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
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
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
• h is XOR-universal, e.g. hk 0 (t) = k 0 ⊗ t (field mult.) • secure up to ∼ 2n/2 queries • related construction XEX [Rog04] uses Ek (t) instead of hk 0 (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]
k x
E
y
• h is XOR-universal, e.g. hk 0 (t) = k 0 ⊗ t (field mult.) • secure up to ∼ 2n/2 queries • related construction XEX [Rog04] uses Ek (t) instead of hk 0 (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]
hk 0 (t) x
k
hk 0 (t)
E
y
• h is XOR-universal, e.g. hk 0 (t) = k 0 ⊗ t (field mult.) • secure up to ∼ 2n/2 queries • related construction XEX [Rog04] uses Ek (t) instead of hk 0 (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]
hk 0 (t) x
k
hk 0 (t)
E
y
• h is XOR-universal, e.g. hk 0 (t) = k 0 ⊗ t (field mult.) • secure up to ∼ 2n/2 queries • related construction XEX [Rog04] uses Ek (t) instead of hk 0 (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]
hk 0 (t) x
k
hk 0 (t)
E
y
• h is XOR-universal, e.g. hk 0 (t) = k 0 ⊗ t (field mult.) • secure up to ∼ 2n/2 queries • related construction XEX [Rog04] uses Ek (t) instead of hk 0 (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
x
f0
f1
fr
k0
k1
kr
n
P1
P2
Pr
y
An r -round key-alternating cipher: • the Pi ’s are public permutations on {0, 1}n • 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
x
f0
f1
fr
k0
k1
kr
n
P1
P2
Pr
y
An r -round key-alternating cipher: • the Pi ’s are public permutations on {0, 1}n • 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
x
f0
f1
fr
k0
k1
kr
n
P1
P2
Pr
y
An r -round key-alternating cipher: • the Pi ’s are public permutations on {0, 1}n • 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
Tweakable Even-Mansour Constructions k f0 x
f1 P1
fr 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 x
f1 P1
fr 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 x
f1 P1
fr 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 x
f1 P1
fr 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 x
f1 P1
fr 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 x
f1 P1
fr P2
P1
y
Pr
qc
qp
···
Pr
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 x
f1 P1
fr P2
P1
y
Pr
qc
qp
···
Pr
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 x
f1 P1
fr P2
P1
y
Pr
qc
qp
···
Pr
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 x
f1 P1
fr P2
P1
y
Pr
qc
qp
···
Pr
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 x
f1 P1
fr P2
y
Pr
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
qp
qc
qp
qc
0/1
0/1
• real world: TEM construction with random master key k e0 independent • ideal world: random tweakable permutation P
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 x
f1 P1
fr P2
y
Pr
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
qp
qc
qp
qc
0/1
0/1
• real world: TEM construction with random master key k e0 independent • ideal world: random tweakable permutation P
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS k
x
k
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS k ⊕t
x
k ⊕t
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
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS P1 y1 = v ⊕ k ⊕ t1
(t1 , x1 ) 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS P1 y1 = v ⊕ k ⊕ t1
(t1 , x1 ) 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS P1 y1 = v ⊕ k ⊕ t1
(t1 , x1 ) x1 ⊕ x2 = t1 ⊕ t2
u
v y2 = v ⊕ k ⊕ t2
(t2 , x2 )
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS P1 y1 = v ⊕ k ⊕ t1
(t1 , x1 ) x1 ⊕ x2 = t1 ⊕ t2
u
v y2 = v ⊕ k ⊕ t2
(t2 , x2 )
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
First Try: One Round, Linear TKS P1 y1 = v ⊕ k ⊕ t1
(t1 , x1 ) x1 ⊕ x2 = t1 ⊕ t2
u
v y2 = v ⊕ k ⊕ t2
(t2 , x2 )
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS k ⊕t
x
• • • •
k ⊕t
P1
k ⊕t
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
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 ) u1
P2 v1
u2
y1 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 ) u1
(t2 , x2 )
P2 v1
y1
u2
v2
u20
v20 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
x3
u10
v10
u20
v20
(t3 , y3 )
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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(t4 , y4 ) y2
x4 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
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(t4 , y4 ) y2
x4 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
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(t4 , y4 ) y2
x4 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
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(t4 , y4 ) y2
x4 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
Second Try: Two Rounds, Linear TKS P1
(t1 , x1 )
P2
y1
(t2 , x2 )
u1
v1
u2
v2
(t3 , y3 )
x3
u10
v10
u20
v20
(t4 , y4 ) y2
x4 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP: Adv(qc , qp ) ≤
6qc qp 4qc2 + . 2n 2n
Proof sketch: • adversary can create collisions at input of P1 or output of P3 • but proba. to create a collision at P2 is . qc2 /2n • no collision at P2
⇒ ∼ single-key security of 1-round EM . qc qp /2n 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP: Adv(qc , qp ) ≤
6qc qp 4qc2 + . 2n 2n
Proof sketch: • adversary can create collisions at input of P1 or output of P3 • but proba. to create a collision at P2 is . qc2 /2n • no collision at P2
⇒ ∼ single-key security of 1-round EM . qc qp /2n 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP: Adv(qc , qp ) ≤
6qc qp 4qc2 + . 2n 2n
Proof sketch: • adversary can create collisions at input of P1 or output of P3 • but proba. to create a collision at P2 is . qc2 /2n • no collision at P2
⇒ ∼ single-key security of 1-round EM . qc qp /2n 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
Theorem ([CS15, FP15]) The 3-round TEM with linear TKS is a strong tweakable PRP: Adv(qc , qp ) ≤
6qc qp 4qc2 + . 2n 2n
Proof sketch: • adversary can create collisions at input of P1 or output of P3 • but proba. to create a collision at P2 is . qc2 /2n • no collision at P2
⇒ ∼ single-key security of 1-round EM . qc qp /2n Yannick Seurin
Constructing TBCs in the RPM
ASK 2015
16 / 36
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Tightness of the Bound k ⊕t x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
e (k, t, x ) = E (k⊕t, x ) where E is the • can be written E
conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: 1. 2. 3. 4.
