Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion

Constructing Tweakable Block Ciphers in the Random Permutation Model

Yannick Seurin

ANSSI, France

September 30, 2015 — ASK 2015

Based on joint work with Benoît Cogliati and Rodolphe Lampe

Yannick Seurin Constructing TBCs in the RPM ASK 2015 1 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 2 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 3 / 36 t

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Block Ciphers (TBCs) k

x Ee y

• tweak t: brings variability to the block cipher • t assumed public or even adversarially controlled • each tweak should give an “independent” permutation • few “natively tweakable” BCs: • Hasty Pudding Cipher [Sch98] • Mercy [Cro00] • Threefish [FLS+10] • CAESAR proposals KIASU, Deoxys, Joltik, (i)SCREAM, Minalpher

Yannick Seurin Constructing TBCs in the RPM ASK 2015 4 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Block Ciphers (TBCs) k

x Ee y

t

• tweak t: brings variability to the block cipher • t assumed public or even adversarially controlled • each tweak should give an “independent” permutation • few “natively tweakable” BCs: • Hasty Pudding Cipher [Sch98] • Mercy [Cro00] • Threefish [FLS+10] • CAESAR proposals KIASU, Deoxys, Joltik, (i)SCREAM, Minalpher

Yannick Seurin Constructing TBCs in the RPM ASK 2015 4 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Block Ciphers (TBCs) k

x Ee y

t

• tweak t: brings variability to the block cipher • t assumed public or even adversarially controlled • each tweak should give an “independent” permutation • few “natively tweakable” BCs: • Hasty Pudding Cipher [Sch98] • Mercy [Cro00] • Threefish [FLS+10] • CAESAR proposals KIASU, Deoxys, Joltik, (i)SCREAM, Minalpher

Yannick Seurin Constructing TBCs in the RPM ASK 2015 4 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Block Ciphers (TBCs) k

x Ee y

t

• tweak t: brings variability to the block cipher • t assumed public or even adversarially controlled • each tweak should give an “independent” permutation • few “natively tweakable” BCs: • Hasty Pudding Cipher [Sch98] • Mercy [Cro00] • Threefish [FLS+10] • CAESAR proposals KIASU, Deoxys, Joltik, (i)SCREAM, Minalpher

Yannick Seurin Constructing TBCs in the RPM ASK 2015 4 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Block Ciphers (TBCs) k

x Ee y

t

• tweak t: brings variability to the block cipher • t assumed public or even adversarially controlled • each tweak should give an “independent” permutation • few “natively tweakable” BCs: • Hasty Pudding Cipher [Sch98] • Mercy [Cro00] • Threefish [FLS+10] • CAESAR proposals KIASU, Deoxys, Joltik, (i)SCREAM, Minalpher

Yannick Seurin Constructing TBCs in the RPM ASK 2015 4 / 36 hk0 (t) hk0 (t)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Generic Constructions of TBCs: LRW

• A generic TBC construction turns a conventional block cipher E into a TBC Ee • example: LRW construction by Liskov et al. [LRW02]

k

x E y

0 • h is XOR-universal, e.g. hk0 (t) = k ⊗ t (field mult.) • secure up to ∼ 2n/2 queries

• related construction XEX [Rog04] uses Ek (t) instead of hk0 (t) (used e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 5 / 36 hk0 (t) hk0 (t)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Generic Constructions of TBCs: LRW

• A generic TBC construction turns a conventional block cipher E into a TBC Ee • example: LRW construction by Liskov et al. [LRW02]

k

x E y

0 • h is XOR-universal, e.g. hk0 (t) = k ⊗ t (field mult.) • secure up to ∼ 2n/2 queries

• related construction XEX [Rog04] uses Ek (t) instead of hk0 (t) (used e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 5 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Generic Constructions of TBCs: LRW

• A generic TBC construction turns a conventional block cipher E into a TBC Ee • example: LRW construction by Liskov et al. [LRW02]

hk0 (t) k hk0 (t)

x E y

0 • h is XOR-universal, e.g. hk0 (t) = k ⊗ t (field mult.) • secure up to ∼ 2n/2 queries

• related construction XEX [Rog04] uses Ek (t) instead of hk0 (t) (used e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 5 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Generic Constructions of TBCs: LRW

• A generic TBC construction turns a conventional block cipher E into a TBC Ee • example: LRW construction by Liskov et al. [LRW02]

hk0 (t) k hk0 (t)

x E y

0 • h is XOR-universal, e.g. hk0 (t) = k ⊗ t (field mult.) • secure up to ∼ 2n/2 queries

• related construction XEX [Rog04] uses Ek (t) instead of hk0 (t) (used e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 5 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Generic Constructions of TBCs: LRW

