
op imize di e en app oaches o enhance he quali y and andomness
o he bi s eams.
4.2. Randomness e alua ion: elemen a y o indus ial-g ade es ing
To e alua e he quali y o a TRNG and he en opy sou ce used o
gene a e such a numbe , a a ie y o me hods a e a ailable, anging
om quick bu less comp ehensi e me ics o mo e igo ous indus y-
g ade es s. Simple me ics o e apid assessmen s by highligh ing
biases o pa e ns in he bi s eam, bu hey do no p o ide a com-
ple e pic u e o andomness. In con as , indus y-g ade me hods o e
g ea e accu acy and eliabili y bu equi e compu a ionally in en-
si e algo i hms o la ge bi s eams (e.g., ens o millions o bi s) o
meaning ul and p ecise analysis. Howe e , op imizing condi ions o
maximize andomness and e iciency o en equi es an i e a i e explo-
a ion, such as sweeping obse a ion windows, a ying he numbe o
de ec s in a ansis o , o es ing di e se combina ions o dwell imes.
While hese app oaches p o ide a deepe e alua ion o andomness,
hei inhe en complexi y and esou ce demands o en make hem
imp ac ical o apid op imiza ion o RTN-based andomness.
In his s udy, wo easy- o-e alua e me ics will be used du ing such
an explo a ion o how o exploi RTN, whe e an e icien iden i ica ion
o he scena ios yielding po en ially ue andom bi s eams is neces-
sa y. These me ics, whose ma hema ical de ini ions can be ound in
Appendix A, a e:
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F.J. Rubio-Ba be o e al.
Algo i hm 1: Mul i-de ec RTN Bi s eam Gene a o
Inpu : 𝑛𝑏𝑖𝑡𝑠 (numbe o bi s o gene a e), 𝑡𝑀𝐶𝐹 (obse a ion window
du a ion), 𝜏𝑒,𝑑 , 𝜏𝑐,𝑑 (emission and cap u e imes o he de ec s),
𝑑 (numbe o de ec s), 𝑡𝑠𝑎𝑚𝑝𝑙 (sampling ime)
Ou pu : bi s eam_RTN (Gene a ed bi s eam)
1S ep 1: Ini ializa ion
2Se 𝑡𝑖𝑚𝑒[𝑑]←0 and andomly ini ialize 𝑠𝑡𝑎𝑡𝑒[𝑑];
3Se bi s eam_RTN ←emp y a ay;
4S ep 2: Simula e E en s
5while Gene a ed bi s < 𝑛𝑏𝑖𝑡𝑠 do
6 o Each de ec 𝑑 do
7while Elapsed ime < 𝑡𝑀𝐶𝐹 do
8i 𝑠𝑡𝑎𝑡𝑒[𝑑] = 1 (high s a e) hen
9Gene a e cap u e e en 𝑡𝑐≥𝑡𝑠𝑎𝑚𝑝𝑙;
10 Upda e 𝑡𝑖𝑚𝑒[𝑑] and se 𝑠𝑡𝑎𝑡𝑒[𝑑]←0;
11 else
12 Gene a e emission e en 𝑡𝑒≥𝑡𝑠𝑎𝑚𝑝𝑙;
13 Upda e 𝑡𝑖𝑚𝑒[𝑑] and se 𝑠𝑡𝑎𝑡𝑒[𝑑]←1;
14 Compu e a bi o he RTN based on e en s;
15 Append he bi o bi s eam_RTN;
16 Adjus 𝑡𝑖𝑚𝑒[𝑑] o align wi h he nex obse a ion window;
17 S ep 3: Finalize Bi s eam
18 Con e bi s eam_RTN o bina y alues (0 o 1);
19 e u n bi s eam_RTN;
1. Hamming Weigh , 𝐻𝑊 . This me ic e alua es whe he he
dis ibu ion o 1s and 0s in he bi s eam is well balanced o a
ue andom numbe , meaning 𝐻𝑊 should con e ge o 50%, o ,
con e sely, biased (i.e., 0%, o an all-ze o bi s eam, o 100%,
in he case o an all-ones bi s eam). This me ic only ocuses on
global balance and does no conside he bi s eam’s s uc u e o
co ela ions.
2. Au oco ela ion, 𝐴𝐶. I measu es he dependency be ween a
bi and i s lagged coun e pa o e he bi s eam. E en i he
Hamming Weigh is balanced (50%), he bi s could s ill exhibi
pe iodici y o pa e ns. Au oco ela ion cap u es hese local de-
pendencies and e eals epea ing pa e ns in a s ing o andom
bi s. The absence o au oco ela ion in a TRN implies ha each
bi is comple ely independen o i s p eceding bi s, so an a acke
would ind i i ually impossible o p edic he subsequen ly
gene a ed bi . A lawless andom bi s eam is expec ed o p esen
an AC alue o ze o. In his pape , unless o he wise s a ed,
AC (A.2) is e alua ed a lag 1 (𝑘= 1), which compu es he
co ela ion ha exis s be ween one bi and he nex .
Rega ding indus y-g ade alida ion, he NIST S a is ical Tes Sui e,
de ailed in Special Publica ion 800-22 [20], p o ides a comp ehensi e
amewo k o assessing he andomness o bina y sequences. This
sui e includes 15 s a is ical es s, each designed o de ec speci ic
ypes o non- andomness. The es s ange om basic checks like he
F equency Tes , which measu es he p opo ion o 0s and 1s, o mo e
complex es s such as he Linea Complexi y Tes , which e alua es he
complexi y o a sequence. The es s ely on a ious ma hema ical mea-
su es, including chi-squa e dis ibu ions and p obabili ies (P- alues), o
alida e whe he a sequence can be conside ed andom. The ypical
minimum sequence leng h is 100 bi s, hough some es s (like he Linea
Complexi y Tes o he Random Excu sions Tes ) equi e signi ican ly
longe sequences (e.g., 103 o 106 bi s wi h a minimum numbe o
55 sequences) o meaning ul esul s. This es sui e is in aluable
o e alua ing andom numbe gene a o s, ensu ing he quali y and
unp edic abili y essen ial o secu e c yp og aphic ope a ions.
I is c ucial o conside an addi ional me ic: he Bi Ra e (𝐵𝑅),
which ep esen s he a e a which uly andom bi s a e gene a ed and
ou pu , ypically measu ed in bi s pe second. Conside ing he bi a e
no only acili a es compa isons wi h o he RTN-based gene a o s, bu
also p o ides a means o quan i y he impac and ade-o s in oduced
by pos -p ocessing echniques.
