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Simple password-based key agreement protocol
Department of Computer Engineering Kyungpook National University
Sung-woon Lee
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Sequence
Related workSecurity requirementsSystem parametersCryptanalysis for SAKA’s variantsSimple password-based key agreement Protocol (SPKA)Security analysis for SPKAConclusion
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Related work (1/3)
Diffie-Hellman key agreement protocol (1976) Session key sharing based on discrete logarithms over
a finite field Vulnerable to man-in-the-middle attack due to not provi
ding authenticationSAKA (Simple authenticated key agreement) protocol (1999) Providing authentication to Diffie-Hellman protocol usin
g a simple way Using a pre-shared password for user authentication
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Related work (2/3)
Tseng’s protocol (2000) Addressed a weakness caused by man-in-
the-middle attack in the key verification steps of SAKA
Improved verification steps of SAKA
Ku and Wang’s protocol (2000) Showed Tseng’s protocol is still vulnerable
to man-in-the-middle attacks Improved verification steps of SAKA
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Related work (3/3)
Sun (2000) Showed that SAKA is vulnerable to man-in-
the-middle attack, password guessing attack, and perfect forward secrecy
Lin et al.’s protocol (2000) Improved the verification steps of SAKA to
overcome the weaknesses pointed out by Sun
Hsieh et al. (2002) Showed Lin et al.’s protocol still suffers from
password guessing attack
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Security requirements (1/3)
Secure to man-in-the-middle attack Although an attacker eavesdrops,
modifies, reflects, or replays messages being transmitted, the session key has to be secure.
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Security requirements (2/3)
Secure to password guessing attack Online
Easily detected by counting authentication fails
Offline Guessing password by intercepting and
using messages being transmitted Due to using the password that a person is
able to memorize
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Security requirements (3/3)
Provide perfect forward secrecy Although the password was
compromised, an attacker should not compute old session keys
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System parameters
A, B Honest entities
g A primitive root modulo n (generator)
n A large prime number
P The password pre-shared between users
Q Integer value derived from P, Ex) the smallest such integer that is greater than P
Q-1 Inverse of Q
a, b Random numbers chosen by A and B
KA, KB Session key of A and B
XY Value computed by user Y
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Additional cryptanalysis for Tseng’s protocol
A B
Key establishment
XA = (ga)Q
XB = (gb)Q
YA = = gb YB = = ga
KA = (YA)a = gab KB = (YB)b = gab
Key verification
Check YA ?= gb
Check YB ?= ga
1
)(Q
BX1
)(Q
AX
Vulnerable to password guessing attack XA ?= (YB)Q = gaQ or XB ?= (YA)Q = gbQ
XA
XB
YA
YB
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Cryptanalysis for Ku and Wang’s protocol
Alice Bob
Key establishment
XA = (ga)Q
XB = (gb)Q
YA = = gb YB = = ga
KA = (YA)a = gab KB = (YB)b = gab
Key verification
VA = (KA)Q = gabQ Check ?= KB
Check YB ?= ga
1
)(Q
BX1
)(Q
AX
Vulnerable to password guessing attack: ?= YB
Not provide perfect forward secrecy: = gab
XA
XB
VA
YB
1
)(Q
AV
1
)(Q
AX1
)(Q
AV
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Weaknesses of SAKA related protocols
Vulnerable to man-in-the-middle attackVulnerable to password guessing attackNot provide perfect forward secrecy
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Simple password-based key agreement protocol (SPKA)
Alice Bob
Key establishment
XA = (ga)Q
XB =
YA = = gb YB = = ga
KA = (YA)a = gab KB = (YB)b = gab
Key verification
VA = = Check VA ?=
Check VB ?= VB = =
1
)(Q
AX
XA
XB
VA
VB
AKAY )( AbKg
BKBY )( BaKg
BKbg )(AKag )(
1
)(Qbg
QBX )(
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Security analysis for SPKA (1/4)
Secure to man-in-the-middle attack If an attacker eavesdrops XA, XB, VA, and
VB, he cannot gain information for session key, gab because of DLP
If an attacker modifies, reflects, or replays XA, XB, VA, and VB, this attack is detected because verification steps confirm both the correctness of XA, XB and the equality of KA, KB
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Security analysis for SPKA (2/4)
Secure to password guessing attack Since a attacker intercepts the
messages, XA, XB, VA, and VB, any way to confirm the correctness of the guessed password P′ does not exist among them.
