security of the fiat-shamir transformation in the quantum … · 2019. 4. 16. · theorem. if s is...

Security of the Fiat-Shamir Transformation in the Quantum Random Oracle Model

Serge FehrCWI and

Leiden University

Jelle DonCWI

Chris MajenzUvA

Chris SchaffnerUvA

Theorem. If S is secure (against a dishonest prover), then FS[S] is secure (against a dishonest prover) in the quantum random oracle model (QROM).

Our Result in Short

Holds for any (reasonable) notion of security.

Proof. Transformation: P attacking FS[S] ⇝ Pʹ attacking S.

Let S be a S-protocol, FS[S] its Fiat-Shamir transformation.

S-Protocols

A S-protocol isan interactive proof of a special formto prove x Î L, i.e., $ w : s.t. (x,w) satisfies relation Rwithout revealing w

Examples (for L): L = {x Î Z | $ w : w2 º x (mod N)} for a composite N .L = {(a,b,c) Î G3 | $ w : a = gw, c = bw} for a group G = ágñ .L = {(G0,G1) Î Graph2 | $ p ÎPerm : G1 = p(G0)}.

S-Protocol for Graph Isomorphism

PROVER P

G0,G1

I want to prove that G0 and G1 are isomorphic without revealing p

VERIFIER V

s ¬ Perm H := s(G0)

cc ¬ {0,1}

t := s ∘ p-c

H = t(Gc)?

Probability can be “boosted” by repeating the proof:

if repeated k times, then V accepts false proof with prob. 1/2k.

Assume that G0 ≄G1, i.e. $⁄ p ÎPerm : G1 = p(G0)

Consider arbitrary (possibly dishonest) prover P

Analysis (of Soundness)

Either H ≄G0 or H ≄G1 (or both)

⇒ P fails to answer c with probability 1/2.

s ¬ PermH := s(G0)

c ¬ {0,1}

t := s ∘ p-c

H = t(Gc)?

c

GI Proof

Probability can be “boosted” by repeating the proof:

if repeated k times, then V accepts false proof with prob. 1/2k.

Assume that G0 ≄G1, i.e. $⁄ p ÎPerm : G1 = p(G0)

Consider arbitrary (possibly dishonest) prover P

Analysis (of Soundness)

Either H ≄G0 or H ≄G1 (or both)

⇒ P fails to answer c with probability 1/2.

s ¬ PermH := s(G0)

c ¬ {0,1}

t := s ∘ p-c

H = t(Gc)?

c

GI Proof

What about privacy (V not learning p)?

V obviously does not obtain p “in the clear”.Can show: V learns no info at all on p (zero-knowledge).

Can show something stronger (proof of knowledge):If P succeeds with good probability then he must know p .

The Fiat-Shamir Transformation

PROVER PVERIFIER V

a

cc ¬ {0,1}n

zV(x,a,c,z) ?

:= H(a)c

x,wx

FS

a z,z) ?V(x,a,H(a)p = ( , )

S

FS[S]


PROVER PVERIFIER V

a

cc ¬ {0,1}n

zV(x,a,c,z) ?

:= H(a)c

x,wx

FS

a z,z) ?V(x,a,H(a)p = ( , )

S

FS[S]

Hope is:if H is a “good” cryptographic hash functionthat behaves like a random functionthen FS[S] inherits security properties of S

Works well in practice - cannot be proven.


PROVER PVERIFIER V

a

cc ¬ {0,1}n

zV(x,a,c,z) ?

:= H(a)c

x,wx

FS

a z,z) ?V(x,a,H(a)p = ( , )

S

FS[S]

Hope is:if H is a “good” cryptographic hash functionthat behaves like a random functionthen FS[S] inherits security properties of S

Works well in practice - cannot be proven.

Side remark: Understanding x as public key and w as secret key, and setting c := H(m,a), a proof p forms a signature on m (can be computed only by someone who knows w).