• A generic TBC construction turns a conventional block cipher E into a TBC Ee • example: LRW construction by Liskov et al. [LRW02]

hk0 (t) k hk0 (t)

x E y

0 • h is XOR-universal, e.g. hk0 (t) = k ⊗ t (field mult.) • secure up to ∼ 2n/2 queries

• related construction XEX [Rog04] uses Ek (t) instead of hk0 (t) (used e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 5 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Other Generic Constructions

Constructions achieving beyond-birthday-bound security: • Minematsu [Min09] tweak length < n/2 • /Cascaded LRW [LST12, LS13] larger key length and block cipher calls • /Mennink [Men15] security proof needs ideal cipher model

Only/ LRW (or rather XEX) is used in practice (e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 6 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Other Generic Constructions

Constructions achieving beyond-birthday-bound security: • Minematsu [Min09] tweak length < n/2 • /Cascaded LRW [LST12, LS13] larger key length and block cipher calls • /Mennink [Men15] security proof needs ideal cipher model

Only/ LRW (or rather XEX) is used in practice (e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 6 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Other Generic Constructions

Constructions achieving beyond-birthday-bound security: • Minematsu [Min09] tweak length < n/2 • /Cascaded LRW [LST12, LS13] larger key length and block cipher calls • /Mennink [Men15] security proof needs ideal cipher model

Only/ LRW (or rather XEX) is used in practice (e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 6 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Other Generic Constructions

Constructions achieving beyond-birthday-bound security: • Minematsu [Min09] tweak length < n/2 • /Cascaded LRW [LST12, LS13] larger key length and block cipher calls • /Mennink [Men15] security proof needs ideal cipher model

Only/ LRW (or rather XEX) is used in practice (e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 6 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Other Generic Constructions

Constructions achieving beyond-birthday-bound security: • Minematsu [Min09] tweak length < n/2 • /Cascaded LRW [LST12, LS13] larger key length and block cipher calls • /Mennink [Men15] security proof needs ideal cipher model

Only/ LRW (or rather XEX) is used in practice (e.g. in the XTS disk encryption mode)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 6 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 7 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion TBCs: Dedicated Designs

Our Goal Provide provable security guidelines to design TBCs “from scratch” (rather than from an existing conventional block cipher).

• “from scratch” → from some lower level primitive • from a PRF: Feistel schemes [GHL+07, MI08] • this talk: SPN ciphers (more gen. key-alternating ciphers)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 8 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion TBCs: Dedicated Designs

Our Goal Provide provable security guidelines to design TBCs “from scratch” (rather than from an existing conventional block cipher).

• “from scratch” → from some lower level primitive • from a PRF: Feistel schemes [GHL+07, MI08] • this talk: SPN ciphers (more gen. key-alternating ciphers)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 8 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion TBCs: Dedicated Designs

Our Goal Provide provable security guidelines to design TBCs “from scratch” (rather than from an existing conventional block cipher).

• “from scratch” → from some lower level primitive • from a PRF: Feistel schemes [GHL+07, MI08] • this talk: SPN ciphers (more gen. key-alternating ciphers)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 8 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion TBCs: Dedicated Designs

Our Goal Provide provable security guidelines to design TBCs “from scratch” (rather than from an existing conventional block cipher).

• “from scratch” → from some lower level primitive • from a PRF: Feistel schemes [GHL+07, MI08] • this talk: SPN ciphers (more gen. key-alternating ciphers)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 8 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Key-Alternating Ciphers

k

f0 f1 fr

k0 k1 kr n x P1 P2 Pr y

An r-round key-alternating cipher:

n • the Pi ’s are public permutations on {0, 1}

• the fi ’s map k to n-bit “round keys” • examples: most SPNs (AES, SERPENT, PRESENT, LED. . . ) • a.k.a. (iterated) Even-Mansour construction

Yannick Seurin Constructing TBCs in the RPM ASK 2015 9 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Key-Alternating Ciphers

k

f0 f1 fr

k0 k1 kr n x P1 P2 Pr y

An r-round key-alternating cipher:

n • the Pi ’s are public permutations on {0, 1}

• the fi ’s map k to n-bit “round keys” • examples: most SPNs (AES, SERPENT, PRESENT, LED. . . ) • a.k.a. (iterated) Even-Mansour construction

Yannick Seurin Constructing TBCs in the RPM ASK 2015 9 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Key-Alternating Ciphers

k

f0 f1 fr

k0 k1 kr n x P1 P2 Pr y

An r-round key-alternating cipher:

n • the Pi ’s are public permutations on {0, 1}

• the fi ’s map k to n-bit “round keys” • examples: most SPNs (AES, SERPENT, PRESENT, LED. . . ) • a.k.a. (iterated) Even-Mansour construction