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Security analysis for SPKA (3/4)
Provide perfect forward secrecy Although password P is compromised,
an attacker does not have any way that produce old session key gab using Q or Q-1 computed from P
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Security analysis for SPKA (4/4)
Protocol
AnalysisSAKA Tseng
Ku and
WangLin et
al. SPKA
Man-in-the-
middle Attack
NS NS S S S
Password
guessing attack
NS NS NS NS S
Perfect forward secrecy
NP P NP P PS: Secure, NS: Not Secure, P: Provide, NP: Not Provide
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Conclusion
Reported the additional weaknesses in the variants of SAKAProposed simple password-based key agreement protocol (SPKA) Secure to man-in-the-middle attack Secure to password guessing attack Provide perfect forward secrecy
Easily implemented in software and hardware because of its simple structure
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Hyun-Sung Kim
Information Security Lab.
Bit-Serial AOP Arithmetic Operators for Modular Exponentiation over
GF(2m)
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Goal
Implement Exponentiation LSB first algorithm
Two multipliers Squarer and multiplier=> Combined squarer and multiplier
MSB first algorithm Power sum (AB2 + C) AB2 multiplier=> New AB2 multiplier
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Index
Crypto SystemModular ExponentiationGalois Field Bit-Serial Arithmetic OperatorsComparisonConclusion
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Crypto system
Elgamal cryptosystem Encryption : C = Mpublic mod p Decryption : M = Cprivate mod p
public*private mod p 1 M, C GF(2m), integer p : irreducible primitive polynomial
Basic operation=>Modular exponentiation
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Modular exponentiation
Basic operation C = ME mod p E = em-12m-1+ em-22m-2+…+ e12+ e0
= [ em-1 em-2 em-3 … e1 e0 ]
Binary method by Knuth LSB-first algorithm MSB-first algorithm
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LSB-first algorithm
Input M,E,p(x)
Output C=ME mod p(x)=Me0(M2)e1(M4)e2 …(M2 )em-1
Step1 C=1, T=M Step2 for i=0 to m-1
T=TT mod p(x)if ei == 1 C=CT mod p(x)
m-1
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LSB-first algorithm
LSB-first algorithm Basic operation
Squaring Multiplication
Traditional implementation Based on two multipliers Based on a multiplier and a squarer
Proposed implementation Based on a combined squarer and
multiplier
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MSB-first algorithm
Input M,E,p(x)
Output C=ME mod p(x)=(Me1…(Mem-2(Mem-1)2)2…)2Me0
Step1 if em-1 == 1 C=M else C=1 Step2 for i=m-2 to 0
if ei == 1 C=MC2 mod p(x)else C=1C2 mod p(x)
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MSB-first algorithm
MSB-first algorithm Basic operation
AB2 multiplication Traditional implementation
Based on Power-sum circuit (AB2+C) Based on AB2 multiplier
Proposed implementation Based on a new AB2 multiplier
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Galois Field GF(2m)
Finite Field GF(2m) Contains 2m elements Canonical basis
{1, , 2, 3,…, m-1} Element representation GF(2m)
a=am-1m-1+am-2m-2+…+a11+a0
Why implement based on GF(2m) Carry free
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Galois Field GF(2m)
AB mod P, B2 mod P, AB2 mod P A, B GF(2m) P : Irreducible polynomial
All one polynomial (AOP) P(x) = xm+xm-1+xm-2+…+x1+1
Property of AOP Let be a root of p(x) p() = 0, m=m-1+m-2+…+1+1 Multiply in both multiplication m+1+1=0 <= use as an modular in extension field
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Galois Field GF(2m)
Extension field Modular m+1+1 Element representation GF(2m+1)
A= amxm+ am-1xm-1+am-2xm-2+…+a1x1+a0
am=0
Why use the extension field Easy modular reduction
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CSM architecture
Basic architecture for LSB first Exp. A2 mod p : Squarer AB mod p : Multiplier
Proposed Architecture Combined Squarer and
Multiplier(CSM)
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CSM architecture
AB mod P multiplication over EF
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CSM architecture
AB mod P multiplication, P = m+1+1
5 +16 +
7 +2 8 +3
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CSM architecture
Ctl = 1m1m-1…100m-1…00
z0 zm-1 zmzm-2z110
b0…bm-1bm
p0…pm-1pm
ym y1 y0y2y3
a0…am-1am
10
10
10
10
10
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CSM architecture
Step 1, ctl=1 for mux
z0 zm-1 zmzm-2z110
b0…bm-1
ym y1 y0y2y3
a0…am-1
10