5. Expe imen al esul s
In e alua ing he andom numbe gene a ion capabili ies o he
a chi ec u e illus a ed in Fig. 3, we will explo e inc easingly complex
de ec scena ios o iden i y sub le ye impac ul ade-o s. We begin
wi h he simples case: a single ansis o exhibi ing a single RTN. By
dissec ing he in ica e in e play o his de ec ’s pa ame e s, we unco e
c i ical challenges ha in luence andomness quali y. D awing on hese
insigh s, we p opose a ge ed s a egies o enhance he andomness o
he gene a ed bi s eam.
Building on he lessons lea ned om his baseline con igu a ion,
we hen examine a mo e complex scena io in which mul iple de ec s
coexis wi hin a single ansis o . This mul i-de ec en i onmen aises
a new se o cons ain s and design conside a ions, di e ging om he
single-de ec case. By con as ing bo h scena ios, we no only elucida e
he co e mechanisms behind hei dis inc s a is ical beha io s, bu also
lay ou p ac ical solu ions o e ine andomness.
Fo all scena ios analyzed, he gene a ed bi s eams using he ap-
p oach desc ibed in Sec ion 4.1 e lec a 65-nm, 1.2-V CMOS echnol-
ogy, a ansis o geome y o 80 nm/60 nm, and biasing ol ages o
𝑉𝐺𝑆 = 0.9V and 𝑉𝐷𝑆 = 0.1V. To accoun o RTN, a dis ibu ion o
de ec s has been conside ed such ha he dwell imes o de ec s a e
uni o mly dis ibu ed be ween 1 ms and 100 s [21].1 Also, and unless
o he wise s a ed, he a e age numbe o de ec s is 𝜆= 4.
5.1. Scena io A.1: one ansis o wi h one RTN de ec
5.1.1. Desc ip ion o expe imen al esul s
The goal o his in es iga ion is o examine he ela ionships among
he de ec ’s dwell imes, he 𝑡MCF window, and he esul ing andom bi -
s eam quali y, le e aging he DMCF concep wi hin he p oposed RTN-
based a chi ec u e shown in Fig. 3. To accomplish his, we gene a ed
one-million-bi bi s eams unde a ying dwell imes, sys ema ically
sweeping ac oss di e en 𝜏𝑐- o-𝜏𝑒 a ios. Fo a mo e s uc u ed and
insigh ul analysis, we in oduce he concep o a cha ac e is ic ime
cons an , 𝜏0.
De ining 𝛽 as 𝜏𝑐∕𝜏𝑒, we can exp ess 𝜏𝑐 and 𝜏𝑒 in e ms o 𝜏0:
𝜏𝑐= (1 + 𝛽)𝜏0and 𝜏𝑒=(1 + 1
𝛽)𝜏0(4)
To co e a wide ange o 𝛽 alues and achie e easonable 𝑡MCF window
leng hs, hus allowing explo a ion o highe bi a es, we se 𝜏0=
0.001 s.
Fig. 5 illus a es he a ious 𝜏𝑐 and 𝜏𝑒 combina ions used. Each
𝛽 alue de e mines he co esponding 𝜏𝑐 and 𝜏𝑒 alues; o each 𝛽,
we sweep he 𝑡MCF window and gene a e a 10𝑒6-bi bi s eam. We
employed 100 equally spaced alues o 𝛽 and 100 equally spaced alues
o 𝑡MCF, esul ing in a o al o 100×100×1M gene a ed bi s. This igo ous
app oach enables us o de e mine he op imal condi ions o p oducing
a uly andom bi s eam om RTN using he MCF en opy ha es ing
unc ion.
The esul s o he pa ame ic explo a ion a e p esen ed in Fig.
6, om which se e al insigh s can be d awn. Fi s , he e is a clea
symme y wi h espec o 𝛽, sugges ing ha he quali y o he andom
bi s eam, in e ms o HW and AC, is in luenced by he la ge o he wo
alues, 𝜏𝑐 and 𝜏𝑒. When 𝑡MCF is signi ican ly la ge (e.g., 𝑡MCF = 0.1s
wi h 𝛽= 1) o signi ican ly smalle (e.g., 𝑡MCF = 1 ms wi h 𝛽= 1k
o 𝛽= 1 m) han max(𝜏𝑐, 𝜏𝑒), he bi s eam consis s en i ely o ze oes
(𝐻𝑊 = 0%) o ones (𝐻𝑊 = 100%), espec i ely.
1The dis ibu ion o de ec s, as gi en by (2) o he 65-nm echnology unde
conside a ion, exhibi s an almos uni o m beha io in he ange be ween 1 ms
and 100 s.
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F.J. Rubio-Ba be o e al.
Fig. 5. Cap u e (𝜏𝑐, solid blue) and emission (𝜏𝑒, dashed ed) ime cons an s as
unc ions o 𝛽=𝜏𝑐
𝜏𝑒
using 𝜏0= 0.001 s. The shaded a ea indica es he explo ed 𝑡MCF
window ange, om 0.001 s o 0.1s.
Fig. 6. Hea maps o he HW and AC me ics o he gene a ed bi s eams o 1
ansis o and 1 de ec . The solid whi e line ep esen s he isoline whe e he ideal
alue 𝐻𝑊 = 50% is a ained.
No ably, he e exis s a speci ic alue o 𝑡MCF a which he bi s eam
achie es pe ec bias, i.e., 𝐻𝑊 = 50%, as indica ed by he isoline
in he lowe pa o Fig. 6. This c i ical alue, deno ed as 𝑡MCF,op , is
essen ial o gene a ing uly andom bi s eams. By analyzing a single
de ec de ice wi h emission and cap u e imes ha ollow exponen ial
dis ibu ions (cha ac e ized by he means 𝜏𝑒 and 𝜏𝑐, espec i ely), i can
be igo ously demons a ed ha 𝑡MCF,op exis s. As de ailed in Appendix
B, his alue can be de e mined using he exp ession p o ided in he
implici Eq. (B.6). In his manne , he alues o 𝑡MCF,op we e nume -
ically e alua ed using (B.6) o each alue o 𝛽, and he esul s ha e
been o e laid on he hea maps as a solid whi e line. As obse ed, he
50%-isoline aligns pe ec ly wi h 𝑡MCF,op , con i ming hei equi alence.
