Mã hóa dữ liệu trên thiết bị di động dùng đặc trưng sinh trắc học

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ik "f詠c"vt‒p"e挨"u荏"f英"nk羽w"違pj"mjw»p o員v"ÐGuugz;6Ñ x "uq"u pj"x噂k"o瓜v"rj逢挨pi"

rj r"8挨p"ik違p"mj e M院v"sw違"vj詠e"pijk羽o"ejq"vj医{"v雨"n羽"n厩k"v逢挨pi"8逢挨pi x噂k"ik違k"vjw壱v"Gkigphcegu"vtw{隠p"vj嘘pi"x "gpvtqr{"荏"o泳e"ej壱r"pj壱p"8逢嬰e0

ABSTRACT

Biometric data are variable by nature, i.e., biometric samples captured from the same object at different time are not identical To be able to use biometrics for cryptography, we need an extraction method so that the extraction results from biometric samples which are captured from same object are identical Based on Eigenfaces algorithm and Error-Correcting theory, this thesis proposes methods to extract stable data from similar face images; this data can be used to generated cryptographic key, protect another key, or combine with other biometrics The extraction methods are implemented and evaluated using ÐGuugz;6Ñ"hceg"fcvcdcug

and compared with another simple method The experimentation shows that error rates are equivalent to traditional Eigenfaces algorithm with good entropy

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1.1 Ik噂k"vjk羽w"8隠"v k."#"pij c"mjqc"j丑e"x "vj詠e"vk宇p 1

1.2 A嘘k"v逢嬰pi"x "rj衣o"xk"pijk‒p"e泳w 3

1.3 V„o"v逸v"p瓜k"fwpi"nw壱p"x<p 6

1.4 Fcpj"o映e"v瑛"xk院v"v逸v 6

Ej逢挨pi"40 E e"pijk‒p"e泳w"nk‒p"swcp 7

2.1 J逢噂pi"vk院r"e壱p"f́pi"o«"u穎c"n厩k 7

2.1.1 Juels & Wattenberg - A Fuzzy Commitment Scheme (1999) 7

2.1.2 Juels & Sudan - A Fuzzy Vault Scheme(2003) 8

2.1.3 Dodis et al Î Fuzzzy Extractor (2008) 9

2.1.4 Pj英pi"e»pi"vt·pj"pijk‒p"e泳w"mj e 9

2.2 J逢噂pi"vk院r"e壱p"mj»pi"f́pi"o«"u穎c"n厩k 11

2.2.1 Colin Soutar et al - Biometric Encryption (1999) 11

2.2.2 E e"rj逢挨pi"rj r"e栄c"Hcdkcp"Oqptqug 14

Ej逢挨pi"50 E挨"u荏"n#"vjw{院v 15

3.1 O«"u穎c"n厩k 15

3.1.1 Ik噂k"vjk羽w 15

3.1.2 Không gian Metric 15

3.1.3 Mjq違pi"e ej"Jcookpi 16

3.1.4 A鵜pj"pij c"o«"u穎c"n厩k 17

3.2 Rj逢挨pi"rj r"rj¤p"v ej"vj pj"rj亥p"ej pj"*REC+ 18

3.2.1 X "f映 19

3.2.2 Vector ri‒pi"x "vt鵜"tk‒pi 23

3.2.3 Rj逢挨pi"rj r"REC 25

3.3 Vjw壱v"vq p"pj壱p"fk羽p"i逢挨pi"o員v"Gkigphcegu 27

3.4 E e" mj k" pk羽o" ÐOkp" Gpvtqr{Ñ." ÐUgewtg" UmgvejÑ." ÐHw||{" GzvtcevqtÑ 30

3.4.1 Entropy t嘘k"vjk吋w 30

3.4.2 Entropy v嘘k"vjk吋w"trung bình 31

3.4.3 Mjq違pi"e ej"v pj 31

3.4.4 J o"Vt ej"zw医v 32

3.4.5 Secure Sketch 32

3.4.6 Fuzzy Extractor 34

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4.1 Ik噂k"vjk羽w 37

4.2 Piw{‒p"n#"z¤{"f詠pi"ÐUgewtg"UmgvejÑ 37

4.3 Taxicab Metric 38

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4.5.1 Dk院p"8鰻k"Jcfcoctf 45

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4.5.4 Pj壱p"zfiv 49

4.6 A瓜"rj泳e"v衣r 50

4.7 A pj"iká entropy 51

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5.1 V益"n羽"n厩k 53

5.2 Entropy 53

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5.4 Vj運k"ikcp"vj詠e"vjk 54

5.5 Uq"u pj"x噂k"rj逢挨pi"rj r"u穎c"n厩k"v瑛pi"vj pj"rj亥p 55

5.5.1 Ik噂k"vjk羽w"rj逢挨pi"rj r 55

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Ej逢挨pi"80 M院v"nw壱p"x "j逢噂pi"rj v"vtk吋p 60

6.1 M院v"nw壱p 60

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Metric 53

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D違pi"7050"Dk院p"vjk‒p"e栄c"gpvtqr{"vjgq"u嘘"ejk隠w"e栄c"xgevqt"8員e"vt逢pi 54

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J·pj"5090"X "f映"x隠"gkigphceg 29

J·pj"50:0"Ugewtg"Umgvej 33

J·pj"50;0"Hw||{"Gzvtcevqt 34

J·pj"50320"永pi"f映pi"e栄c"Hw||{"Gzvtcevqt 35

J·pj"50330"Z¤{"f詠pi"Hw||{"Gzvtcevqt"v瑛"Ugewtg"Umgvej"x "j o"vt ej"zw医v 35

J·pj"6030"Eqfg"Qhhugv"eqpuvtwevkqp"hqt"Ugewtg"Umgvej 37

J·pj"6040"V益"n羽"n厩k"mjk f́pi"mjq違pi"e ej"Gwenkf 40

J·pj"6050"V益"n羽"n厩k"mjk"f́pi"mjq違pi"e ej"Vczkecd 41

J·pj"6060"Uq"u pj"v益"n羽"n厩k"ik英c"rj逢挨pi"rj r"mk吋o"vtc"f́pi"mjq違pi"e ej" Gwnekf"x噂k"rj逢挨pi"rj r"mk吋o"vtc"f́pi"mjq違p"e ej"Vczkecd 41