Yannick Seurin Constructing TBCs in the RPM ASK 2015 9 / 36 ( , t)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Even-Mansour Constructions

k

f0 f1 fr

x P1 P2 Pr y

• let the round keys depend on the key and the tweak t • ⇒ “tweakable” Even-Mansour (TEM) construction(s)

• fi ’s = “tweak and key schedule” (TKS) • high-level abstraction of the TWEAKEY constructions [JNP14] • analysis in the Random Permutation Model

Yannick Seurin Constructing TBCs in the RPM ASK 2015 10 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Even-Mansour Constructions

(k, t)

f0 f1 fr

x P1 P2 Pr y

• let the round keys depend on the key and the tweak t • ⇒ “tweakable” Even-Mansour (TEM) construction(s)

• fi ’s = “tweak and key schedule” (TKS) • high-level abstraction of the TWEAKEY constructions [JNP14] • analysis in the Random Permutation Model

Yannick Seurin Constructing TBCs in the RPM ASK 2015 10 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Even-Mansour Constructions

(k, t)

f0 f1 fr

x P1 P2 Pr y

• let the round keys depend on the key and the tweak t • ⇒ “tweakable” Even-Mansour (TEM) construction(s)

• fi ’s = “tweak and key schedule” (TKS) • high-level abstraction of the TWEAKEY constructions [JNP14] • analysis in the Random Permutation Model

Yannick Seurin Constructing TBCs in the RPM ASK 2015 10 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Even-Mansour Constructions

(k, t)

f0 f1 fr

x P1 P2 Pr y

• let the round keys depend on the key and the tweak t • ⇒ “tweakable” Even-Mansour (TEM) construction(s)

• fi ’s = “tweak and key schedule” (TKS) • high-level abstraction of the TWEAKEY constructions [JNP14] • analysis in the Random Permutation Model

Yannick Seurin Constructing TBCs in the RPM ASK 2015 10 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tweakable Even-Mansour Constructions

(k, t)

f0 f1 fr

x P1 P2 Pr y

• let the round keys depend on the key and the tweak t • ⇒ “tweakable” Even-Mansour (TEM) construction(s)

• fi ’s = “tweak and key schedule” (TKS) • high-level abstraction of the TWEAKEY constructions [JNP14] • analysis in the Random Permutation Model

Yannick Seurin Constructing TBCs in the RPM ASK 2015 10 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The Random Permutation Model (RPM)

(k, t)

f0 f1 fr

y x P1 P2 Pr P1 ··· Pr

qc qp qp

• the Pi ’s are modeled as public random permutation oracles (adversary can only make black-box queries) • adversary cannot exploit any weakness of the Pi ’s ⇒ generic attacks • complexity measure of the adversary: • qc = # construction queries = pt/ct pairs (data D) • qp = # queries to each internal permutation oracle (time T ) • but otherwise computationally unbounded • ⇒ information-theoretic proof of security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 11 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The Random Permutation Model (RPM)

(k, t)

f0 f1 fr

y x P1 P2 Pr P1 ··· Pr

qc qp qp

• the Pi ’s are modeled as public random permutation oracles (adversary can only make black-box queries) • adversary cannot exploit any weakness of the Pi ’s ⇒ generic attacks • complexity measure of the adversary: • qc = # construction queries = pt/ct pairs (data D) • qp = # queries to each internal permutation oracle (time T ) • but otherwise computationally unbounded • ⇒ information-theoretic proof of security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 11 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The Random Permutation Model (RPM)

(k, t)

f0 f1 fr

y x P1 P2 Pr P1 ··· Pr

qc qp qp

• the Pi ’s are modeled as public random permutation oracles (adversary can only make black-box queries) • adversary cannot exploit any weakness of the Pi ’s ⇒ generic attacks • complexity measure of the adversary: • qc = # construction queries = pt/ct pairs (data D) • qp = # queries to each internal permutation oracle (time T ) • but otherwise computationally unbounded • ⇒ information-theoretic proof of security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 11 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The Random Permutation Model (RPM)

(k, t)

f0 f1 fr

y x P1 P2 Pr P1 ··· Pr

qc qp qp

• the Pi ’s are modeled as public random permutation oracles (adversary can only make black-box queries) • adversary cannot exploit any weakness of the Pi ’s ⇒ generic attacks • complexity measure of the adversary: • qc = # construction queries = pt/ct pairs (data D) • qp = # queries to each internal permutation oracle (time T ) • but otherwise computationally unbounded • ⇒ information-theoretic proof of security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 11 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Formalization of the Security Experiment