In he special case whe e 𝜏𝑒=𝜏𝑐 (i.e., 𝛽= 1), an in e es ing scena io
a ises: he wo o e lapping exponen ial dis ibu ions (𝜏𝑒 and 𝜏𝑐) a e
sampled simul aneously. Unde hese condi ions, 𝑡MCF,op co esponds
o he median o 𝜏𝑒 (o equi alen ly 𝜏𝑐), which simpli ies o 𝑡MCF,op =
𝜏𝑒⋅ln 2.
Howe e , bi bias is no he only condi ion o gene a ing ue
andom bi s eams. Achie ing e y low au oco ela ion (AC) is also a
Fig. 7. Di e en cases o o e lapping 𝑡𝑒 and 𝑡𝑐 exponen ial dis ibu ions: (a) Full
O e lapping, (b) Pa ial O e lapping and (c) Non-o e lapping.
c i ical equi emen . As shown in Fig. 6, low alues o AC a e p ima ily
ob ained when 𝛽= 1, ega dless o he alue o 𝑡MCF,op . Addi ionally,
low AC alues can also be obse ed o ex eme alues o 𝛽, p o ided
ha 𝑡MCF,op is su icien ly la ge. No ably, se ing 𝑡MCF =𝑡MCF,op
(indica ed by he solid whi e line in he op hea map) does no , by i sel ,
ensu e a low AC alue, excep in he speci ic cases whe e 𝛽= 1, 𝛽 ≫ 1,
o 𝛽 ≪ 1.
The obse a ion window 𝑡MCF,op is closely ied o he pa ame e
𝛽, which di ec ly a ec s he andomness me ics. In an ideal andom
bi s eam, he chances o he nex bi being ‘‘1’’ o ‘‘0’’ a e equal.
This ideal scena io co esponds o a 50% p obabili y o obse ing a
cap u e e en immedia ely a e an emission e en (and ice e sa).
Such balance is achie ed only when he exponen ial dis ibu ions o
𝑡𝑒 (emission imes) and 𝑡𝑐 (cap u e imes) ully o e lap, which occu s
when 𝛽= 1 (as shown in Fig. 7a). When 𝛽≠1, he o e lap be ween he
dis ibu ions becomes pa ial (Fig. 7b). In his case, he op imal obse -
a ion window 𝑡MCF,op shi s o a alue be ween 𝜏𝑐 and 𝜏𝑒. This pa ial
o e lap dis up s he ideal andomness, educing he 50% p obabili y o
al e na ing e en s and in oducing au oco ela ion, which deg ades he
quali y o he andom bi s eam. As he dis ibu ions sepa a e u he
(𝛽 ≫ 1 o 𝛽 ≪ 1), hey no longe o e lap (Fig. 7c). In his si ua ion,
𝑡MCF,op is domina ed by he slowe dis ibu ion ( he one wi h a la ge
𝜏), since he p obabili y o de ec ing he as e e en ( he one wi h
a smalle 𝜏) app oaches 100%. As a esul , sampling ocuses almos
en i ely on he slowe dis ibu ion, p oducing beha io simila o he
ideal o e lap scena io shown in Fig. 7a.
Fig. 8 p o ides h ee dis inc pe spec i es o he hea maps p esen ed
in Fig. 6. In Fig. 8(a) (le ), i is shown ha bo h 𝑡MCF,op and he
au oco ela ion (AC) inc ease as 𝛽 de ia es om he ideal alue o
𝛽= 1. Fig. 8(b) (cen e ) illus a es he beha io o he Hamming Weigh
(HW) and AC as he a io 𝛼=𝑡MCF∕𝑡MCF,op mo es away om uni y,
highligh ing he deg ada ion in andomness quali y when 𝑡MCF is no
op imally se . Finally, Fig. 8(c) ( igh ) e eals ha e en when 𝛼= 1
(i.e., 𝑡MCF =𝑡MCF,op ) and HW app oaches he ideal alue o 50%, he
au oco ela ion s ill depends on he speci ic dwell imes 𝜏𝑒 and 𝜏𝑐 o he
de ec om which en opy is ex ac ed.
A sample se o 65,000 bi s was used o gene a e he bi maps in
Fig. 9 (black pixels ep esen 𝟷-bi s) o ou alues o 𝛽: 1, 5, 10, and
0.01. The 1.38 ms obse a ion window used in all o hese cases o
𝛽 co esponds o he ansis o ’s op imal window, 𝑡MCF,op , wi h 𝛽=
𝜏𝑐∕𝜏𝑒= 0.002 s∕0.002 s = 1, indica ed by a g ay dashed line in Fig.
8(a). As shown in Fig. 9(b), no disce nible epea ing pa e ns, la ge
con iguous blocks, o ex ended lines o a single colo a e appa en ,
implying negligible bi bias. Ins ead, he balanced dis ibu ion o black
and whi e pixels sugges s a high deg ee o andomness in ha sequence.
In con as , he o he bi maps inc easingly exhibi signs o bi bias
and ele a ed au oco ela ion, highligh ing he impac o 𝛽 on bi s eam
andomness.
To u he e alua e andomness and ge a mo e accu a e insigh ,
hese 4 ansis o s (co esponding o 4 di e en alues 𝛽) o ha e been
selec ed, as indica ed in 8(c), and he co esponding bi s eams ha e
been p ocessed by he NIST SP800-22 es sui e [20]. This sui e, de-
eloped by he Na ional Ins i u e o S anda ds and Technology (NIST)
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
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F.J. Rubio-Ba be o e al.
Fig. 8. (a) HW and AC o selec ed alues o 𝛽. (b) HW and AC o selec ed alues o 𝛽 a ound he op imal obse a ion window. (c) HW and AC o selec ed alues o 𝛽 a he
op imal obse a ion window.
Fig. 9. 256 × 256 bi maps o 𝑡MCF = 0.00138 s and ansis o s wi h (a) 𝛽= 1, (b) 𝛽= 5,(c) 𝛽= 10, and (d) 𝛽= 0.01.
Table 1
Pass a es esul s om he NIST SP800-22 es s a selec ed alues o 𝛽.