J·pj"6070"Z¤{"f詠pi"ÐUgewtg"UmgvejÑ"ejq"Vczkecd"Ogvtke 42

J·pj"6080"Ik違k"o«"Vczkecd"vt‒p"mj»pi"ikcp"4"ejk隠w 45

J·pj"7030"V益"n羽"n厩k e栄c"rj逢挨pi"rj r"u穎c"n厩k"v瑛pi"vj pj"rj亥p 57

J·pj"7040"Uq"u pj"v益"n羽"n厩k"e栄c"rj逢挨pi"rj r"u穎c"n厩k"v瑛pi"vj pj"rj亥p"x " rj逢挨pi"rj r"u穎c"n厩k"f́pi"mjq違pi"e ej"Taxicab 57

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ghi nh噂 chúng tr荏 p‒p"mj„"mj<p"8嘘i v噂k"pi逢運i s穎 d映ng, d磯p"8院n vi羽c anh ta

s胤 ijk"p„"tc"8¤w"8„"x "e„"mj違 p<pi"o壱t kh育u s胤 b鵜 l瓜

‚ Mã hóa dばa trên token: Token là m瓜t thi院t b鵜 ph亥n c泳ng có kh違 p<pi"n逢w"vt英

khóa và th詠c hi羽n các thao tác mã hóa, gi違i mã d詠c"vt‒p"mj„c"n逢w"vt英 Các

thi院t b鵜 này vj逢運ng là các th飲 thông minh ti院p xúc ho員c không ti院p xúc, trong

8„."vq p"d瓜 h羽 th嘘ng 8逢嬰c tích h嬰p trong m瓜t chip có m ej"vj逢噂c vài milimet,

vi羽c can thi羽p vào h羽 th嘘pi"8吋 8丑e"8逢嬰e"mj„c"n逢w"vt英 vtqpi"8„"n "i亥p"pj逢"

không th吋 Rj逢挨pi"rj r"p {"8逢嬰c s穎 d映ng ph鰻 bi院n trong các h羽 th嘘ng yêu

c亥w"8瓜 b違o m壱t cao Tuy nhiên, eqp"pi逢運k"nw»p"n "8k吋m y院u nh医t trong b医t k

h羽 th嘘ng b違o m壱t nào, pi逢運k"f́pi"vj逢運ng c違m th医y b医t ti羽n khi ph違i luôn

mang token theo bên mình và vi羽e"8 pj"o医t ho員e"8吋 sw‒p"vqmgp"e pi"vj逢運ng

xuyên x違y ra M瓜t s嘘 token yêu c亥u ph違k"8逢嬰c kích ho衣t b茨ng m瓜t mã PIN bí

m壱v"8吋 k飲 8 pj"e逸r"e pi"mj»pi"u穎 d映pi"8逢嬰c token Tuy nhiên vi羽c này l衣i

làm nhi隠w"pi逢運i c違m th医y b医t ti羽n và h丑 ghi luôn mã PIN lên token

E e"rj逢挨pi"rj r"p„k"vt‒p"8隠w"mj»pi"e„"u詠"e¤p"d茨pi"ik英c"u泳e"o衣pj"e栄c"mj„c"x "8瓜"

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Technologies (Toronto, Canada) [1, p 22]0"Xk羽e" r"f映pi"pj英pi"vj pj"sw違"pijk‒p"e泳w"vtqpi"n pj"x詠e"p {"x q"vj詠e"v院"e p"mj "j衣p"ej院0"Vjgq"pj逢"pj英pi"i·"j丑e"xk‒p"v·o"jk吋w"8逢嬰e"vj·"e»pi"pij羽"rtkxKFTM e栄c"Rjknkru"n "u違n rj育o"vj逢挨pi"o衣k"8亥w"vk‒p"

x "fw{"pj医v"v pj"8院p"vj運k"8k吋o"p {" r"f映pi"m悦"vjw壱v"o«"j„c"f́pi"8員e"vt逢pi"ukpj"vt逸e"j丑e [1, p 22]

Pi {"pc{."e e"vjk院v"d鵜"e "pj¤p"pj逢"8k羽p"vjq衣k"vj»pi"okpj."o {"v pj"d違pi"pi {"o瓜v"rj鰻"dk院p"x "o衣pj"o胤0"Pj英pi"vjk院v"p {"e„"u泳e"o衣pj"i亥p"v逢挨pi"8逢挨pi"x噂k"o瓜v"o {"

tính cá nhân, e„"mj違"p<pi"n逢w"vt英"x "z穎"n#"j亥w"j院v"o丑k"nq衣k"f英"nk羽w"v瑛"j·pj"違pj."fcpj"d衣."ijk"ej¿"e "pj¤p"8院p"pj英pi"f英"nk羽w"pj衣{"e違o"pj逢"gockn."v k"mjq違p"pi¤p"j pi0"O瓜v"u嘘"pj¤p"xk‒p"e p"n逢w"vt英"e違"pj英pi"v k"nk羽w"o壱v"e栄c"e»pi"v{"vt‒p"8k羽p"vjq衣k"fk"8瓜pi"e栄c"o·pj0"Vtqpi"mjk"8„."x医p"8隠"d違q"o壱v"vt‒p"e e"vjk院v"d鵜"fk"8瓜pi"x磯p"ej逢c"pj壱p"8逢嬰e"u詠"swcp"v¤o"e亥p"vjk院v"v瑛"rj c"pi逢運k"f́pi."o瓜v"u嘘"e違o"vj医{"rjk隠p"rj泳e."mj»pi"vjw壱p"vk羽p"mjk"rj違k" r"f映pi"e e"ik違k"rj r"d違q"o壱v="xk羽e"p {"dk院p"e e"vjk院v"d鵜"

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o "8逢嬰c x穎 lý qua m瓜v"d逢噂c g丑k"n "Ðtrích xuＸv"xgevqt"8ｐe"vt⇔piÑ, vi羽c x穎 lý,

so sánh d詠c"vt‒p"e e"xgevqt"8員e"vt逢pi"p {"f宇 dàng và hi羽u qu違 j挨p"uq"x噂i d英

li羽w"vj»"dcp"8亥u Ví d映<"xgevqt"8員e"vt逢pi"e栄a d医u vân tay là m瓜t t壱r"e e"8k吋m 8亥u mút c栄c"8逢運ng vân tay trong 違nh ch映p d医u vân tay