Real world Ideal world

(k, t)

f0 f1 fr

y x P1 P2 Pr P1,..., Pr Pe0 P1,..., Pr

qc qp qc qp

0/1 0/1

• real world: TEM construction with random master key k

• ideal world: random tweakable permutation Pe0 independent from P1,..., Pr

• RPM: D has oracle access to P1,..., Pr in both worlds

Yannick Seurin Constructing TBCs in the RPM ASK 2015 12 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Formalization of the Security Experiment

Real world Ideal world

(k, t)

f0 f1 fr

y x P1 P2 Pr P1,..., Pr Pe0 P1,..., Pr

qc qp qc qp

0/1 0/1

• real world: TEM construction with random master key k

• ideal world: random tweakable permutation Pe0 independent from P1,..., Pr

• RPM: D has oracle access to P1,..., Pr in both worlds

Yannick Seurin Constructing TBCs in the RPM ASK 2015 12 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 13 / 36 P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

k k

x P1 y

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

k ⊕ t k ⊕ t

x P1 y

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 x u v y

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k

x1 ⊕ x2 = t1 ⊕ t2 x y

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1 (t1, x1) y1 = v ⊕ k ⊕ t1

u v

k ⊕ t1

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k

x y

y2 = v ⊕ k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2)

k ⊕ t1 k ⊕ t2

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k

x y

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k

x y

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 k k

x y

P1

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion First Try: One Round, Linear TKS

P1 (t1, x1) y1 = v ⊕ k ⊕ t1

x1 ⊕ x2 = t1 ⊕ t2 u v

(t2, x2) y2 = v ⊕ k ⊕ t2

k ⊕ t1 k ⊕ t2

Check that y1 ⊕ y2 = t1 ⊕ t2 (∗)

• 2 queries to the encryption oracle, 0 queries to P1 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 14 / 36 P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

k ⊕ t k ⊕ t k ⊕ t

x P1 P2 y

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3) x y 0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

(t2, x2) (t3, y3) x y 0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

u1 v1 u2 v2

k ⊕ t1

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

(t3, y3) x y 0 0 x3 u1 v1 (t4, y4)

x4

k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2

0 0 u2 v2

y2

k ⊕ t1 k ⊕ t2

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y (t4, y4)

x4

k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2

y2

k ⊕ t1 k ⊕ t2 k ⊕ t3

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

x4

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 k ⊕ t k ⊕ t k ⊕ t

x y

P1 P2

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Second Try: Two Rounds, Linear TKS

P1 P2 (t1, x1) y1

(t2, x2) u1 v1 u2 v2 (t3, y3)

0 0 0 0 x3 u1 v1 u2 v2 (t4, y4)

x4 y2

k ⊕ t1 k ⊕ t2 k ⊕ t3 k ⊕ t4

t1 ⊕ t2 ⊕ t3 ⊕ t4 = 0

Check that x3 ⊕ x4 = t3 ⊕ t4 (∗)

• 4 queries to the enc/dec oracle, 0 queries to P1, P2 • (∗) holds with proba. 1 for the TEM construction • (∗) holds with proba. 2−n for a random tweakable permutation • works for any linear TKS

Yannick Seurin Constructing TBCs in the RPM ASK 2015 15 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Security for Three Rounds

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP:

6q q 4q2 Adv(q , q ) ≤ c p + c . c p 2n 2n

Proof sketch:

• adversary can create collisions at input of P1 or output of P3 2 n • but proba. to create a collision at P2 is . qc /2 • no collision at P2 n ⇒ ∼ single-key security of 1-round EM . qc qp/2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 16 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Security for Three Rounds

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP:

6q q 4q2 Adv(q , q ) ≤ c p + c . c p 2n 2n

Proof sketch:

• adversary can create collisions at input of P1 or output of P3 2 n • but proba. to create a collision at P2 is . qc /2 • no collision at P2 n ⇒ ∼ single-key security of 1-round EM . qc qp/2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 16 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Security for Three Rounds

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP:

6q q 4q2 Adv(q , q ) ≤ c p + c . c p 2n 2n

Proof sketch:

• adversary can create collisions at input of P1 or output of P3 2 n • but proba. to create a collision at P2 is . qc /2 • no collision at P2 n ⇒ ∼ single-key security of 1-round EM . qc qp/2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 16 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Security for Three Rounds

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP:

6q q 4q2 Adv(q , q ) ≤ c p + c . c p 2n 2n

Proof sketch:

• adversary can create collisions at input of P1 or output of P3 2 n • but proba. to create a collision at P2 is . qc /2 • no collision at P2 n ⇒ ∼ single-key security of 1-round EM . qc qp/2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 16 / 36 Question Construction with less permutations?