Tes name 𝛽 alues
0.01 1 5 10
F equency 55/55 (100%) 55/55 (100%) 55/55 (100%) 54/55 (98.1818%)
Block equency 54/55 (98.18%) 55/55 (100%) 50/55 (90.91%) 52/55 (94.5455%)
# Cumula i e Sumsa55/55 (100%) 54.5/55 (99.09%) 54.5/55 (99.09%) 54/55 (98.1818%)
Runs 0/55 (0%)b55/55 (100%) 0/55 (0%) 0/55 (0%)
Longes un 52/55 (94.55%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
Rank 54/55 (98.18%) 55/55 (100%) 55/55 (100%) 55/55 (100%)
FFT 55/55 (100%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
# non O e lapping Templa e 53.452/55 (97.19%) 54.487/55 (99.07%) 3.743/55 (6.81%) 8.839/55 (16.0688%)
O e lapping empla e 50/55 (90.91%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
Uni e sal 51/55 (92.73%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
App oxima e en opy 0/55 (0%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
# RandomExcu sions 37.125/38 (97.7%) 38.75/39 (99.36%) 11.625/26.875 (43.26%) 26.125/33 (79.1667%)
# Random excu sions a ian 37.722/38 (99.27%) 38.6111/39 (99%) 29.5556/30 (98.52%) 32.833/33 (99.4948%)
# Se ial 0/55 (0%) 55/55 (100%) 0/55 (0%) 0/55 (0%)
Linea complexi y 53/55 (96.36%) 55/55 (100%) 55/55 (100%) 54/55 (98.1818%)
a Symbol # indica es es s wi h wo o mo e sub- es s; he Pass a es in hese cases a e he a e age alues o he sub- es s.
b The bold alues indica e es s ha ailed a passing he es (Pass a e ≤95%).
in he Uni ed S a es, se es as a comp ehensi e se o guidelines o
in o ma ion secu i y and has gained widesp ead ecogni ion as an
au ho i a i e esou ce and a global benchma k in he ield o secu i y
s anda ds. The NIST SP800-22 sui e comp ises 15 dis inc es s, each
designed o assess a ious aspec s o andomness in a sequence. These
es s p oduce wo c i ical ou comes: a minimum 𝑃- alue and a pass
a e. A es is conside ed success ul i he 𝑃- alue exceeds 0.01 and
he pass a e is g ea e han 95%, e lec ing compliance wi h igo ous
andomness c i e ia. Acco ding o NIST ecommenda ions, a leas 55
sequences o 1 million bi s each mus be p ocessed o de i e s a is i-
cally meaning ul esul s o he uni o mi y o he a ained P- alues.
The e o e, o each ansis o (i.e., each alue o 𝛽), 55 × 106−𝑏𝑖𝑡
bi s eams ha e been gene a ed. The speci ic se ing pa ame e s used
du ing he NIST es uns can be ound in Appendix D. The esul s o
his e alua ion a e p esen ed in Table 1.
No e ha each bi s eam submi ed o he NIST es sui e is gen-
e a ed wi h 𝛼= 1, ha is, a he obse a ion window 𝑡MCF =𝑡MCF,op ,
which p o ides an op imal Hamming weigh (HW) o 50%. Main aining
his 50% HW is essen ial: any de ia ion would cause he bi s eam
o ail he F equency es ( he i s in he NIST es sui e), he eby
in alida ing he subsequen es s. The able shows he Pass a es o
each es ( he numbe o sequences whose 𝑃- alue exceeds 0.01 and
hus is conside ed uly andom). These a es e eal ha 𝛽= 1 (i.e., 𝜏𝑒=
𝜏𝑐) consis en ly yields he highes Pass a es in all he NIST SP800-
22 es s, indica ing ha bi s eams gene a ed unde hese condi ions
exhibi he s onges s a is ical p ope ies. In con as , a 𝛽= 0.01, 𝛽=
5, and 𝛽= 10, mul iple es s ail, which could indica e how de ia ions
om 𝛽= 1 deg ade he andomness quali y o he ou pu . Howe e ,
a mo e de ailed inspec ion e eals ha he lowes pass a es occu a
𝛽= 5. As 𝛽 shi s away om his alue, he pass a es s eadily imp o e.
This end is illus a ed in Fig. 10, whe e a 55 × 106-bi bi s eam was
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
7
F.J. Rubio-Ba be o e al.
Fig. 10. NIST Pass Ra es (NPRs) a di e en 𝛽 alues.
Fig. 11. AC le els o di e en 𝛽 alues.
gene a ed and e alua ed o 100 dis inc alues o 𝛽, spanning 𝛽= 0.1m
o 𝛽= 10𝑘.
This g aph epo s he NIST pass a es (o NPRs) by a e aging and
no malizing o 1, o each alue o 𝛽, he pass a es ob ained om he
15 NIST es s. As illus a ed, he wo egions (Pass and Fail) a e clea ly
di e en ia ed; he NPR peaks nea 𝛽= 1 (T ansis o A), indica ing
ha he gene a ed bi s eams pass all andomness es s, sugges ing an
op imal 𝜏𝑐∕𝜏𝑒 a io, beyond which de ia ions p og essi ely deg ade he
s a is ical quali y o he ou pu bi s eams.
Howe e , once 𝛽 su passes h esholds o 1k o 1m, he NPR alues
again indica e beha io consis en wi h uly andom bi s eams, as i
has been poin ed ou in Fig. 7 whe e o alues close o 𝛽 close o 1
bu di e en (non-o e lapping) he wo s esul s o AC a e ound ou ,
p ope ly ansla ed o NIST sui e es s. This can also be seen in Fig. 11,
showing how alues o AC below 0.001 seem o be a good p edic o
(as long as HW emains close o 50%) o a high NPR.
The bi a e o he p oposed app oach o single-de ec ansis o s
is con ingen on he obse a ion window 𝑡MCF, as one bi is p oduced
pe obse a ion when he alue o 𝐷MCF is eached. As p e iously
discussed, o ensu e he gene a ion o uly andom bi s eams, he
obse a ion window mus be se o i s op imal alue, 𝑡MCF =𝑡MCF,op ,
which is a unc ion o he de ec ’s dwell imes ( e e o Appendix
B). No ably, sho e dwell imes esul in a highe bi a e. Thus, o
a gi en in eg a ion echnology whe e andom eleg aph noise (RTN)
is p esen , i is c ucial o iden i y he de ec wi h he sho es dwell
imes in o de o maximize he bi a e o he p oposed RTN-based
TRNG. Conside a echnology whe e RTN de ec s exhibi dwell imes
uni o mly dis ibu ed be ween 10 μs and 1ms. In his scena io, he
minimum achie able 𝑡MCF,op occu s when he cha ac e is ic imes 𝜏𝑒
and 𝜏𝑐 a e bo h equal and app oxima ely 10 μs, esul ing in a bi
a e o app oxima ely 1
𝑡MCF,op ≈1
7𝜇𝑠 ≈ 140 kbi s/s. The likelihood o
iden i ying such a ansis o is in luenced by he size o he ansis o
a ay employed (see Fig. 3). I can be shown, as elabo a ed in Appendix
C, ha an a ay o 1000 ansis o s is su icien o achie e a 90%
p obabili y o inding a ansis o wi h 𝜏𝑒=𝜏𝑐≈ 10 μs.