2 B2: Các vector 8員e"vt逢pi"p {"x磯p"ej逢c"rj違i là d英 li羽u chính xác hoàn toàn,

v磯p"ej逢c"vj吋 f́pi"8逢嬰c cho các gi違i thu壱v"o«"j„c0"D逢噂c này s胤 th詠c hi羽n

quá trình s穎a l厩i, lo衣i b臼 nh英ng thành ph亥n sai s嘘 và trích xu医t ra thông tin c嘘 8鵜nh N院u d英 li羽w"8亥u vào là k院t qu違 thu th壱r"8逢嬰c trên cùng m瓜v"8嘘k"v逢嬰ng,

k院t qu違 8亥u ra ph違i gi嘘ng nhau

3 K院t qu違 c栄a B2 có th吋 8逢嬰e" f́pi" 8吋 sinh ra khóa ho員c b違o v羽 m瓜t khóa

khác, ho員c k院t h嬰p v噂i k院t qu違 trích xu医t c栄a m瓜v"8員e"vt逢pi"ukpj"vt逸c h丑c

mj e"8吋 mã hóa d英 li羽u

A隠"v k"v壱r"vtwpi"pijk‒p"e泳w"x "8隠"zw医v"rj逢挨pi"rj r"e„"vj吋" r"f映pi"8逢嬰e"ejq"

B2 O映e"vk‒w"e栄c"8隠"v k"n "z¤{"f詠pi"rj逢挨pi"rj r"8吋"e„"vj吋" r"f映pi"8逢嬰e"vt‒p"vjk院v"d鵜"fk"8瓜pi0"Jk羽p"v衣k."8嘘k"x噂k"j亥w"j院v"e e"vjk院v"d鵜"fk"8瓜pi"vj»pi"okpj."ej雨"e„"xk羽e"ej映r"違pj"i逢挨pi"o員v"x "ijk"¤o"ik丑pi"p„k"n mj違"vjk."e e"8員e"vt逢pi"ukpj"vt逸e"j丑e"mj e"pj逢"f医w"x¤p"vc{"jc{"o嘘pi"o逸v"n "i亥p"pj逢"mj»pi"vj吋0"Fq"8„."8隠"v k"n詠c"ej丑p"違pj"

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x q"j羽"vj嘘pi"o«"j„c [1, p 21]0"Vtqpi"e ej"vk院r"e壱p"p {."mj„c"8逢嬰e"ukpj"tc"o瓜v"

cách pi磯w"pjk‒p."ucw"8„"m院v"j嬰r"x噂k"8員e"vt逢pi"ukpj"vt逸e"j丑e"8吋"d違q"x羽"mj„c"8„0

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1.3 V„o"v逸v"p瓜k"fwpi"nw壱p"x<p

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‚ Ej逢挨pi"4 s胤 gi噂i thi羽u các công trình nghiên c泳u tiêu bi吋w"nk‒p"swcp"8院n v医n 8隠 mã hóa d英 li羽w"f́pi"8員e"vt逢pi"ukpj"vt逸c h丑c

‚ Ej逢挨pi"5 tóm t逸v"e e"e挨"u荏 lý thuy院t và gi違i thu壱t 8逢嬰c s穎 d映ng tr詠c ti院p

trong vi羽c xây d詠pi"rj逢挨pi"rj r"x "8 pj"ik "e栄a lu壱p"x<p0

‚ 0 là ph亥n chính, di宇n gi違k"rj逢挨pi"rj r"o "nw壱p"x<p"8隠 xu医t

‚ Ej逢挨pi"7 trình bày k院t qu違 th詠c nghi羽m và so sánh v噂i m瓜v"rj逢挨pi"rj r"8挨p"ik違n khác

‚ Ej逢挨pi"8 nêu k院t lu壱p"x "8隠 xu医v"e e"j逢噂ng phát tri吋n ti院p theo c栄c"8隠 tài

1.4 Fcpj"o映e"v瑛"xk院v"v逸v

FAR False Acceptance Rate<"V益"n羽"ej医r"pj壱p"uck."n "v益"n羽"o瓜v"o磯w"ukpj"

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j丑e"j嬰r"n羽"*e栄c"8嘘k"v逢嬰pi"8cpi"z e"vj詠e+"d鵜"v瑛"ej嘘k

EER Equal Error Rate<"V益"n羽"n厩k"e¤p"d茨pi."vtqpi"e e"j羽"vj嘘pi"d違q"o壱v."

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Ej逢挨pi"40 E e"pijk‒p"e泳w"nk‒p"swcp

2.1 J逢噂pi"vk院r"e壱p"f́pi"o«"u穎c"n厩k

A.Juels và M.Wattenberg [2]."8«"泳pi"f映pi"o«"u穎c"n厩k"x q"xk羽e"d違q"x羽"mj„c"d茨pi"8員e"vt逢pi"ukpj"vt逸e"j丑e0

Sw "vt·pj"8<pi"m#<

‚ Gi違 s穎 m磯u sinh tr逸c h丑e" 8逢嬰c bi吋u di宇n b茨ng m瓜t chu厩i n bit nh鵜 phân

捲 樺 岶ど な岼津

‚ 系岫券 倦 穴岻 là m瓜t mã s穎a l厩i có chi隠u dài n bit, k chi隠u và kho違ng cách t嘘i

thi吋u là d Gi違 s穎 khóa c亥p"8逢嬰c b違o v羽 là 潔 樺 系

‚ K院t qu違 c栄a quá trình 8<pi"m#"*vjw壱t ng英 g嘘e"vtqpi"d k"d q"n "ÐcommitÑ+ là

‚ N宇u 捲旺 khác 捲 không quá 岫穴 伐 な岻 に dkv." pij c" n " 潔旺 khác 潔 không quá

岫穴 伐 な岻 に bit Ta có th吋 gi違i mã 潔旺 8吋 vjw"8逢嬰c 潔 dcp"8亥u

Pj壱p"zfiv<

‚ A¤{"n "o瓜t trong nh英pi"rj逢挨pi"rj r"8亥u tiên 泳ng d映ng mã s穎a l厩i vào vi羽c

b違o v羽 d英 li羽u b茨pi" 8員e" vt逢pi" ukpj" vt逸c h丑c, dùng mã s穎a l厩k" 8吋 lo衣i b臼

nh英ng sai s嘘 vtqpi"8員c vt逢pi"ukpj"vt逸c h丑c Tuy nhiên, không ph違k"8員e"vt逢pi"

sinh tr逸c h丑e"p q"e pi"e„"vj吋 bi吋u di宇p"f逢噂i d衣ng m瓜t chu厩i bit nh鵜 phân và

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A.Juels và M.Sudan [3] gi噂i thi羽u m瓜t khái ni羽m g丑k"n "ÐHw||{"XcwnvÑ M瓜t bí m壱t