10
10
10
10
am
bm
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CSM architecture
Step 2, ctl=1 for mux
z0 zm-1 zmzm-2z110
b0…bm-3bm-2
ym y1 y0y2y3
a0…am-3am-2
10
10
10
10
10
am-1
bm-1
am
bm
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CSM architecture
Step m+1, ctl=1 for mux
z0 zm-1 zmzm-2z110
pm
ym y1 y0y2y310
10
10
10
10
a3
b0
am
b1
a1a2 a0
bm-2 bm-1 bm
am×b0 a3×bm-3 a2×bm-2 a1×bm-1 a0×bm
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CSM architecture
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CSM architecture
Step m+2, ctl=0 for mux
z0 zm-1 zmzm-2z110
pm-1pm
ym y1 y0y2y310
10
10
10
10
a3
bm
am
b0
a1a2 a0
bm-3 bm-2 bm-1
am×bm a3×bm-4 a2×bm-3 a1×bm-2 a0×bm-1
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CSM architecture
A2 mod P=(amm+am-1m-1+…+a1+a0)
2
=am2m+am-12(m-1)+…+a24+a12+a0
=am/2m+amm-1+…+a12+am/2+1+a0
m+1 = 1, m+2 = , m+3 = 2, m+4 = 3
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CSM architecture
Example over GF(24)=(a44+a33+a22+a1+a0)
2
=a48+a36+a24+a12+a0
= a24+a43+a12+a31+a0
5 = 1, 6 = , 7 = 2, 8 = 3
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CSM architecture
Squarer over GF(24)
10
x1 x0
y1 y010
x2
y210
x3
y3
b0b1b2b3b4
10
x4
y4
s0s1s2s3s4
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CSM architecture
Step 4, ctl = 1 for mux
10
x1 x0
y1 y010
x2
y210
x3
y3
b0
10
x4
y4
b4 b2b3 b1 b0
b4 b3b2 b1b0
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CSM architecture
Proposed CSM Architecture
z0 zm-1 zmzm-2z110
b0…bm-1bm
p0…pm-1pm
ym
01
y1 y0
xm x1 x001
01
y2
x201
y3
x3
a0…am-1am
10
10
10
10
10
smsm-1…s0
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POM architecture
Basic architecture for MSB first Exp. Multiplier for AB2 mod p Power-Sum circuit
Proposed Architecture New Power Multiplier (POM)
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POM architecture
AB2 mod P multiplication over EF
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POM architecture
AB mod P multiplication, P = m+1+1
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POM architecture
Proposed POM Architecture
xm
10
x1 x0
ym y1 y010
10
x2
y210
x3
y310
z0 zm-1 zmzm-2z110
b0…bm-1bm
a0…am-1am
p0…pm-1pm
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POM architecture
Step m, ctl=1 for mux
xm
10
x1 x0
ym y1 y010
10
x2
y210
x3
y310
z0 zm-1 zmzm-2z110
a0
b0
a3
b1
a4
b2
a1a2 a0
bm-` bm
am
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Two architectures
Input A and B m bits
Output for AB multiplication, squaring, and AB2 multiplication m+1 bits Computed over extended field
Need to reduce the output => m bits
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MCSM architecture
Output with m bits A : m+1 bits over extended field a : m bits ai = Ai + Am, 0 i m-1
Example over GF(24) A : 10011 p: 11111 a = 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0Am
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MCSM architecture
Pre-compute the most significant bit of result p4=a4b0+a3b1+a2b2+a1b3+a0b4, b4=a4=0
=a3b1+a2b2+a1b3
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MCSM architecture
Proposed MCSM Architecture
z3 z4z2
a0a1a2a3
z0 z110
b0b1b2b3
01
y1 y0
x1 x001
y2
x201
y3
x3
10
10
10
10
x410
s3s2s1s0
z510
p0p1p2p3
must be initialized before computation operation start
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MCSM architecture
Step 3, ctl = 1, p4 = a3b1+a2b2+a1b3
z3 z4z2
a0
z0 z110
b0
01
y1 y0
x1 x001
y2
x201
y3
x3
10
10
10
10
x410
z510
a3 a2 a1
b1 b2 b3
a3
a2a1
b1
b2
b3
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MCSM architecture
Step 4, ctl = 1, p4 = a3b1+a2b2+a1b3
z3 z4z2z0 z110
01
y1 y0
x1 x001
y2
x201
y3
x3
10
10
10
10
x410
s3
z510
p3
a3 a2 a1
b1 b2 b3
p4
a0
0
s4
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MPOM architecture
Pre-compute the most significant bit of result p4=a4b0+a3b3+a2b1+a1b4+a0b2, b4=a4=0
=a3b3+a2b1+a0b2
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MPOM architecture
Proposed MPOM Architecture
10
x1 x0
y0 y410
x2
y110
x3
y210
z0 z3 z4z2z110
b0b1b2b3
a0a1a2a3
p0p1p2p3
z510
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5842
Comparison
CSM and MCSM architecture with Fenn’s architecture
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Comparison
POM and MPOM architecture with Liu’s architecture
2m+1 2m-1
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Conclusion
Proposed 4 multipliers Computes squaring and multiplication Computes AB2 multiplication
Could be used for exponentiation, inversion, and division architecturesEasy to implement VLSI