I is aluable o compa e he bi a es achie able wi h he p o-
posed app oach o hose o simila RTN-based TRNGs, pa icula ly
Table 2
Maximum bi a es (bi s/s) compa ison.
[11] [10] This wo k
1.5
2⋅𝜋⋅𝜏0
1
2.2⋅𝜏0
1
𝑡MCF,op
hose in [10,11]. The maximum bi a es epo ed in hese s udies
a e summa ized in Table 2. When applying he echnology desc ibed
ea lie , he esul ing bi a es a e shown in he hea maps in Fig. 12. As
demons a ed, he RTN-based TRNG p esen ed in his wo k achie es
highe bi a es han hose epo ed in he e e enced s udies.
5.1.2. Scena io A.1: Discussion o esul s.
Se e al conclusions can be d awn om he p e iously epo ed
esul s on he implemen a ion o he TRNG using ansis o s wi h
single-de ec RTN (1T1D). The esul s demons a e ha he use o 1T1D
holds conside able p omise, o e ing a obus mechanism o gene a ing
high-quali y, uly andom bi s eams. This me hod no only shows
signi ican po en ial in e ms o andomness quali y bu also o e s
a p ac ical app oach o ha dwa e-based andom numbe gene a ion
in c yp og aphic applica ions. Howe e , despi e he p omising pe -
o mance, se e al challenges ha e eme ged ha need o be ca e ully
add essed o op imize he o e all sys em pe o mance. These challenges
include: (A) he di icul ies in achie ing he co ec 𝛽=𝜏𝑐∕𝜏𝑒 alue o
he de ec , (B) he p ecision equi ed in se ing he obse a ion window
𝑡MCF,op , and, (C) he ela i ely low p obabili y o inding a ansis o
wi h exac ly one de ec and he igh dwell imes o boos he bi a e.
The impo ance o conside ing and o e coming hese challenges will be
discussed and deal wi h in his pape , as hey a e c ucial o enhancing
he eliabili y, e iciency, and bi a e o he TRNG, ensu ing i s iabili y
o eal-wo ld c yp og aphic applica ions.
The i s challenge in ol es he di icul y in inding he de ec
wi h he co ec alue o he 𝛽 a io. Fo op imal andom numbe
gene a ion, 𝛽 mus be wi hin speci ic anges: ei he close o 1, less
han o equal o 1 ms, o g ea e han o equal o 1k. This is because
de ia ions om hese anges can lead o au oco ela ion (AC), whe e
he e a e epea ing pa e ns o dependencies be ween consecu i e bi s
in he bi s eam. This au oco ela ion deg ades he quali y o he an-
dom numbe s gene a ed by he TRNG, making hem unsui able o
c yp og aphic applica ions.
The second signi ican challenge ela es o he di icul y in accu-
a ely se ing and iden i ying he op imal obse a ion window,
𝑡MCF,op . The obse a ion window 𝑡MCF plays a c i ical ole in de e min-
ing he andomness o he gene a ed bi s eam. When 𝑡MCF is se o i s
op imal alue, 𝑡MCF,op , he bi s eam exhibi s a 50% Hamming weigh ,
meaning ha he e a e an equal numbe o 0s and 1s, a necessa y
condi ion o ue andomness. Howe e , se ing 𝑡MCF p ecisely o his
op imal alue p esen s wo main challenges. Fi s , i is echnically
di icul o se 𝑡MCF wi h he equi ed p ecision. The RTN signal is
inhe en ly noisy and luc ua es o e ime, so accu a ely measu ing he
maximum cu en luc ua ion o e a speci ic ime window equi es
sophis ica ed equipmen and high p ecision. E en small de ia ions in
he window could esul in a signi ican educ ion in he quali y o
he bi s eam, causing he bi s eam o become biased (i.e., all 0s o
all 1s) o o exhibi pe iodic pa e ns. Second, iden i ying 𝑡MCF,op is a
non- i ial ask and de e mining he op imal window o each ansis o
may equi e empi ical measu emen and ine- uning which can be ime
consuming and compu a ionally expensi e, pa icula ly when dealing
wi h la ge a ays o ansis o s.
Fo una ely, bo h challenges can be add essed, as demons a ed
la e , by employing mo e ad anced bi s eam gene a ion echniques
and pos -p ocessing me hods ha mi iga e he impac o de ia ions
om he ideal alues o 𝛽 and 𝑡MCF.
The las challenge iden i ied is he low p obabili y o inding a
ansis o wi h exac ly one de ec and p ope dwell imes. This is
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
8
F.J. Rubio-Ba be o e al.
Fig. 12. (a) Maximum bi a e om [11] . (b) Maximum bi a e om [10]. (c) Bi a e in his wo k.
a signi ican issue, as he eliabili y and e iciency o he RTN-based
TRNG depend hea ily on he p esence o a single de ec wi hin he
ansis o .
In eal-wo ld scena ios, he densi y o de ec s wi hin a ansis o
ypically ollows a Poisson dis ibu ion, which can a y depending
on he manu ac u ing p ocess and he speci ic cha ac e is ics o he
echnology used. The dis ibu ion o de ec s (i.e., how p obable a e
he dwell imes 𝜏𝑐 and 𝜏𝑐) a e also dependen on he manu ac u ing
p ocess (bo h he ansis o ’s biasing ol ages and he empe a u e ha e
an impac as well, bu , o he sake o simplici y, hese wo ac o s will
no be conside ed he e). Essen ially, he de ec s mus exhibi cap u e
and emission imes ha a e sui able o achie ing he desi ed bi a e.
Fo example, i bi a es in he kilobi -pe -second ange a e equi ed,
a ansis o wi h a single RTN de ec and dwell imes on he o de
o milliseconds mus be iden i ied. As explained in Appendix C, he
p obabili y o inding such a ansis o wi hin a gi en a ay can be
de e mined. This p obabili y is in luenced by he echnology (in e ms
o bo h de ec densi y and de ec dis ibu ion), he size o he ansis o
a ay, and he desi ed dwell imes.