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Quá trình khóa bí m壱t 愁 d茨pi"v壱r"C<

‚ Bí m壱t 煮 8逢嬰c bi吋u di宇n b荏i m瓜t vector 煮" 噺 " 岫倦待 倦怠 倦津貸怠岻

‚ Bi吋u di宇n 煮 f逢噂i d衣pi""8c"vj泳c:"喧岫捲岻 噺 デ倦伐な件噺ど 倦件捲件

‚ 迎 噺 岶岫捲 喧岫捲岻 捲 樺 畦岼 8逢嬰c g丑i là t壱r"e e"Ð8k吋o"ej pj"fcpjÑ

‚ Ta thêm vào R m瓜t s嘘 8k吋m, g丑k"n "Ð8k吋m gây nhi宇wÑ:

迎 華 迎 姦 岶岫捲 検 捲 樺 戟 伐 畦 検 塙 喧岫捲岻岼

E e"Ð8k吋m gây nhi宇wÑ"p {"ej pj"n "vj pj"rj亥n t衣o nên tính b違o m壱t c栄a h羽

th嘘ng, k飲 t医n công s胤 không bi院v"8逢嬰e"8¤w"n "Ð8k吋m gây nhi宇wÑ."8¤w"n "Ð8k吋m

ej pj" fcpjÑ." u嘘 n逢嬰pi" Ð8k吋m gây nhi宇wÑ" e pi" pjk隠u, tính b違o m壱t c栄a h羽

th嘘pi"e pi"ecq."pj逢pi"d́"n衣k."8 k"j臼k"mj»pi"ikcp"n逢w"vt英 càng l噂n

Quá trình m荏 khóa bí m壱t 煮 d茨pi"v壱r"B:

‚ 芸 噺 岶岫捲 検岻 捲 樺 稽球岫捲 検岻 樺 迎岼

‚ N院u ph亥n giao nhau gi英a t壱p B và t壱p A n "8栄 l噂n, Q s胤 ch泳a m瓜t s嘘 n逢嬰ng Ð8k吋o"ej pj"fcpjÑ"8栄 l噂n, lúc này, ta dùng mã Reed-Uqnqoqp"8吋 gi違i mã Q 8吋 tính 愁0"Pi逢嬰c l衣i, quá trình gi違i mã Reed-Solomon s胤 cho ra k院t qu違 khác

愁

2.1.3 Dodis et al Î Fuzzzy Extractor (2008)

Y.Dodis et al [4] 8«" ik噂k" vjk羽w" jck" mj k" pk羽o" ÐUgewtg" UmgvejÑ" x " ÐHw||{"GzvtcevqtÑ."fq"rj逢挨pi"rj r"8逢嬰e"8隠"zw医v"vtqpi"nw壱p"x<p"p {"8逢嬰e"z¤{"f詠pi"f詠c"vt‒p"mj k"pk羽o"ÐUgewtg"UmgvejÑ."rj亥p"3.4.5 u胤"vt·pj"d {"ejk"vk院v"mj k"pk羽o"p {, bài

d q"e pi"8逢c"tc"rj逢挨pi"rj r"8 pj"ik "o泳e"8瓜"d違q"o壱v"e栄c"ÐUgewtg"UmgvejÑ"x "ÐHw||{"GzvtcevqtÑ"f詠c"vt‒p"mj k"pk羽o"ÐOkp Gpvtqr{Ñ."mj k"pk羽o"p {"e pi"u胤"8逢嬰e"vt·pj"d {"ejk"vk院v"vtqpi"rj亥p3.4.1

D k" d q" z¤{" f詠pi" ÐUgewtg" UmgvejÑ" ejq" 5" mj»pi" ikcp" ogvtke" mj e" pjcw<"ÐJcookpiÑ." ÐUgv" FkhhgtgpvÑ" x " ÐGfkvÑ." vtqpi" 8„" ÐUgewtg" UmgvejÑ" f詠c" vt‒p"ÐJcookpiÑ"n "rj逢挨pi"rj r"p隠p"v違pi."ÐUgv"FkhhgtgpvÑ"x "ÐGfkvÑ"8逢嬰e"dk院p"8鰻k"8吋"8逢c"x隠"ÐJcookpiÑ"x "u穎"f映pi"e e"rj逢挨pi"rj r o«"u穎c"n厩k"f詠c"vt‒p"mjq違pi"e ej"ÐJcookpiÑ E»pi"vt·pj"e栄c"Y.Fqfku"vt荏"vj pj"p隠p"v違pi ejq"e e"pijk‒p"e泳w"x隠"ucw0

U.Martini và S.Beinlich [5] 8逢c"tc"o»"j·pj"f詠c"vt‒p"n#"vjw{院v"o«"u穎c"n厩k"pj逢"n "o瓜v"rj亥p"e栄c"f詠" p"ÐXktvwcn"RKP"Dcugf"qp"Dkqogvtkeu"*XkRDqD+Ñ"8逢嬰e"v k"vt嬰"d荏k"Nk‒p"

Minh Châu Âu (EU) J·pj"2.1 o»"v違"rj逢挨pi"rj r"e栄c"Octvkpk."x隠"e挨"d違p."p„"ik嘘pi"pj逢"ÐHw||{"EqookvogpvÑ"e栄c"A.Juels và M.Wattenberg [2] Tuy nhiên, U.Martini 8«"ik噂k"vjk羽w"o瓜v"rj逢挨pi"rj r"dk院p"8鰻k"f医w"x¤p"vc{"vj pj"o瓜v"ejw厩k"dkv"8吋"e„"vj吋"8逢嬰e" r"f映pi cho o«"u穎c"n厩k0