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tightness of the Bound

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

• can be written Ee(k, t, x) = E(k⊕t, x) where E is the conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: ∗ n/2 1. query ci = Eek∗ (ti , 0) = E(k ⊕ ti , 0) for 2 tweaks ti 0 n/2 2. compute cj = Eekj (0, 0) = E(kj , 0) for 2 keys kj 0 3. look for a collision ci = cj ∗ 4. w.h.p., the real key is k = ti ⊕ kj • ⇒ increasing the number of rounds does not improve security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 17 / 36 Question Construction with less permutations?

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tightness of the Bound

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

• can be written Ee(k, t, x) = E(k⊕t, x) where E is the conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: ∗ n/2 1. query ci = Eek∗ (ti , 0) = E(k ⊕ ti , 0) for 2 tweaks ti 0 n/2 2. compute cj = Eekj (0, 0) = E(kj , 0) for 2 keys kj 0 3. look for a collision ci = cj ∗ 4. w.h.p., the real key is k = ti ⊕ kj • ⇒ increasing the number of rounds does not improve security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 17 / 36 Question Construction with less permutations?

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tightness of the Bound

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

• can be written Ee(k, t, x) = E(k⊕t, x) where E is the conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: ∗ n/2 1. query ci = Eek∗ (ti , 0) = E(k ⊕ ti , 0) for 2 tweaks ti 0 n/2 2. compute cj = Eekj (0, 0) = E(kj , 0) for 2 keys kj 0 3. look for a collision ci = cj ∗ 4. w.h.p., the real key is k = ti ⊕ kj • ⇒ increasing the number of rounds does not improve security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 17 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Tightness of the Bound

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

• can be written Ee(k, t, x) = E(k⊕t, x) where E is the conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: ∗ n/2 1. query ci = Eek∗ (ti , 0) = E(k ⊕ ti , 0) for 2 tweaks ti 0 n/2 2. compute cj = Eekj (0, 0) = E(kj , 0) for 2 keys kj 0 3. look for a collision ci = cj ∗ 4. w.h.p., the real key is k = ti ⊕ kj • ⇒ increasing the number of rounds does not improve security

Question Construction with less permutations?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 17 / 36 k0 k0 Non-Linear Tweakable Even-MansourP (NL-TEM) construction

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

k ⊗ t k0 k ⊗ t

x E y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 Non-Linear Tweakable Even-Mansour (NL-TEM) construction

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

k0 k0 P

k ⊗ t k0 k ⊗ t

x E y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 k0 k0 Non-Linear Tweakable Even-MansourP (NL-TEM) construction

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

(k ⊗ t) ⊕ k0 (k ⊗ t) ⊕ k0

x P y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 k0 k0 Non-Linear Tweakable Even-MansourP (NL-TEM) construction

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

(k ⊗ t) ⊕ k0 (k ⊗ t) ⊕ k0

x P y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 k0 k0 Non-Linear Tweakable Even-MansourP (NL-TEM) construction

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

k ⊗ t k ⊗ t

x P y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 k0 k0 P

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Back to LRW • instantiate E with the 1-round Even-Mansour construction

Non-Linear Tweakable Even-Mansour (NL-TEM) construction

k ⊗ t k ⊗ t

x P y

• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:

q2 2q q Adv(q , q ) ≤ c + c p . c p 2n 2n • t 6= 0 ⇒ k0 is superfluous (k ⊗ t unif. random for any t 6= 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 18 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Birthday-Bound Security: Wrap-up

Two constructions provably secure up to the birthday bound: 1. linear TKS

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

2. nonlinear TKS

k ⊗ t k ⊗ t

x P y

Question Constructions secure beyond the birthday-bound?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 19 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Birthday-Bound Security: Wrap-up

Two constructions provably secure up to the birthday bound: 1. linear TKS

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

2. nonlinear TKS

k ⊗ t k ⊗ t

x P y

Question Constructions secure beyond the birthday-bound?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 19 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Birthday-Bound Security: Wrap-up

Two constructions provably secure up to the birthday bound: 1. linear TKS

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

2. nonlinear TKS

k ⊗ t k ⊗ t

x P y

Question Constructions secure beyond the birthday-bound?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 19 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Birthday-Bound Security: Wrap-up

Two constructions provably secure up to the birthday bound: 1. linear TKS

k ⊕ t k ⊕ t k ⊕ t k ⊕ t

x P1 P2 P3 y

2. nonlinear TKS

k ⊗ t k ⊗ t

x P y

Question Constructions secure beyond the birthday-bound?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 19 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 20 / 36 0 0 k2 ⊗ t kr ⊗ t

Ek2 Ekr y

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the LRW Construction

0 k1 ⊗ t

x Ek1

0 0 • k1,..., kr and k1,..., kr independent keys ⇒ total key-length = r(κ + n) • 2 rounds: provably secure up to ∼ 22n/3 queries [LST12] rn • r rounds, r even: provably secure up to ∼ 2 r+2 queries [LS13] • NB: only assuming E is a PRP (standard security notion, no ideal model)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 21 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the LRW Construction