The Random Oracle Model - in the context here

PROVER PVERIFIER V

c := H(a)

x,wx

p = (a,z)V(x,a,H(a),z)

FS[S]Idea:not let H be fixed function known to P and V

accessible (only) via an oracle - the random oracle (RO)

but insteadlet H be random function unknown to P and V

RO

a

H(a)a

H(a)

S versus FS[S] in the RO Model

PROVER P

x,w

FS[S]

RO

a

H(a)

p = (a,z)

PROVER P

x,wa

c

z

versus

S

S versus FS[S] in the RO Model

PROVER PFS[S]

RO

p = (a,z)

PROVER P

a

c

z

versus

S

Dishonest prover P can query the RO many times

Theorem. If S is secure then FS[S] is secure in the ROM.

Security of FS[S] in the (classical) RO Model

Security is w.r.t. any notion regarding a dishonest prover P: secure Î {comp./stat. sound, comp./stat. proof-of-knowledge}

Classical result:

The RO heuristic then suggests security in real life if using a “good enough” cryptographic hash function(Cannot be proven, but works well in practice)

Remark: This RO methodology is used throughout crypto (to avoid no-go results, obtain more efficient schemes)

The Proof

a1

H(a1)

ai

aq

a,z

aʹcʹ

zʹ

……

Pr[V(x,a,H(a),z) ] ³ d

choose i ←{1,..,q}

set aʹ := ai

from i-th query on answer with H ʹ where

H (́ai) = H (́aʹ) := cʹ and H ʹ := H otherwise P

Pʹset zʹ := z

Transformation: P attacking FS[S] in ROM ⇝ Pʹ attacking S

H (́ai)

H (́aq)

The Proof

a1

H(a1)

ai

aq

a,z

aʹcʹ

zʹ

……



set aʹ := ai



Pʹset zʹ := z

Pr[V(x,a ,́c ,́zʹ) ] > d/q~

Transformation: P attacking FS[S] in ROM ⇝ Pʹ attacking S

Need to make sure: a = ai

Pr[V(x,a,H (́a),z) ] ³ d

Easy to see: holds with prob. »1/q

H (́ai)

H (́aq)

Part II:

The Quantum Random Oracle Model

FS[S] in the Quantum RO Model (QROM)

PROVER PFS[S]

RO

p = (a,z)

Dishonest prover P can query the RO many times

and in quantum superposition, i.e.

åba|añ ↦ åba|añ|H(a)ña a

If dishonest P is equipped with a quantum computerin real life: P can compute H in quantum superpositionin RO model: must allow P superposition queries

Theorem. If S is secure then FS[S] is secure in the ROM.

Bit Question

(here: security always refers to dishonest prover P)

Classical result:

Is that still true in the Quantum ROM???

Known Results

Partial positive results [Unruh15 & 17]: Security of another, less efficient, transformationS is statistically sound ⇒ FS[S] computationally sound

in the QROM.

Negative claims: [DFG13] claim impossibility for proof-of-knowledge[Unruh17] claims necessity of stat-to-comp degradation

(both claims have unconvincing reasoning)

Theorem. If S is secure then FS[S] is secure in the QROM.

Our Result

Also here: security is w.r.t. secure Î {comp./stat. sound, comp./stat. proof-of-knowledge}

or any reasonable notion regarding a dishonest prover P.

Small caveat: Our security reduction is less tight: q2 loss, rather than q.

Remark: In independent work, Lie & Zhandry showed the same kind of result, but with a q9 loss.

Intuition


a,z

……


choose i ←{1,..,q}set aʹ := ai obtained by measuring

from i-th query on answer with H ʹ

Recall: H (́ai) = H (́aʹ) := cʹ and H ʹ := H otherwise

P

Pʹset zʹ := z

Natural approach:

åbai|aiñ

|aiñ|cʹñ

åbaq|aqñ


aʹcʹ

zʹ

Intuition


a,z

……





P

Pʹset zʹ := z

Natural approach:

åbai|aiñ

|aiñ|cʹñ

åbaq|aqñ


aʹcʹ

zʹ

But: such detection destroys info gained on H !