J·pj"2.20"Vjwｆv"vq p"dkｘp"8ごk"xgevqt"8ｐe"vt⇔pi"eてc"[cik|"Uwvew

E e"rj逢挨pi"rj r"e栄c"Y.Dodis [4] f詠c"vt‒p"f英"nk羽w"t運k"t衣e *u嘘"piw{‒p+"vtqpi"mjk"e e"xgevqt"8員e"vt逢pi"e栄c"f英"nk羽w"ukpj"vt逸e"j丑e"vj逢運pi"荏"f衣pi"nk‒p"v映e *u嘘"vj詠e+."fq"8„"vc"e亥p"rj違k"dk院p"8鰻k"ej¿pi"x隠"f衣pi"t運k t衣e"d茨pi"e ej"n逢嬰pi"v穎"j„c"ej¿pi0"Q.Li [7], 8«"8逢c" tc" mj k" pk羽o" Ðtgncvkxg" gpvtqr{" nquuÑ" 8吋" 8 pj" ik " o瓜v" e ej" v逢挨pi" 8嘘k" n逢嬰pi"

gpvtqr{"d鵜"o医v"fq"j o"n逢嬰pi"v穎"j„c0"Ðtgncvkxg"gpvtqr{"nquuÑ"ejq"dk院v"ik噂k"j衣p"e栄c"n逢嬰pi"gpvtqr{"e瓜pi"vj‒o"p院w"vc"f́pi"o瓜v"j o"n逢嬰pi"v穎"j„c"v嘘k"逢w"o員e"f́"t医v khó 8吋"z e"8鵜pj"j o"n逢嬰pi"v穎"j„c"v嘘k"逢w"p {0

Feng Hao et al [8] 8«"u穎"f映pi"o«"Tggf-Solomon và mã Hadamard."vt ej"zw医v"mj„c"e„"ejk隠w"f k"362"dkv"v瑛"f英"nk羽w"o嘘pi"o逸v0

Felix Fontein et al [9] 8«"z¤{"f詠pi"o«"u穎c"n厩k"e„"vj吋"u穎c"8逢嬰e"pj英pi"n厩k"z«{"tc"vt‒p"o瓜v"e映o"e e"dkv"i亥p"pjcw"e pi"pj逢"n厩k"z違{"tc"vt‒p"e e"dkv"tk‒pi"dk羽v0 Vt‒p"vj詠e"v院."pj英pi"n厩k"vjgq"e映o"mk吋w"p {"vj逢運pi"z違{"tc"fq"o瓜v"v e"8瓜pi"o衣pj"p q"8„"v瑛"o»k"vt逢運pi."x "f映"x羽v"o運"vt‒p"ecogtc."jq員e"o瓜v"zwpi"8k羽p"e„"vj吋"n o"vjc{"8鰻k"ik "vt鵜"e栄c"pjk隠w"dkv"p茨o"i亥p"pjcw0

2.2 J逢噂pi"vk院r"e壱p"mj»pi"f́pi"o«"u穎c"n厩k

O瓜v" vtqpi" pj英pi" ik違k" rj r" o«" j„c" f́pi" 8員e" vt逢pi" ukpj" vt逸e" j丑e" 8亥w" vk‒p" 8逢嬰e"

C.Soutar et al [10] ik噂k"vjk羽w"vào p<o"3;;; Rj逢挨pi"rj r"8逢嬰e"ik噂k"vjk羽w"vtqpi"

d k"d q"e„"vj吋"8逢嬰e"v„o"v逸v"vtqpi"4"j·pj"ucw<

J·pj"2.3 Sw "vt·pj"o«"j„c"fばc"vt‒p"vj»pi"vkp"ukpj"vtｈe"jがe"eてc"Dkquet{rv"Kpe0

Quá trinh mã hóa trong J·pj"2.3 e„"vj吋"o»"v違"x逸p"v逸v"pj逢"ucw<

‚ T壱p d英 li羽u sinh tr逸c h丑c sau quá trình x穎 lý, k院t h嬰p v噂i m瓜t dãy s嘘 ng磯u

nhiên s胤 sinh ra 2 thành ph亥n: Hstored(u) là thành ph亥n công khai s胤 8逢嬰e"n逢w"

vào Bioscypt, c0(x) là thành ph亥n bí m壱v"8逢嬰e"f́pi"8吋 liên k院t v噂i khóa

‚ Gi違i thu壱t liên k院t khóa liên k院t c0(x) v噂i khóa k0, k院t qu違 8逢嬰e" n逢w" x q"

Bioscrypt

‚ D k"d q"8逢c"tc"o瓜t mô hình hoàn ch雨nh, x英 lý d詠a trên hình 違pj"dcp"8亥u c栄a

d医w"x¤p"vc{"*mj»pi"t¿v"vt ej"xgevqt"8員e"vt逢pi+."f́pi"dk院p"8鰻i Fourier cùng

m瓜t mô hình toán h丑c khá hoàn ch雨pj"ejq"ikck"8q衣n E-1 và V-30"Fq"8逢嬰c k院t

h嬰p v噂i m瓜t dãy s嘘 ng磯u nhiên, nên tính bí m壱t c栄a d英 li羽u sinh tr逸c h丑c 8逢嬰e"8違m b違o

‚ Tuy nhiên gi違i thu壱t liên k院t khóa và nh壱n l衣i khóa d詠c"vt‒p"rj逢挨pi"rj r"

b亥u ch丑n s嘘 8»pi."d k"d q"ej逢c"8逢c"tc"8逢嬰c mô hình toán h丑c ch泳ng minh 8逢嬰e"8瓜 cp"vq p"x "8 pj"ik " mj違 p<pi"ej鵜u nhi宇u c栄a gi違i thu壱t này Bên

c衣pj"8„."e e"m院t qu違 FAR x "HTT"e pi"ej逢c"8逢嬰e"8q"8衣v"x "8 pj"ik 0

2.2.2 E e"rj逢挨pi"rj r"e栄a Fabian Monrose

F.Monrose et al [11] 8«"ik噂k"vjk羽w"o瓜v"rj逢挨pi"rj r"8吋"v<pi"e逢運pi"d違q"o壱v"ejq"j羽"vj嘘pi"8<pi"pj壱r"d茨pi"o壱v"mj育w f詠c"vt‒p"xk羽e"rj¤p"v ej"vjcq"v e"i "rj o"e栄c"pi逢運k"f́pi"n¿e"8<pi"pj壱r0"N¿e"8<pi"m#."vjcq"v e"i "rj o"e栄c"pi逢運k"f́pi"u胤"8逢嬰e"rj¤p"