0 0 0 k1 ⊗ t k2 ⊗ t kr ⊗ t

x Ek1 Ek2 Ekr y

0 0 • k1,..., kr and k1,..., kr independent keys ⇒ total key-length = r(κ + n) • 2 rounds: provably secure up to ∼ 22n/3 queries [LST12] rn • r rounds, r even: provably secure up to ∼ 2 r+2 queries [LS13] • NB: only assuming E is a PRP (standard security notion, no ideal model)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 21 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the LRW Construction

0 0 0 k1 ⊗ t k2 ⊗ t kr ⊗ t

x Ek1 Ek2 Ekr y

0 0 • k1,..., kr and k1,..., kr independent keys ⇒ total key-length = r(κ + n) • 2 rounds: provably secure up to ∼ 22n/3 queries [LST12] rn • r rounds, r even: provably secure up to ∼ 2 r+2 queries [LS13] • NB: only assuming E is a PRP (standard security notion, no ideal model)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 21 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the LRW Construction

0 0 0 k1 ⊗ t k2 ⊗ t kr ⊗ t

x Ek1 Ek2 Ekr y

0 0 • k1,..., kr and k1,..., kr independent keys ⇒ total key-length = r(κ + n) • 2 rounds: provably secure up to ∼ 22n/3 queries [LST12] rn • r rounds, r even: provably secure up to ∼ 2 r+2 queries [LS13] • NB: only assuming E is a PRP (standard security notion, no ideal model)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 21 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the LRW Construction

0 0 0 k1 ⊗ t k2 ⊗ t kr ⊗ t

x Ek1 Ek2 Ekr y

0 0 • k1,..., kr and k1,..., kr independent keys ⇒ total key-length = r(κ + n) • 2 rounds: provably secure up to ∼ 22n/3 queries [LST12] rn • r rounds, r even: provably secure up to ∼ 2 r+2 queries [LS13] • NB: only assuming E is a PRP (standard security notion, no ideal model)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 21 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the NL-TEM Construction

• k1, k2 independent n-bit keys

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Theorem ([CLS15]) The 2-round NL-TEM construction is secure up to ∼ 22n/3 queries in the RPM: √ 34q3/2 30 q q Adv(q , q ) ≤ c + c p . c p 2n 2n

Yannick Seurin Constructing TBCs in the RPM ASK 2015 22 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Cascading the NL-TEM Construction

• k1, k2 independent n-bit keys

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Theorem ([CLS15]) The 2-round NL-TEM construction is secure up to ∼ 22n/3 queries in the RPM: √ 34q3/2 30 q q Adv(q , q ) ≤ c + c p . c p 2n 2n

Yannick Seurin Constructing TBCs in the RPM ASK 2015 22 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Proof Technique: H-coefficients

Real world Ideal world

k1 ⊗ t k2 ⊗ t kr ⊗ t

y x P1 P2 Pr P1,..., Pr Pe0 P1,..., Pr

qc qp qc qp

1. consider the transcript of all queries of D to the construction and to the inner permutations 2. define bad transcripts and show that their probability is small (in the ideal world) 3. show that good transcripts are almost as probable in the real and the ideal world

Yannick Seurin Constructing TBCs in the RPM ASK 2015 23 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Proof Technique: H-coefficients

Real world Ideal world

k1 ⊗ t k2 ⊗ t kr ⊗ t

y x P1 P2 Pr P1,..., Pr Pe0 P1,..., Pr

qc qp qc qp

1. consider the transcript of all queries of D to the construction and to the inner permutations 2. define bad transcripts and show that their probability is small (in the ideal world) 3. show that good transcripts are almost as probable in the real and the ideal world

Yannick Seurin Constructing TBCs in the RPM ASK 2015 23 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Proof Technique: H-coefficients

Real world Ideal world

k1 ⊗ t k2 ⊗ t kr ⊗ t

y x P1 P2 Pr P1,..., Pr Pe0 P1,..., Pr

qc qp qc qp

1. consider the transcript of all queries of D to the construction and to the inner permutations 2. define bad transcripts and show that their probability is small (in the ideal world) 3. show that good transcripts are almost as probable in the real and the ideal world

Yannick Seurin Constructing TBCs in the RPM ASK 2015 23 / 36 q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 q q2 (t, x) u2 v2 c p proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

u1 v1

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 2 (t, x) qcqp proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

u1 v1 u2 v2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 2 qcqp proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

(t, x) u1 v1 u2 v2

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 (t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 (t, x) 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 2 proba ≤ qc (t0, x 0) 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