Problem: May be detected by P !

Can this be turned into a rigorous proof ? No!

Thus: should work if i is the query where P learns H(a)

A Small Tweak


a,z

……



P

Pʹset zʹ := z

Our approach:

åbai|aiñ

|aiñ|cʹñ or |aiñ|H(ai)ñ

åbaq|aqñ


aʹcʹ

zʹ



or (i +1)-st

Remark: The uniformly random H can be dealt with by replacing it with a 2q-wise independent function.

Part III:

The Proof

For fixed a ,́ let

PH = å |a ñ́áa |́Ä|H(a )́ñáH(a )́|Ä|zñáz|

be the projection/measurement that measures a,h,z and checks if a = a ,́ h = H(a) and V(x,a,H(a),z) .

PreliminariesAssume wlog:

P makes all measurements at the end

Let|f0ñ be P’s initial state

……

…

a h z

|f0ñ

|fiñP

z :V(x,a ,́H(a )́,z)

|fqñ

be unitary from query j to i Î {0,...,q -1}UHj�i

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sha1_base64="JmYTAx3kr0KXFHo7DLJVmPsVabM=">AAACJ3icdZDNSgMxFIUz9a/Wv6pLN6lFcCFlRvBvV3TTZQWnLXRqyaRpG5uZDMkdZRi69llcuNXHcCe69Al8BdNaoUo9EPg4517IPX4kuAbbfrcyc/MLi0vZ5dzK6tr6Rn5zq6ZlrChzqRRSNXyimeAhc4GDYI1IMRL4gtX9wcUor98ypbkMryCJWCsgvZB3OSVgrHa+4HqF69Q7qAzb6Q32dF8qULzXB6KUvMN8iNv5ol2yx8JTcGQ7Z8cOdiZOEU1Ubec/vY6kccBCoIJo3XTsCFopUcCpYMOcF2sWETogPdY0GJKA6VY6PmWI94zTwV2pzAsBj93pjZQEWieBbyYDAn39NxuZMzMNAVGJ6swKmzF0T1spD6MYWEi/f9GNBQaJR6XhDleMgkgMEKq4OQTTPlGEgqk2Zxr6qQH/D7XDkmP40imWzyddZdEO2kX7yEEnqIwqqIpcRNE9ekRP6Nl6sF6sV+vtezRjTXa20S9ZH1/YV6Zc</latexit><latexit 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sha1_base64="xoA1aDiCskN+OBTLVRdx/PlgTJc=">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</latexit><latexit 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sha1_base64="xoA1aDiCskN+OBTLVRdx/PlgTJc=">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</latexit><latexit 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UHj�i


|fiñ := |f0ñ be P’s state at query iUH0�i

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(the “continuation” of P)⇧Hi�q := ⇧HUH

i�q<latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit>

outputs h = H(a) as well.

For fixed a ,́ let

PH = å |a ñ́áa |́Ä|H(a )́ñáH(a )́|Ä|zñáz|

be the projection/measurement that measures a,h,z and checks if a = a ,́ h = H(a) and V(x,a,H(a),z) .

PreliminariesAssume wlog:

P makes all measurements at the end

Let|f0ñ be P’s initial state

……

…

a h z

|f0ñ

|fiñP

z :V(x,a ,́H(a )́,z)