v ej"x "dk院p"8鰻k"vj pj"o瓜v"ejw厩k"dkv"pj鵜"rj¤p."ejw厩k"dkv"pj鵜"rj¤p"p {"8逢嬰e" pj"z衣"

vj pj"o瓜v"mj„c"d "o壱v." pj"z衣"p {"8逢嬰e"dk吋w"fk宇p"f逢噂k"f衣pi"d違pi"x "8逢嬰e"o«"j„c"d茨pi ej pj" o壱v" mj育w" pi逢運k" f́pi" pj壱r" x q0" N¿e" 8<pi" pj壱r." p院w" pi逢運k" f́pi" i "8¿pi"o壱v"mj育w."d違pi" pj"z衣"vt‒p"u胤"8逢嬰e"ik違k"o«."ucw"8„"j羽"vj嘘pi"u胤"rj¤p"v ej"vjcq"

v e"i "rj o"e栄c"pi逢運k"f́pi."p院w"ik嘘pi"x噂k"n亥p"8亥w"vk‒p"pi逢運k"f́pi"i "n¿e"8<pi"m#."mj„c"u胤"8逢嬰e"v k"v衣q"n衣k"ej pj"z e0"P„k"v„o"n衣k."pi逢運k"f́pi"mj»pi"pj英pi"rj違k"i "8¿pi"o壱v"mj育w"o "vjcq"v e"i "e pi"rj違k"ik嘘pi"pj逢"n¿e"8<pi"m#0

F詠c"vt‒p"rj逢挨pi"rj r"8„, F.Monrose et al [12] 8«"rj v"vtk吋p"o瓜v"rj逢挨pi"rj r"v逢挨pi"v詠"pj逢pi"vjc{"x·"f́pi"vjcq"v e"i "rj o"e栄c"pi逢運k"f́pi."rj逢挨pi"rj r"o噂k"f́pi"ik丑pi"p„k0"V p"jk羽w"ik丑pi"p„k"e栄c"pi逢運k"f́pi"u胤"8逢嬰e"ejkc"tc"vj pj"v瑛pi"8q衣p."ucw"8„"rj¤p"v ej"8吋"ukpj"tc"o瓜v"f«{"dkv"pj鵜"rj¤p"*mjq違pi"37"dkv+."f«{"dkv"pj鵜"rj¤p"

p {"n衣k"8逢嬰e" pj"z衣"vj pj"o瓜v"mj„c"v逢挨pi"v詠"pj逢"rj逢挨pi"rj r"vt‒p0

rj r"ejq"rjfir"rj v"jk羽p"n厩k"z違{"tc"vt‒p"m‒pj"vtw{隠p"x "rj映e"j欝k f英"nk羽w dcp"8亥w"o瓜v"

e ej"jk羽w"sw違0

F英"nk羽w"vj逢運pi"8逢嬰e"vtw{隠p"8k"f逢噂k"f衣pi"o瓜v"ejw厩k"dkv"pj鵜"rj¤p."fq"8„"n#"vjw{院v"o«"8逢嬰e"z¤{"f詠pi"i逸p"nk隠p"x噂k"ÐJcookpi"ogvtkeÑ"*mjq違pi"e ej"ÐJcookpiÑ"vj吋"jk羽p"u詠"mj e"pjcw"ik英c"4"ejw厩k"dkv+0"Vw{"pjk‒p."piw{‒p"n#"e挨"d違p"e栄c"xk羽e"u穎c"n厩k"e„"vj吋"8逢嬰e" r"f映pi"ejq"e e"mj»pi"ikcp"ogvtke"mj e0

A鵜pj"pij c"3.1: Không gian Metric 嫉 n "o瓜v"d瓜"4"vj pj"rj亥p"*M,d+."vtqpi"8„<

‚ M là m瓜t t壱p h嬰p (h英u h衣n hay vô h衣n)

‚ d: M ×M s"温 là hàm tính kho違ng cách gi英a 2 ph亥n t穎 b医t k c栄a M th臼a các 8k隠u ki羽n:

o 穴岫捲 検岻 髪 穴岫検 権岻 半 穴岫捲 権岻褐捲 検 権 樺 警 (b医v"8鰯ng th泳c tam giác) (3.4) Vtqpi"vj詠e"v院."e„"t医v"pjk隠w"j o"v pj"mjq違pi"e ej"mj e"pjcw"v́{"x q"8員e"v pj"e栄c"8嘘k"v逢嬰pi"e亥p"mj違q"u v0"Vj»pi"vj逢運ng, v‒p"e栄c"j o"v pj"mjq違pi"e ej"e pi là tên e栄c không gian Metric

馨 岶件 捲沈 塙 権沈岼 判 岶件 捲沈 塙 検沈岼 髪 岶件 検沈 塙 権沈岼

馨 穴岫捲 検岻 髪 穴岫検 権岻 半 穴岫捲 権岻褐捲 検 権 樺 警

3.1.4 A鵜pj"pij c"o«"u穎a l厩i

‚ Cho m瓜t t壱p h英u h衣n A: g丑i là t壱p ký t詠 *vj逢運ng là t壱p các ký s嘘 nh鵜 phân

‚ n là chi隠u dài c栄a mã

A鵜pj" pij c" 3.3: Mjq違pi" e ej" Jcookpi" e栄c" o瓜v" o«" 8逢嬰e" 8鵜pj" pij c" n " mjq違pi"

e ej"Jcookpi"v嘘k"vjk吋w"ik英c"4"v瑛"o«<

穴陳沈津岫系岻 "噺 "兼件券岶穴張岫捲 検岻 "捲 検" 樺 "系 捲" 塙 "検岼 (3.6)

Mjq違pi"e ej"Jcookpi"e„"#"pij c"swcp"vtqpi"vtqpi"xk羽e"rj v"jk羽p"n厩k"x "u穎c"n厩k"vtqpi"o»"j·pj"vtw{隠p"vj»pi"e栄c"Ujcpqp."o瓜v"o«"e„"mjq違pi"e eh Hamming v嘘k"vjk吋w

dminu胤"e„"mj違"p<pi"rj v"jk羽p"v嘘k"8c"*dminÎ3+"n厩k"x "u英c"v嘘k"8c"**dminÎ3+14+"n厩k0"