As a oy example, Fig. 13 illus a es he size o he ansis o
a ay equi ed o iden i y a ansis o wi h exac ly one RTN de ec ,
gi en a ying alues o he echnology’s de ec densi y, ep esen ed by
he mean alue 𝜆. The de ec dis ibu ion in his example echnology
assumes ha dwell imes a e uni o mly dis ibu ed be ween 1 ms and
1 s, wi h he desi ed de ec ha ing a dwell ime be ween 1 ms and
1.25 ms ( he lowes a ailable in he echnology) o ensu e adequa e bi
a es. These s ingen condi ions lead o a ying a ay sizes necessa y
o achie e a 95% p obabili y o loca ing such a ansis o , as shown in
Fig. 13(a). Fo de ec densi ies o 𝜆= 3,4, an a ay o 1000 ansis o s
is su icien , while o 𝜆= 2, a leas 1600 ansis o s a e needed and
o 𝜆= 1 an a ay wi h a leas 3700 ansis o s would be equi ed.
I is in e es ing o no e ha , o loca e a ansis o wi h exac ly
one de ec (and no mo e), he obse a ion window can be adjus ed
o maximize he de ec ion p obabili y. This is due o he na u e o
he numbe o RTN de ec s in a ansis o , which ollows a Poisson
dis ibu ion, and hei uni o mi y o dwell imes. By changing he
obse a ion window, om he smalles one ( om 1 ms o 1.25 ms)
o he ull window ( om 1 ms o 1 s), using he example abo e,
he p obabili y o de ec ing a ansis o wi h exac ly one de ec can
be maximized, eaching a peak o ≈37%. This is illus a ed in Fig.
13(b), whe e he x-axis shows he a io o he window size o he ull
window. When 𝜆= 1, he ull window mus be used o achie e he
highes de ec ion p obabili y. Howe e , o 𝜆= 20, he window mus
be na owed o maximize he p obabili y. This sugges s ha , o any
alue o 𝜆, he maximum de ec ion p obabili y o ≈37% can be achie ed
by p ope ly adjus ing he obse a ion window. A key ad an age o
his adjus men is ha , o he peak de ec ion p obabili y o 37%, he
minimum a ay size needed o gua an ee a de ec ion p obabili y o 95%
is educed o a ound 10 de ices (see Appendix C). Howe e , adjus ing
he window size comes wi h a ade-o in bi a e. Fo example, when
Fig. 13. (a) P obabili y o inding single-de ec RTN e sus ansis o a ay size. (b)
A ay size o achie e a 95% p obabili y o inding 1 single-RTN ansis o .
𝜆= 1, using he ull window minimizes he a ay size o 10 de ices, bu
i signi ican ly impac s he bi a e, which can d op, depending on he
de ec ’s speci ic dwell imes, om 1000 b/s o jus 1 b/s. I he window
is na owed o he ange om 1 ms o 1.25 ms, he a ay size inc eases
o o e 3700 de ices, while he bi a e can sa ely emain be ween 1000
b/s and 800 b/s.
These p obabili y alues in oduce a p ac ical challenge, as inding
a ansis o ha mee s he ideal condi ions is c ucial o op imal TRNG
ope a ion. Al e na i ely, since he p obabili y o inding ansis o s
wi h mul iple RTN de ec s is always highe han he p obabili y o
inding ansis o s wi h exac ly one de ec , a me hod o handling
mul iple de ec s, each wi h i s own dwell imes, wi hou comp omising
he andomness o he bi s eam is essen ial. This app oach will be
explo ed in subsequen sec ions.
In summa y, he issues discussed abo e (A) inding he co ec
𝛽 alue, (B) accu a ely se ing he obse a ion window 𝑡MCF, and
(C) inding a ansis o wi h exac ly one de ec — ep esen signi ican
challenges in he design and implemen a ion o RTN-based TRNGs.
Al hough hese challenges p esen obs acles, hey also o e an oppo -
uni y o inno a e and e ine he app oach. In he ollowing sec ions,
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
9
F.J. Rubio-Ba be o e al.
Fig. 26. Resul ing bi a es om he bi s eams wi h he alues o 𝑡MCF,op o Fig. 21.
as he co e me ic, we showed ha ca e ul uning o he sampling
window yields bi s eams ha main ain excellen andomness ac oss
bo h single-de ec and mul i-de ec scena ios. Th oughou ou expe -
imen s, alida ed using simple me ics such as Hamming weigh and
au oco ela ion, as well as he NIST SP 800-22 sui e, we encoun-
e ed h ee cen al conside a ions. Fi s , while single-de ec ansis o s
p o ide s aigh o wa d en opy ex ac ion, mul i-de ec de ices dom-
ina e in eal ab ica ion, and we showed ha simple pos -p ocessing
and well-chosen obse a ion windows can accommoda e hese o e -
lapping de ec dynamics wi hou comp omising andomness. Second,
inding he igh balance be ween bi a e and bi quali y is essen ial:
sho e windows accele a e bi gene a ion bu can induce bias, whe eas
longe windows imp o e andomness a he cos o lowe h ough-
pu , so design lexibili y is c ucial o s ike an e ec i e comp omise.
Finally, ensu ing p ac ical scalabili y elies on ecognizing ha mul i-
de ec ansis o s appea mo e equen ly, which educes he size o he
ansis o a ay needed o iden i y sui able RTN sou ces, hus p omo -
ing easible in eg a ion in o ad anced echnology nodes. O e all, he
p oposed scheme compa es a o ably o exis ing RTN-based TRNGs,
o e ing no able gains in bi - a e lexibili y, ha dwa e simplici y, and
de ice-le el adap abili y. The indings he ein ein o ce he idea ha
RTN, p ope ly measu ed and managed, can se e as a na u al, eadily
a ailable sou ce o en opy, pa ing he way o e icien and secu e
on-chip andom numbe gene a ion.
Looking ahead, ou nex s ep is o explo e XOR-based se ups ha
me ge RTN om mul iple ansis o s, sus aining high bi a es while
educing he impac o de ice a iabili y. By ca e ully combining pa al-
lel s eams o andomness, we an icipa e a mo e powe ul and scalable
RTN-based TRNG capable o e icien , obus ope a ion.
CRediT au ho ship con ibu ion s a emen
F.J. Rubio-Ba be o: W i ing – e iew & edi ing, W i ing – o iginal
d a , Visualiza ion, Valida ion, So wa e, Me hodology, In es iga ion,
Fo mal analysis, Da a cu a ion, Concep ualiza ion. F. de los San os-
P ie o: W i ing – e iew & edi ing, W i ing – o iginal d a , Valida ion,
So wa e, In es iga ion, Fo mal analysis. R. Cas o-Lopez: W i ing
– e iew & edi ing, Visualiza ion, Valida ion, Supe ision, So wa e,
Resou ces, P ojec adminis a ion, Me hodology, In es iga ion, Fund-
ing acquisi ion, Fo mal analysis, Da a cu a ion, Concep ualiza ion. E.
