Semantic Security and Indistinguishability in the

Quantum World

Tommaso Gagliardoni1, Andreas Hülsing2, Christian Schaffner3

1 IBM Research, Swiss; TU Darmstadt, Germany2 TU Eindhoven, The Netherlands

3University of Amsterdam, CWI, QuSoft, The Netherlands

Crypto Working Group, Utrecht, NL24/03/2017

Introduction

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Symmetric encryption E = (Kg, Enc, Dec)

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EncPlaintext𝒎

𝒓Randomness

Secret key 𝐤

Ciphertext

DecCiphertext Plaintext

𝒎

Adversaries I: Classical Security

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E

Adversary = probabilistic polynomial time (PPT) algorithm

Adversaries II: Post-Quantum Security

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E

Adversary = bounded-error quantum polynomial time (BQP) algorithm

Adversaries III: Quantum Security

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E

Adversary = bounded-error quantum polynomial time (BQP) algorithm

Why should we care?

1. Use in protocols

2. Quantum cloud

3. Quantum obfuscation

4. Side-channel attacks that trigger some measurable quantum behaviour

5. Oh, and because we can!

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Semantic security (SEM)

• Simulation-based security notion

• Captures intuition:

It should not be possible to learn anything about the plaintext given the ciphertext which you could not also have learned without the ciphertext.

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Semantic security (SEM): Challenge phase

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(𝑺𝒏, 𝒉, 𝒇)

𝒎 ⟵ 𝑺𝒏,𝒄 = 𝑬𝒏𝒄𝒌 𝒎 ,(𝒄, 𝒉(𝒎))

(𝒇(𝒎))

A C

A cannot do significantly better in the above game than a simulator S that does not receive 𝑐.

Indistinguishability (IND)(of ciphertexts)• Pure game-based notion (no simulator)

• Easier to work with than SEM

• Intuition:

You cannot distinguish the encryptions of two messages of your choice

• Shown to be equivalent to SEM!

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Indistinguishability (IND): Challenge phase

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(𝒎𝟏,𝒎𝟐)𝒃 ⟵𝑹 {𝟎, 𝟏},𝒄 = 𝑬𝒏𝒄𝒌 𝒎𝒃 ,𝒄

𝒃

A C

A cannot output correct b with significantly bigger probability than guessing.

Chosen plaintext attacks (CPA)

• Adversary might learn encryptions of known messages

• To model worst case: Let adversary chose messages

• Can be combined with both security notions – IND & SEM

• Normally: Learning phases before & after challenge phase

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CPA Learning phase

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𝒎

𝒄 = 𝑬𝒏𝒄𝒌 𝒎𝒄

A C

A can ask 𝑞 ∈ 𝑝𝑜𝑙𝑦(𝑛) queries in all learning phases.

IND-CPA

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(𝒎𝟏,𝒎𝟐)𝒃 ⟵𝑹 {𝟎, 𝟏},𝒄 = 𝑬𝒏𝒄𝒌 𝒎𝒃 ,

𝒄

𝒃

AC

A cannot output correct b with significantly bigger probability than guessing.

𝒎𝒄 = 𝑬𝒏𝒄𝒌 𝒎 ,𝒄

𝒎𝒄 = 𝑬𝒏𝒄𝒌 𝒎 ,𝒄

Learning I

Learning II

Challenge

Finish

Quantum security notions

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Previous work

[BZ13] Boneh, Zhandry: "Secure Signatures and Chosen Ciphertext Security in a Quantum Computing World", CRYPTO'13

Model encryption as unitary operator defined by:

𝑥,𝑦 𝑥, 𝑦 →

𝑥,𝑦 𝑥, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥)

(where 𝐸𝑛𝑐𝑘(∙) is a classical encryption function)

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Indistinguishability under quantum chosen message attacks (IND-qCPA)

• Give adversary quantum access in learning phase

• Classical challenge phase

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𝑥, 𝑦

𝑥, 𝑦 → 𝑥, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥)

IND-qCPA

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(𝒎𝟏,𝒎𝟐)𝒃 ⟵𝑹 {𝟎, 𝟏},𝒄 = 𝑬𝒏𝒄𝒌 𝒎𝒃 ,

𝒄

𝒃

AC

A cannot output correct b with significantly bigger probability than guessing.

𝑥, 𝑦

𝑥, 𝑦 → 𝑥, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥)

Indistinguishability under quantum chosen message attacks (IND-qCPA)

• Give adversary quantum access in learning phase

• Classical challenge phase

• Can be proven strictly stronger than IND-CPA

• Why would you do this?

• If we assume adversary has quantum access, why not also when it tries to learn something new?

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Fully-quantum indistinguishability under quantum chosen message attacks (fqIND-qCPA)

• Give adversary quantum access in learning phase

• Quantum challenge phase

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𝑥, 𝑦

𝑥, 𝑦 → 𝑥, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥)

fqIND-qCPA

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𝑏 ⟵𝑅 {0,1}, 𝑥1, 𝑥2, 𝑦 → 𝑥1, 𝑥2, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥𝑏)

𝒃

A C

A cannot output correct b with significantly bigger probability than guessing.

𝑥, 𝑦

𝑥, 𝑦 → 𝑥, 𝑦 ⨁𝐸𝑛𝑐𝑘(𝑥)

𝑥1, 𝑥2, 𝑦

fqIND is unachievable [BZ13]

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fqIND is unachievable [BZ13]

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fqIND is unachievable [BZ13]

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[BZ13] & our contribution

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[BZ13] & our contribution

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How to define qIND-qCPA?

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How to define qIND-qCPA?

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How to define qIND-qCPA?

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How to define qIND-qCPA?

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Model: (O) vs (C)

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(O) (C)

Model: (Q) vs (c)

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(Q) (c)

Model: Type (1) vs type (2)

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Type (1) Type (2)

Quantum indistinguishability (qIND)

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Quantum indistinguishability (qIND)

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Separation example

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Separation example

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Separation example

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Impossibility result

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Impossibility result

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The attack

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The attack

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The attack

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The attack

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The attack

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The attack

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The attack

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The solution

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The solution

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The solution

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The solution

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The solution

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Secure Construction

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Conclusion

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https://eprint.iacr.org/2015/355

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Thank you!

Questions?