Không gian mã An x " mjq違pi" e ej" Jcookpi" dH v衣q" vj pj" mj»pi" ikcp" Ogvtke"

Hamming 嫉=( An, dH)0"Ik違"u穎"vc"vtw{隠p"8k"o瓜v"v瑛"o«"c 樺 C."p院w"n厩k"z違{"tc"vt‒p"d

dkv."d‒p"pj壱p"u胤"pj壱p"8逢嬰e"eÓ x噂k"dH (c,cÓ+?d P院w"d ø dminÎ1 thì cÓ鞄 C."d‒p"pj壱p"u胤"dk院v"8逢嬰e"e„"n厩k"z違{"tc"vt‒p"8逢運pi"vtw{隠p0

Mj違"p<pi"u穎c"n厩k"e栄c"o«"E"8逢嬰e"o»"rj臼pi"d荏k"j·pj"x胤"ucw:

J·pj"3.10"O»"rjぎpi"mjＶ"p<pi"uぬc"nざk"eてc"oじv"o«

Ik違"u穎"C chia không gian mã An thành các v壱r"eqp."o厩k"v壱r"eqp dcq"i欝o"o瓜v"v瑛"o«"

c x "e e"v瑛"cÓ"ucq"ejq"dH (c,cÓ+"ø"*dminÎ1)/2."vc"f宇"f pi"pj壱p"vj医{"e e"v壱r"eqp này

không giao nhau Pj逢"x壱{"o厩k"v瑛"cÓ"u胤"vjw瓜e"x隠"pjk隠w"pj医v"o瓜v"v壱r"eqp0

Mjk"d‒p"pj壱p"pj壱p"8逢嬰e"cÓ."p院w"vc"ik違"u穎"n厩k"z違{"tc"vt‒p"pjk隠w"pj医v"n "*dminÎ1)/2 bit vj·"v瑛"o«"dcp"8亥w"d‒p"i穎k"i穎k"8k"n "v瑛"o«"fw{"pj医v"e栄c"v壱r"eqp"ej泳c"cÓ

Sw "vt·pj"ik違k"o«"8逢嬰e"8鵜pj"pij c"n "sw "vt·pj"mj»k"rj映e"n衣k"v瑛"o«"dcp"8亥w"d‒p"i穎k"i穎k"8k"v瑛"ejw厩k"dkv"o "d‒p"pj壱p"pj壱p"8逢嬰e"x噂k"ik違"8鵜pj n厩k"z違{"tc"vt‒p"pjk隠w"pj医v"n "

(dminÎ1)/2 bit

3.2 Rj逢挨pi"rj r"rj¤p"v ej"vj pj"rj亥p"ej pj"*REC+

Vtqpi"e e"vj詠e"pijk羽o"mjqc"j丑e"e pi"pj逢"e e"vj嘘pi"m‒."e e"dk院p"e嘘"swcp"u v"8逢嬰e"e„"vj吋"e„"u嘘"ejk隠w"mj n噂p."xk羽e"rj¤p"v ej"f英"nk羽w"vt‒p"v壱r"f英"nk羽w"p {"i員r"pjk隠w"mj„"mj<p0 Vw{"pjk‒p."ik "vt鵜"vt‒p"e e"ejk隠w"e栄c"e e"dk院p"e嘘"p {"vj逢運pi"e„"o嘘k"v逢挨pi"swcp" x噂k" pjcw0 Rj¤p" v ej" vj pj" rj亥p" ej pj" *REC+" n " o瓜v" rj逢挨pi" rj r" vq p" j丑e"dk院p"8鰻k"o瓜v"v壱r"e e"dk院p"e嘘"8c"ejk隠w"ucpi"o瓜v"j羽"vt映e"v丑c"8瓜"vt詠e"ikcq"o噂k"ucq"ejq"

ik "vt鵜"vt‒p"e e"ejk隠w"n "mj»pi"v逢挨pi"swcp"vw{院p"v pj"x噂k"pjcw."u嘘"ejk隠w"vtqpi"mj»pi"ikcp"o噂k"e„"vj吋 pj臼"j挨p"u嘘"ejk隠w"vtqpi"mj»pi"ikcp"dcp"8亥w Vtqpi"mj»pi"ikcp"o噂k"

p {" vc" e pi" e„" vj吋" ej丑p" d臼" 8k" o瓜v" x k" vj pj" rj亥p" *ejk隠w+" o " mj»pi" d鵜" o医v" sw "pjk隠w"vj»pi"vkp."cj雨"ik英"n衣k"e e"vj pj"rj亥p"ej pj."ik¿r"ejq"xk羽e"rj¤p"v ej"f英"nk羽w"8逢嬰e"f宇"f pi"j挨p0

Rj亥p"p {"u胤"ik噂k"vjk羽w"e e"mj k"pk羽o"nk‒p"swcp"x "#"pij c"e栄c"rj逢挨pi"rj r"REC."mj»pi"8k"u¤w"x q"ejk"vk院v"jk羽p"vj詠e0

J·pj"3.30"Dkあw"fkいp"fの"nkうw"swcp"u v"8⇔ぢe"ucw"mjk"vおpj"vkｘp"iぐe"vがc"8じ

Ik違"u穎"vc"ej丑p"e e"8逢運pi"vj鰯pi"pfiv"8泳v"n o"j羽"vt映e"v丑c"8瓜"o噂k."v壱r"f英"nk羽w"8逢嬰e"dk吋w"fk宇p"pj逢"j·pj"ucw<

‚ Giá tr鵜 cách thành ph亥n x và y không có m嘘k"v逢挨pi"swcp"x噂i nhau

‚ Giá tr鵜 trên tr映c x bi院n thiên nhi隠w"j挨p"ik "vt鵜 trên tr映c y

Rj逢挨pi"rj r"REC"8逢嬰e"f́pi"8吋"uw{"ik違o"u嘘"ejk隠w"e栄c"v壱r"f英"nk羽w."P院w"vc"nq衣k"d臼"

vj pj"rj亥p"y."vc"e pi"mj»pi"o医v"sw "pjk隠w"vj»pi"vkp0"F英"nk羽w"ucw"mjk"nq衣k"d臼"vj pj"rj亥p"y."ucw"8„"dk院p"8鰻k"x隠"j羽"vt映e"v丑c"8瓜"dcp"8亥w"8逢嬰e"dk吋w"fk宇p"vtqpi J·pj"3.5 Ta

vj医{"f英"nk羽w"o噂k"mj»pi"sw "mj e"*mj»pi"o医v"sw "pjk隠w"vj»pi"vkp+"uq"x噂k"v壱r"f英"nk羽w"i嘘e"dcp"8亥w