|fqñ

be unitary from query j to i Î {0,...,q -1}UHj�i


UHj�i


|fiñ := |f0ñ be P’s state at query iUH0�i

<latexit sha1_base64="ijAGK+xwX7t3/lKNg6EzsbGaTdY=">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</latexit><latexit sha1_base64="Q7yAtojrEOFEEFbFzHHD5TJ7FHU=">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</latexit><latexit sha1_base64="Q7yAtojrEOFEEFbFzHHD5TJ7FHU=">AAACJ3icdZDLSgMxFIYz9VZbL62uxE1qEVxImRG87YpuuqxgL9CpJZOmbWhmMiRnlDJ07bO4cKuP4U50JT5BX8G0VahSfwj8fP85kPN7oeAabPvdSiwsLi2vJFdT6bX1jc1MdquqZaQoq1AppKp7RDPBA1YBDoLVQ8WI7wlW8/qX47x2y5TmMriGQciaPukGvMMpAYNamVzFzd3E7mFp2Ipt7OqeVKB4twdEKXmH+RC3Mnm7YE+EZ8yx7ZyfONj5Jvlifucjq0bpciszctuSRj4LgAqidcOxQ2jGRAGngg1TbqRZSGifdFnD2ID4TDfjySlDvG9IG3ekMi8APKGzGzHxtR74npn0CfT032wM52YafKIGqj0vbETQOWvGPAgjYAGd/qITCQwSj0vDba4YBTEwhlDFzSGY9ogiFEy1KdPQTw34f1M9KjjGXzn54gWaKol20R46QA46RUVUQmVUQRTdo0f0hJ6tB+vFerXepqMJ63tnG/2S9fkFmhio+g==</latexit><latexit sha1_base64="trF1w5OTrZze2mbKOop3QJLNBMs=">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</latexit>

(the “continuation” of P)⇧Hi�q := ⇧HUH

i�q<latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit><latexit sha1_base64="8lHU1fQJWB0AzKXPZ8ktowxALBk=">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</latexit>

We are interested in

where A = |aʹñáa ́|, and

H (́a )́ := c ́and H (́a) := H(a) for a ¹ aʹ

i,b,cʹE��⇧H0

i+b�qUHi�i+bA|�ii

��2<latexit sha1_base64="0qpphJ0BW5oFntUflKvipKLNWfA=">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</latexit><latexit sha1_base64="h7hbcnyRgfUbmCq0n5PVaFejf9o=">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</latexit><latexit sha1_base64="h7hbcnyRgfUbmCq0n5PVaFejf9o=">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</latexit><latexit sha1_base64="rgaIcmdWj871irPbGz9B3tfFhj8=">AAACdXicdZFPb9QwEMWd0EJZ/i3liCqNWApIVCWp1EJvhapSj4vEtpXW28jxehOrjp3aE9AqzcfsgU/BkQOXOtsgFSgjWXp6P481fpOWSjqMou9BeGdp+e69lfu9Bw8fPX7Sf7p65ExluRhxo4w9SZkTSmoxQolKnJRWsCJV4jg922/58VdhnTT6C85LMSlYpuVMcobeSvqGFgzzNK0PGqAbQFOZ0Qs6lKf14esmqeXbFKjLjUUrsxyZteYbnDcw8rzF/0Lf0cBHuKBlLhNJLdOZEt27p1uQ9AfRZrQouCG2o3h3J4a4cwakq2HS/0GnhleF0MgVc24cRyVOamZRciWaHq2cKBk/Y5kYe6lZIdykXgTTwLp3pjAz1h+NsHBvdtSscG5e+InX2xjc36w1b2UOC2bndnobHFc4+zCppS4rFJpfTzGrFKCBdgUwlVZwVHMvGLfSfwR4zizj6BfV8wn9jgH+L462NmOvP8eDvU9dVivkOXlB3pCYvCd75JAMyYhwckl+BUvBcvAzXAtfhq+ur4ZB1/OM/FHhuyuzysBE</latexit>

Want: E��⇧H0


��2 &��⇧H

0�q|�0i��2

<latexit sha1_base64="/qcGRGG5z6+tb8luVkfmr4/hl/o=">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</latexit><latexit sha1_base64="HKFwpGw8IiUzZQhZEFn+Zs23g6Q=">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</latexit><latexit sha1_base64="HKFwpGw8IiUzZQhZEFn+Zs23g6Q=">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</latexit><latexit sha1_base64="fjcPkyXRaNjtMOy/KAh5c3JjI7g=">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</latexit>

outputs h = H(a) as well.