(t, x)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 2 qc proba ≤ 22n

Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

(t, x) (t0, x 0)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Bad Transcripts

• one needs to avoid “two-fold” collisions:

k1 ⊗ t k2 ⊗ t

x P1 P2 y

q q2 (t, x) u1 v1 u2 v2 c p proba ≤ 22n

(t, x) 2 proba ≤ qc (t0, x 0) 22n

Yannick Seurin Constructing TBCs in the RPM ASK 2015 24 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The Ten “Bad Collision” Cases

P1 P2

(t, x) u1 v2 (t, y)

(t, x) u1 v1 u2 v1 u2 v2 (t, y)

(t, x) (t, y) (t0, x 0) (t00, y 00)

(t, x) (t, y) (t0, x 0) (t0, y 0)

(t, x) u1 (t, y) (t, x) v2 (t, y) (t0, y 0) (t0, x 0)

(t, x) u1 v1 u2 v2 (t, y)

0 0 0 0 0 0 0 0 (t , x ) u1 v1 u2 v2 (t , y )

Yannick Seurin Constructing TBCs in the RPM ASK 2015 25 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Distribution of Good Transcripts

P1 P2 • assuming there are no bad collisions, show that

QU1 Uf2 Vf2 U1 V1 the answers of the TEM construction are close to answers of a random QV2 Uf1 Vf1 U2 V2 tweakable permutation • for each query, there is a “fresh” value of P or U0 V 0 0 0 1 QX 1 1 U2 V2 P2 which randomizes the output

00 00 U00 V 00 QY U1 V1 2 2

Q0

Yannick Seurin Constructing TBCs in the RPM ASK 2015 26 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Distribution of Good Transcripts

P1 P2 • assuming there are no bad collisions, show that

QU1 Uf2 Vf2 U1 V1 the answers of the TEM construction are close to answers of a random QV2 Uf1 Vf1 U2 V2 tweakable permutation • for each query, there is a “fresh” value of P or U0 V 0 0 0 1 QX 1 1 U2 V2 P2 which randomizes the output

00 00 U00 V 00 QY U1 V1 2 2

Q0

Yannick Seurin Constructing TBCs in the RPM ASK 2015 26 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Longer Cascades of the NL-TEM Construction

k1 ⊗ t k2 ⊗ t kr ⊗ t

x P1 P2 Pr y

• r rounds, r even, with independent keys k1,..., kr secure up to

rn (r/2)n ∼ 2 r+2 = 2 (r/2)+1 queries

• proof: 1. non-adaptive security for r/2 rounds (coupling technique) 2. adaptive security for r rounds (“two weak make one strong” composition theorem) rn • conjecture: secure up to ∼ 2 r+1 queries

Yannick Seurin Constructing TBCs in the RPM ASK 2015 27 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Longer Cascades of the NL-TEM Construction

k1 ⊗ t k2 ⊗ t kr ⊗ t

x P1 P2 Pr y

• r rounds, r even, with independent keys k1,..., kr secure up to

rn (r/2)n ∼ 2 r+2 = 2 (r/2)+1 queries

• proof: 1. non-adaptive security for r/2 rounds (coupling technique) 2. adaptive security for r rounds (“two weak make one strong” composition theorem) rn • conjecture: secure up to ∼ 2 r+1 queries

Yannick Seurin Constructing TBCs in the RPM ASK 2015 27 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Longer Cascades of the NL-TEM Construction

k1 ⊗ t k2 ⊗ t kr ⊗ t

x P1 P2 Pr y

• r rounds, r even, with independent keys k1,..., kr secure up to

rn (r/2)n ∼ 2 r+2 = 2 (r/2)+1 queries

• proof: 1. non-adaptive security for r/2 rounds (coupling technique) 2. adaptive security for r rounds (“two weak make one strong” composition theorem) rn • conjecture: secure up to ∼ 2 r+1 queries

Yannick Seurin Constructing TBCs in the RPM ASK 2015 27 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Longer Cascades of the NL-TEM Construction

k1 ⊗ t k2 ⊗ t kr ⊗ t

x P1 P2 Pr y

• r rounds, r even, with independent keys k1,..., kr secure up to

rn (r/2)n ∼ 2 r+2 = 2 (r/2)+1 queries

• proof: 1. non-adaptive security for r/2 rounds (coupling technique) 2. adaptive security for r rounds (“two weak make one strong” composition theorem) rn • conjecture: secure up to ∼ 2 r+1 queries

Yannick Seurin Constructing TBCs in the RPM ASK 2015 27 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion BBB Security with a Linear TKS

• k1, k2 independent n-bit keys

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

Theorem (B. Cogliati, Y.S., AC 2015) The 4-round TEM with “alternating” linear TKS is secure up to ∼ 22n/3 queries in the RPM.