Roca: W i ing – e iew & edi ing, P ojec adminis a ion, Funding
acquisi ion. F.V. Fe nandez: W i ing – e iew & edi ing, Resou ces,
P ojec adminis a ion, Funding acquisi ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
This wo k was suppo ed by g an TED2021-131240B-I00 unded
by MICIU/AEI/10.13039/501100011033 and by he ‘‘Eu opean Union
Nex Gene a ionEU/PRTR’’. The wo k was also suppo ed by g an
PID2022-136949OB-C21 unded by MICIU/AEI/10.13039/5011000
11033 and by ‘‘ERDF/EU’’, by g an P oyExcel_00536 unded by Con-
seje ía de Uni e sidad, In es igación e Inno ación o Jun a de An-
dalucía, and by Minis e io de Asun os Económicos y T ans o mación
Digi al h ough g an TSI-069100-2023-1 o PERTE Chip Chai p o-
g am, unded by Eu opean Union - Nex Gene a ionUE. F. J. Rubio-
Ba be o was suppo ed by g an PREP2022-000765 unded by MI-
CIU/AEI/10.13039/501100011033 and by ‘‘FSE+’’. A p elimina y e -
sion o his wo k is published in he 2024 edi ion o he In e na ional
Con e ence on Syn hesis, Modeling, Analysis and Simula ion Me hods,
and Applica ions o Ci cui Design (SMACD) [24].
Appendix A. Basic andomness me ics o mulae
To measu e he quali y o he andom numbe , some me ics can be
used o assess his andomness. Wi h he ollowing exp essions, a i s
analysis can be done compu a ionally as e han di ec ly e alua ing
he NIST SP 800-22 sui e.
•Hamming Weigh : In heo y, a pe ec ly andom bi s eam has an
HW alue o 50%. To e alua e he HW in a andom sequence, he
ollowing exp ession is used:
𝐻𝑊 (%) = 1
𝑛
𝑛
∑
𝑖=1
𝑏𝑖⋅100 (A.1)
whe e 𝑛 is he o al numbe o bi s and 𝑏𝑖 deno es he alue, 0 o
1, o he 𝑖 h bi .
•Au oco ela ion: A lawless andom bi s eam is expec ed o
p esen an AC alue o ze o. AC can be de e mined by using he
subsequen exp ession:
𝑟k=∑𝑛−𝑘
𝑖=1 (𝑥𝑖−𝑥)⋅(𝑥𝑖+𝑘−𝑥)
∑𝑛
𝑖=1 (𝑥𝑖−𝑥)2(A.2)
whe e 𝑥𝑖 and 𝑥𝑖+𝑘 co espond o he 𝑖 bi and he 𝑘-lagged bi
espec i ely. The compu ed 𝑟k co esponds o he au oco ela ion
unc ion a he 𝑘 lagged alue o he bi s eam.
Appendix B. Op imum 𝒕MCF alues
The p obabili y o an ini ially emi ed (cap u ed) de ec o unde go-
ing a cap u e (emission) in a ime in e al 𝑡MCF is gi en by he ollowing
exp ession [25]:
𝑃𝑐∕𝑒(𝑡MCF)= 1 − exp (−𝑡MCF
𝜏𝑐∕𝑒)(B.1)
The p obabili y o an e en occu ence (and he de ec ion o he
co esponding cu en luc ua ion) in a de ec ha can be emi ed o
cap u ed is
𝑃𝑑𝑒𝑓 (𝑡MCF)=𝑃𝑜𝑐𝑐 ⋅𝑃𝑒(𝑡MCF)+ (1 − 𝑃𝑜𝑐𝑐 )⋅𝑃𝑐(𝑡MCF)(B.2)
whe e 𝑃𝑜𝑐𝑐 is he occupa ion p obabili y o he de ec , which, in he
s eady-s a e, is gi en by 𝜏𝑒∕(𝜏𝑒+𝜏𝑐). Using his alue o 𝑃𝑜𝑐𝑐 and
eo de ing he exp ession,
𝑃𝑑𝑒𝑓 (𝑡MCF)= 1 − 𝜏𝑐
𝜏𝑒+𝜏𝑐
⋅exp (−𝑡MCF
𝜏𝑐)−𝜏𝑒
𝜏𝑒+𝜏𝑐
⋅exp (−𝑡MCF
𝜏𝑒)(B.3)
Conside ing a si ua ion whe e ime cons an s a e equal (i.e., 𝛽= 1)
o signi ican ly di e en om each o he , i can be demons a ed ha
he la ge alue o 𝜏 de e mines he beha io . Thus, he in luence o he
smalle 𝜏 can be neglec ed. Typical scena ios o his a e when 𝛽 > 1000
AEUE - In e na ional Jou nal o Elec onics and Communica ions 196 (2025) 155801
16

F.J. Rubio-Ba be o e al.
o 𝛽 < 0.001. Consequen ly, he exp ession abo e can be app oxima ed
by
𝑃𝑑𝑒𝑓 (𝑡MCF)= 1 − exp (−𝑡MCF
𝜏𝑒𝑞 )(B.4)
whe e 𝜏𝑒𝑞 is de ined as
𝜏𝑒𝑞 =𝜏𝑒+𝜏𝑐−𝜏𝑒
1 + exp (−𝜏𝑐−𝜏𝑒
𝜏𝑐+𝜏𝑒)(B.5)
To achie e an unbiased bi s eam in a one-de ec ansis o , a p obabil-
i y o 1/2 can be imposed ei he in exp ession (B.3) o ob ain Eq. (B.6),
𝜏𝑐
𝜏𝑒+𝜏𝑐
⋅exp (−𝑡MCF,op
𝜏𝑐)+𝜏𝑒
𝜏𝑒+𝜏𝑐
⋅exp (−𝑡MCF,op
𝜏𝑒)=1
2(B.6)
o in Eq. (B.4) o ob ain he app oxima ed op imum alue 𝑡MCF,op =
ln(2) ⋅𝜏𝑒𝑞. In he mul i-de ec case, e e y de ec can p o oke he an-
sis o cu en luc ua ion independen ly, and he inclusion–exclusion
p inciple can be used o accoun o he p obabili y o an e en .