It thus holds that

= Pr[a = a ́ÙV(x,a,H(a),z) ] ��⇧H

0�q|�0i��2

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The Math

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<latexit sha1_base64="2EdDCzzWMdAY/E9yy4D7IYcMOU4=">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</latexit><latexit sha1_base64="2EdDCzzWMdAY/E9yy4D7IYcMOU4=">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</latexit><latexit sha1_base64="2EdDCzzWMdAY/E9yy4D7IYcMOU4=">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</latexit><latexit sha1_base64="2EdDCzzWMdAY/E9yy4D7IYcMOU4=">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</latexit>

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i�q|�ii<latexit sha1_base64="YmcXndvS8HlaQiLcpRcK/0eOydY=">AAACNXicdZDNThsxFIU9KbQhQDu0SzYWEYJVmKn6u4voJssgERIpE0Z3HCdj4bGn9p2iaJrH4FlYsG0foovuqm7zCjghSGkVjmTp0zn3Sr4nyaWwGAS/vMqzjc3nL6pbte2d3Zev/L3XF1YXhvEO01KbXgKWS6F4BwVK3ssNhyyRvJtcfZnn3W/cWKHVOU5yPshgrMRIMEBnxf5J1BaXZetoGpeCRjbVBo0YpwjG6Gv6dUq/R3kqYhEZUGPJaezXg0awEF2B90H4+UNIw6VTJ0u1Y38WDTUrMq6QSbC2HwY5DkowKJjk01pUWJ4Du4Ix7ztUkHE7KBeHTemhc4Z0pI17CunCXd0oIbN2kiVuMgNM7f/Z3FybWczATMxwXdgvcPRpUAqVF8gVe/jFqJAUNZ1XSIfCcIZy4gCYEe4QylIwwNAVXXMNPdZAn4aLt43Q8dm7evN02VWV7JMDckxC8pE0SYu0SYcwckPuyA/y07v1fnt/vL8PoxVvufOG/CNvdg8g16yq</latexit><latexit sha1_base64="YmcXndvS8HlaQiLcpRcK/0eOydY=">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</latexit><latexit sha1_base64="YmcXndvS8HlaQiLcpRcK/0eOydY=">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</latexit><latexit sha1_base64="YmcXndvS8HlaQiLcpRcK/0eOydY=">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</latexit>

Adding up from i = 0 to q -1, we get

By rearranging terms, taking norms, applying ∆-inequality: X

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= ⇧H0|�qi+

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<latexit sha1_base64="17brCmokxdY8yL6AA10QCCaTnF0=">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</latexit><latexit sha1_base64="17brCmokxdY8yL6AA10QCCaTnF0=">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</latexit><latexit sha1_base64="17brCmokxdY8yL6AA10QCCaTnF0=">AAACunicdVHbbtNAEF2bWwmXBvpGX0aNEEUVlY1aLhJIKby0PAWJtJXiYK03G3uVvbi7Y6rI5K/4mX5Ff4FN6kgBwkgrHZ0zZzR7JiulcBhFV0F46/adu/c27rcePHz0eLP95OmpM5VlvM+MNPY8o45LoXkfBUp+XlpOVSb5WTb5PNfPfnDrhNHfcFryoaK5FmPBKHoqbf/6CElPfK+PX8zgZ1IWIr1ILNW55LAHiatUKiDJRC53l21p7RlXGItW5AVSa80lXMzgqPGLpf8VwIpnL17r6nt9/Ujv+HfofBX7Mm13ov1oUbACDqP4/ZsY4obpkKZ6afs6GRlWKa6RSercII5KHNbUomCSz1pJ5XhJ2YTmfOChpoq7Yb1IdwbPPTOCsbH+aYQFu+qoqXJuqjLfqSgW7m9tTq7VHCpqp3a0ThxUOH43rIUuK+Sa3WwxriSggfkdYSQsZyinHlBmhf8IsIJaytBfu+UTWsYA/wenr/djj78edLqfmqw2yDbZIbskJm9JlxyTHukTFjwLusFJ8CX8EGahCCc3rWHQeLbIHxXiby4I2Hc=</latexit><latexit sha1_base64="17brCmokxdY8yL6AA10QCCaTnF0=">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</latexit>