Proof idea:

• exclude bad events related to P1 and P4

• “reduction” to 2-round NL-TEM security based on (P2, P3)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 28 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion BBB Security with a Linear TKS

• k1, k2 independent n-bit keys

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

Theorem (B. Cogliati, Y.S., AC 2015) The 4-round TEM with “alternating” linear TKS is secure up to ∼ 22n/3 queries in the RPM.

Proof idea:

• exclude bad events related to P1 and P4

• “reduction” to 2-round NL-TEM security based on (P2, P3)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 28 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion BBB Security with a Linear TKS

• k1, k2 independent n-bit keys

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

Theorem (B. Cogliati, Y.S., AC 2015) The 4-round TEM with “alternating” linear TKS is secure up to ∼ 22n/3 queries in the RPM.

Proof idea:

• exclude bad events related to P1 and P4

• “reduction” to 2-round NL-TEM security based on (P2, P3)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 28 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Outline

Background: Tweakable Block Ciphers

Tweakable Even-Mansour Constructions

Birthday-Bound Secure Constructions

Beyond-Birthday-Bound Secure Constructions

Conclusion and Perspectives

Yannick Seurin Constructing TBCs in the RPM ASK 2015 29 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Conclusion

22n/3-secure constructions: 1. linear TKS

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

2. nonlinear TKS

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Open problems: rn 1. prove tight 2 r+1 -security for r-round NL-TEM, r ≥ 3 2. propose a construction with linear TKS and security > 22n/3 3. reduce key length for BBB-security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 30 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Conclusion

22n/3-secure constructions: 1. linear TKS

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

2. nonlinear TKS

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Open problems: rn 1. prove tight 2 r+1 -security for r-round NL-TEM, r ≥ 3 2. propose a construction with linear TKS and security > 22n/3 3. reduce key length for BBB-security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 30 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Conclusion

22n/3-secure constructions: 1. linear TKS

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

2. nonlinear TKS

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Open problems: rn 1. prove tight 2 r+1 -security for r-round NL-TEM, r ≥ 3 2. propose a construction with linear TKS and security > 22n/3 3. reduce key length for BBB-security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 30 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Conclusion

22n/3-secure constructions: 1. linear TKS

k1 ⊕ t k2 ⊕ t k1 ⊕ t k2 ⊕ t k1 ⊕ t

x P1 P2 P3 P4 y

2. nonlinear TKS

k1 ⊗ t k2 ⊗ t

x P1 P2 y

Open problems: rn 1. prove tight 2 r+1 -security for r-round NL-TEM, r ≥ 3 2. propose a construction with linear TKS and security > 22n/3 3. reduce key length for BBB-security

Yannick Seurin Constructing TBCs in the RPM ASK 2015 30 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Link with the TWEAKEY Framework

• proposed by Jean, Nikolić, and Peyrin [JNP14] • Superposition TWEAKEY (STK) constructions:

t g g g

k f f f

x P1 P2 Pr y

• sufficient conditions on f and g to have provable beyond-birthday-bound security in the RPM? • NB: f = g linear does not work since Ee(k, t, x) = E(k ⊕ t, x)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 31 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Link with the TWEAKEY Framework

• proposed by Jean, Nikolić, and Peyrin [JNP14] • Superposition TWEAKEY (STK) constructions:

t g g g

k f f f

x P1 P2 Pr y

• sufficient conditions on f and g to have provable beyond-birthday-bound security in the RPM? • NB: f = g linear does not work since Ee(k, t, x) = E(k ⊕ t, x)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 31 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Link with the TWEAKEY Framework

• proposed by Jean, Nikolić, and Peyrin [JNP14] • Superposition TWEAKEY (STK) constructions:

t g g g

k f f f

x P1 P2 Pr y

• sufficient conditions on f and g to have provable beyond-birthday-bound security in the RPM? • NB: f = g linear does not work since Ee(k, t, x) = E(k ⊕ t, x)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 31 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion Link with the TWEAKEY Framework

• proposed by Jean, Nikolić, and Peyrin [JNP14] • Superposition TWEAKEY (STK) constructions:

t g g g

k f f f

x P1 P2 Pr y

• sufficient conditions on f and g to have provable beyond-birthday-bound security in the RPM? • NB: f = g linear does not work since Ee(k, t, x) = E(k ⊕ t, x)

Yannick Seurin Constructing TBCs in the RPM ASK 2015 31 / 36 Tweakable BC Tweakable EM Birthday Security BBB Security Conclusion The end. . .

Thanks for your attention!

Comments or questions?

Yannick Seurin Constructing TBCs in the RPM ASK 2015 32 / 36 References ReferencesI

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