𝑃𝑒𝑣𝑒𝑛𝑡(𝑡MCF) =
𝑛
∑
𝑖=1
𝑃𝑑𝑒𝑓,𝑖 −∑
𝑖<𝑗
𝑃𝑑𝑒𝑓,𝑖 ⋅𝑃𝑑𝑒𝑓,𝑗 + …
+(−1)𝑛−1 ∑
𝑖<⋯<𝑛
𝑃𝑑𝑒𝑓,𝑖 ⋅𝑃𝑑𝑒𝑓,𝑗 ⋅…⋅𝑃𝑑𝑒𝑓,𝑛
(B.7)
whe e 𝑛 is he numbe o de ec s in he ansis o . I he app oxima ion
(B.4) is alid o e e y de ec , he exp ession abo e can be simpli ied.
Fo ins ance, when 𝑛= 2, he exp ession can be p o en o be
𝑃𝑒𝑣𝑒𝑛𝑡(𝑡MCF) = 1 − exp (−𝑡MCF
(𝜏𝑒𝑞,1∥𝜏𝑒𝑞,2))(B.8)
whe e (𝜏𝑒𝑞,1∥𝜏𝑒𝑞,2) = 𝜏𝑒𝑞1𝜏𝑒𝑞2
𝜏𝑒𝑞1+𝜏𝑒𝑞2. I can be seen ha he exp ession
ob ained is equi alen o Eq. (B.3) wi h an e ec i e ime cons an o
(𝜏𝑒𝑞,1∥𝜏𝑒𝑞,2). Thus, he gene aliza ion o 𝑛 de ec s is s aigh o wa d.
𝑃𝑒𝑣𝑒𝑛𝑡(𝑡MCF) = 1 − exp (−𝑡MCF
(𝜏𝑒𝑞,1∥𝜏𝑒𝑞,2‖…‖𝜏𝑒𝑞,𝑛))(B.9)
Again, we can impose an e en p obabili y o 1/2 and ei he sol e
nume ically Eq. (B.7) o use he app oxima ed op imum alue 𝑡MCF,op =
ln(2) ⋅(𝜏𝑒𝑞,1∥𝜏𝑒𝑞,2‖…‖𝜏𝑒𝑞,𝑛).
Appendix C. Finding he app op ia e ansis o
Assuming ha 𝑃𝑡𝑡𝑜 is he p obabili y ha a ansis o has he app o-
p ia e de ec cha ac e is ics, he p obabili y o inding a leas one such
ansis o in an a ay o 𝑁 de ices is gi en by
𝑃𝑎𝑟𝑟𝑎𝑦 = 1 − (1 − 𝑃𝑡𝑡𝑜)𝑁(C.1)
The p obabili y 𝑃𝑡𝑡𝑜 can be b oken down by conside ing he key ea u es
o de ec s in a ansis o : he numbe o de ec s, hei ime cons an s,
and hei ampli udes. Subsequen ly, i is assumed ha he p obabili y
o ha ing 𝑘 de ec s is deno ed by 𝑃𝑘, he p obabili y ha a de ec has
he app op ia e ime cha ac e is ics is 𝑃𝜏, and he p obabili y ha i has
sui able ampli udes is 𝑃𝛥𝐼 . Since he key ea u es in ol e de ec s wi h
he app op ia e ime cons an s, he numbe o o al de ec s in he de ice
is no signi ican . The e o e, he p obabili y o a ansis o ha ing 𝑛
de ec s wi h he desi ed ime and ampli ude cha ac e is ics is gi en by
𝑃𝑡𝑡𝑜,𝑛 =
∞
∑
𝑘=𝑛
𝑃𝑘⋅(𝑘
𝑛)⋅(𝑃𝜏)𝑛⋅(1 − 𝑃𝜏)𝑘−𝑛⋅(𝑃𝛥𝐼 )𝑛(C.2)
whe e he binomial e m a ises om he p obabili y o selec ing 𝑛
de ec s wi h he app op ia e cha ac e is ics ou o 𝑘 o al de ec s.
I we a e in e es ed in a ansis o con aining se e al de ec s wi hin
a ange, he p obabili y can be ob ained by summing he equi ed
e ms o he equa ion abo e. Fo ins ance, he p obabili y o a ansis o
Table D.5
Con igu a ion alues used o un he NIST SP 800-22 sui e.
NIST es pa ame e s
Pa ame e De aul Used
Block F equency Tes - block leng h (M) 128 100k
NonO e lapping Templa e Tes - block leng h (m) 9 10
O e lapping Templa e Tes - block leng h (m) 9 10
App oxima e En opy Tes - block leng h (m) 10 2
Se ial Tes - block leng h (m) 16 2
Linea Complexi y Tes - block leng h (M) 500 500
ha ing a leas one bu no mo e han 𝑛𝑚𝑎𝑥 de ec s wi h he desi ed ime
cons an s and ampli udes is
𝑃𝑡𝑡𝑜 =
𝑛𝑚𝑎𝑥
∑
𝑛=1
𝑃𝑡𝑡𝑜,𝑛 (C.3)
In hese equa ions, 𝑃𝑘, 𝑃𝜏, and 𝑃𝛥𝐼 a e quan i ies ha can be
e alua ed acco ding o he echnology-dependen dis ibu ion o he
numbe o de ec s, ime cons an s, and ampli udes, espec i ely. Using
he alues o hese quan i ies and Eq. (C.1) in combina ion wi h (C.2)
o (C.3), an app oxima ed alue o he a ay size equi ed o gua an ee
he p esence o app op ia e ansis o s can be ob ained.
Appendix D. NIST es sui e used pa ame e s
All he 15 NIST es s a e execu ed wi h he speci ied bi s eam
leng h, while he es o he pa ame e s a e adjus ed acco ding o he
inpu leng h as indica ed in [20]. Mo e speci ically, he es pa ame e s
u ilized a e indica ed in Table D.5, which mus be ca e ully selec ed o
comply wi h he guidelines gi en in he NIST documen . This p ocedu e
also in ol es he selec ion o he minimum numbe o sequences, which
mus be a leas 𝑛𝑠𝑒𝑞 = 55, ha mus be p ocessed o de i e s a is ically
meaning ul esul s o he sake o uni o mi y o he 𝑝- alue analysis.
Addi ionally, and acco ding o he minimum inpu da a o some es s,
some o hem need a la ge numbe o bi s han o he s (i.e. se ial
es s, andom excu sions es and andom excu sions a iable), being
he minimum equi ed o hose o be a leas 106 bi s. This esul s in
an ex a challenge in e ms o compu ing cha ge o e alua e NIST and
alida ing passing he es s only unde ce ain condi ions.
Da a a ailabili y
Da a will be made a ailable on eques .
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