⇧H0

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= ⇧H0|�qi+

X

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<latexit sha1_base64="a+8auC2vFONoY8F3Wt3KYK/ISR4=">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</latexit><latexit sha1_base64="a+8auC2vFONoY8F3Wt3KYK/ISR4=">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</latexit><latexit sha1_base64="a+8auC2vFONoY8F3Wt3KYK/ISR4=">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</latexit><latexit sha1_base64="a+8auC2vFONoY8F3Wt3KYK/ISR4=">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</latexit>

The Math Want: E��⇧H0


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⇝ unlikely for random cʹ

The Math= state P produces

when interacting with H

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The Math

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= success probability of P when interacting with H ʹ

(from the start)

(for random H & c )́

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= 14q2

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The Math

QED

Remarks

Above considers fixed instance x given to P. Extends to adaptive P that can choose x himself(that’s why one should set c := H(x,a) instead c := H(a))

Works beyond classical S-protocols: z may be quantum.

Implications for Signatures

and S is a S-protocol for L thatis a proof of knowledge (against quantum P)is a (honest verifier) zero knowledgesatisfies some additional mild properties,

then signature scheme Sig[S], defined by setting pk = x (for random x Î L) and sk = w signing m by means of p = (a,z) where c = H(m,x,a)

is a secure (EUF-CMA) in the QROM.

If L is hard on average for quantum algorithms: for random x Î L it is hard to find w s.t. (x,w) Î R

Part IV:

On quantum proof-of-knowledge

Proof of Knowledge

A S-protocol for L is a proof of knowledge if: any (dishonest) P must “know” w to have V accept x

Formalized by requiring existence of an extractor that extracts w from any successful (possibly dishonest) P

Typically (i.e. classically) this extraction involves rewinding P

Problematic if P is quantum: measurements are irreversible

Quantum Proof of Knowledge

So far know [Unruh12,ARU14]:

for S to be a quantum proof-of-knowledge (QPoK) itneeds to have unique responses:

i.e., "x Î L "a,c exists unique z : V(x,a,c,z) .

Intuition: z unique ⇒ measuring it does not disturb state Caveat: often not satisfied

Computational unique responses, where it is hard to compute more than one z,

is not sufficient (even for a computational QPoK)Pity: often is (or seems) satisfied

Strong Computational Unique ResponsesWe can:

circumvent (or relativize) negative result bystrengthening notion of computational unique responsesin the spirit of Unruh’s collapsing property

Caveat 1: again property that is hard to prove

On the other hand: no (convincing) separation examplesThus, for typical schemes: expect non-strong ⇒ strong

Caveat 2: still require S to be special sound

[LieZhandry19]: S underlying Dilithium has strong comp... ⇒ security of NIST candidate Dilithium

Conclusion - and Comparison We (and [LieZhandry19]) show: a particular extract-and-reprogram technique in QROM

Implies security of Fiat-Shamir in the QROM: whatever security S satisfies, FS[S] satisfies it as well(we lose a factor q2, [LieZhandry19] a factor q9)

Implies: S QPoK (& ...) ⇒ Sig[S] secure

We (and [LieZhandry19]) show: new notion of computational unique responses ⇒ QPoK

We conjectured, [LieZhandry19] prove: S underlying Dilithium satisfies new notionImplies: security of Dilithium (in QROM)

Open Problems

Is the q2 loss optimal or can it be reduced to q ?

Find new and better technical tools for proving our new and stronger notion of computational unique responses.

Be able to relax the special soundness requirement.

security of the fiat-shamir transformation in the quantum … · 2019. 4. 16. · theorem. if s is...

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