J·pj"3.50"Fの"nkうw"8⇔ぢe"mj»k"rjつe"ucw"ejえ"ikの"nＴk"vj pj"rjＺp"ej pj

Pj逢"x壱{."ucw"sw "vt·pj"dk院p"8鰻k"vt詠e"ikcq"x "nq衣k"d臼"8k"o瓜v"u嘘"vj pj"rj亥p"e„"vt丑pi"u嘘"vj医r."ej雨"ik英"n衣k"pj英pi"vj pj"rj亥p"e„"vt丑pi"u嘘"ecq"*vj pj"rj亥p"ej pj+."vc"e„"8逢嬰e"o瓜v"v壱r"f英"nk羽w"o噂k"x噂k"u嘘"ejk隠w" v"j挨p"u嘘"ejk隠w"dcp"8亥w"o "mj»pi"o医v"sw "pjk隠w"vj»pi"vkp."ik¿r"ejq"xk羽e"rj¤p"v ej"f英"nk羽w"8逢嬰e"f宇"f pi"j挨p0

Rj亥p"ucw"u胤"vt·pj"d {"e e"d逢噂e"e映"vj吋"e栄c"rj逢挨pi"rj r"REC

3.2.2 Vector riêng và tr 鵜 riêng

Vc"zfiv"o瓜v"x "f映"ucw"8¤{<

峙に ぬに な峩 抜 峙なぬ峩 噺 峙ななの 峩 (3.7)

Vtqpi" x " f映" *509+" vc" vj医{" m院v" sw違" e栄c" rjfir" pj¤p" oc" vt壱p" mj»pi" rj違k" n " d瓜k" e栄c"xgevqt"dcp"8亥w0"Pi逢嬰e"n衣k."vtqpi"x "f映"*50:+"vc"vj医{"m院v"sw違"rjfir"pj¤p"n "d瓜k"e栄c"xgevqt"dcp"8亥w Vector 峙ぬに峩 8逢嬰e"i丑k"n "o瓜v"xgevqt"tk‒pi"(eigen vector) e栄c"oc"vt壱p"

x "6"n "vt鵜"tk‒pi"(eigen value) v逢挨pi"泳pi0

‚ N院u m瓜t ma tr壱n vuông n×n có vector riêng thì nó có n vector riêng

‚ N院u v là m瓜t vector riêng thì c×v (c樺温+"e pi"n "o瓜t vector riêng có cùng tr鵜

riêng v噂i c

‚ Các vector riêng tr詠c giao v噂i nhau

Xk羽e"vjc{"8鰻k"8瓜"f k"*mj»pi"vjc{"8鰻k"j逢噂pi+"e栄c"o瓜v"xgevqt"mj»pi"n o"違pj"j逢荏pi"8院p"v pj"ej医v"Ðxgevqt"tk‒piÑ"e栄c"p„."vtqpi"vq p"j丑e."vc"vj逢運pi"n o"xk羽e"x噂k"xgevqt"

riêng e„"8瓜"f k"n "3."i丑k"n "Ðxgevqt"tk‒pi"ejw育p"j„cÑ jc{"Ðxgevqt"tk‒pi"8挨p"x鵜Ñ

X "f映."xgevqt"tk‒pi"8挨p"x鵜"v逢挨pi"泳pi"x噂k 峙ぬに峩 là

琴欽欽

欣ぬ斑ヂなぬにヂなぬ

斑筋禽禽禁

Xk羽e"v·o"xgevqt"tk‒pi"8嘘k"x噂k"e e"oc"vt¤p"e„"m pj"vj逢噂e"n噂p"*n噂p"j挨p"5×5+"8 k"j臼i

nhi隠u tính toán ph泳c t衣p [13] Lu壱p"x<p"p {"u胤 mj»pi"vt·pj"d {"rj逢挨pi"rj r"v·o"

vector riêng Trong th詠c t院 có r医t nhi隠w"vj逢"xk羽n có th吋 giúp ta làm vi羽c này

3.2.3 Rj逢挨pi"rj r"REC

Rj亥p"p {"u胤"vt·pj"d {"e e"d逢噂e"e栄c"rj逢挨pi"rj r"REC0

Ik違"u穎"vc"e„"v壱r"f英"nk羽w"n ejk隠w 8逢嬰e"dk吋w"fk宇p"d荏k"*x1, x2, , xn)

D逢噂e"3< V pj"ik "vt鵜"m "x丑pi"x "oc"vt壱p"jk羽r rj逢挨pi"uck

潔剣懸 噺 頒

潔剣懸岫捲違怠 捲違怠岻 潔剣懸岫捲違怠 捲違態岻 潔剣懸岫捲違怠 捲違津岻潔剣懸岫捲違態 捲違怠岻 潔剣懸岫捲違態 捲違態岻 潔剣懸岫捲違態 捲違津岻

riêng

Pj逢"8«"vt·pj"d {"vtqpi"rj亥p"3.2.2, các vector viêng n {"vt詠e"ikcq"x噂k"pjcw."pj逢pi"swcp"vt丑pi"j挨p."p„"ejq"vc"vj医{"e e"vj pj"rj亥p"e栄c"f英"nk羽w"vjw"vj壱r"8逢嬰e0 O厩k"o瓜v"xgevqt"tk‒pi"v逢挨pi"泳pi"x噂k"o瓜v"vj pj"rj亥p."xgevqt"tk‒pi"e„"vt鵜"tk‒pi"e pi"n噂p."vj pj"rj亥p"8„"e pi"ej泳"pjk隠w"vj»pi"vkp0

Oc"vt壱p"xgevqt"tk‒pi"v逢挨pi"泳pi"x噂k"f英"nk羽w"x "f映"vtqpi"rj亥p"3.2.1 pj逢"ucw:

Vtqpi"8„"系貸怠 n "oc"vt壱p"pij鵜ej"8違q"e栄c"oc"vt壱p"C0"P院w"C = eigenvector thì C-1=CT

dùng Eigenfaces