ek ∗ (ti , 0) = E (k ∗ ⊕ ti , 0) for 2n/2 tweaks ti query ci = E ek (0, 0) = E (kj , 0) for 2n/2 keys kj compute cj0 = E j look for a collision ci = cj0 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
e (k, t, x ) = E (k⊕t, x ) where E is the • can be written E
conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: 1. 2. 3. 4.
ek ∗ (ti , 0) = E (k ∗ ⊕ ti , 0) for 2n/2 tweaks ti query ci = E ek (0, 0) = E (kj , 0) for 2n/2 keys kj compute cj0 = E j look for a collision ci = cj0 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
e (k, t, x ) = E (k⊕t, x ) where E is the • can be written E
conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: 1. 2. 3. 4.
ek ∗ (ti , 0) = E (k ∗ ⊕ ti , 0) for 2n/2 tweaks ti query ci = E ek (0, 0) = E (kj , 0) for 2n/2 keys kj compute cj0 = E j look for a collision ci = cj0 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 x
k ⊕t P1
k ⊕t P2
k ⊕t P3
y
e (k, t, x ) = E (k⊕t, x ) where E is the • can be written E
conventional 3-round EM cipher with trivial key-schedule • ⇒ secure up to 2n/2 queries at best by a simple collision attack: 1. 2. 3. 4.
ek ∗ (ti , 0) = E (k ∗ ⊕ ti , 0) for 2n/2 tweaks ti query ci = E ek (0, 0) = E (kj , 0) for 2n/2 keys kj compute cj0 = E j look for a collision ci = cj0 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Back to LRW • instantiate E with the 1-round Even-Mansour construction
k ⊗t x
k0
k ⊗t y
E
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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
Back to LRW • instantiate E with the 1-round Even-Mansour construction
k0
k0 P
k ⊗t x
k0
k ⊗t y
E
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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
Back to LRW • instantiate E with the 1-round Even-Mansour construction
(k ⊗ t) ⊕ k 0 x
(k ⊗ t) ⊕ k 0 y
P
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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
Back to LRW • instantiate E with the 1-round Even-Mansour construction
(k ⊗ t) ⊕ k 0 x
(k ⊗ t) ⊕ k 0 y
P
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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
Back to LRW • instantiate E with the 1-round Even-Mansour construction
k ⊗t x
k ⊗t y
P
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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
Back to LRW • instantiate E with the 1-round Even-Mansour construction
Non-Linear Tweakable Even-Mansour (NL-TEM) construction k ⊗t x
k ⊗t y
P
• provably secure in the RPM up to ∼ 2n/2 queries [FP15, CLS15]:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 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 x
k ⊕t P1
k ⊕t
k ⊕t
P2
P3
y
2. nonlinear TKS k ⊗t x
k ⊗t 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 x
k ⊕t P1
k ⊕t
k ⊕t
P2
P3
y
2. nonlinear TKS k ⊗t x
k ⊗t 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 x
k ⊕t P1
k ⊕t
k ⊕t
P2
P3
y
2. nonlinear TKS k ⊗t x
k ⊗t 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 x
k ⊕t P1
k ⊕t
k ⊕t
P2
P3
y
2. nonlinear TKS k ⊗t x
k ⊗t 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
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Cascading the LRW Construction k10 ⊗ t
x
Ek 1
• k1 , . . . , kr and k10 , . . . , kr0 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 k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
x
y
• k1 , . . . , kr and k10 , . . . , kr0 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 k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
x
y
• k1 , . . . , kr and k10 , . . . , kr0 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 k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
x
y
• k1 , . . . , kr and k10 , . . . , kr0 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 k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
x
y
• k1 , . . . , kr and k10 , . . . , kr0 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
P1
P2
x
y
Theorem ([CLS15]) The 2-round NL-TEM construction is secure up to ∼ 22n/3 queries in the RPM: 3/2
34qc Adv(qc , qp ) ≤ 2n Yannick Seurin
√ 30 qc qp + . 2n
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
P1
P2
x
y
Theorem ([CLS15]) The 2-round NL-TEM construction is secure up to ∼ 22n/3 queries in the RPM: 3/2
34qc Adv(qc , qp ) ≤ 2n Yannick Seurin
√ 30 qc qp + . 2n
Constructing TBCs in the RPM
ASK 2015
22 / 36
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Proof Technique: H-coefficients Real world
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
qc
Ideal world
y
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
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
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
qc
Ideal world
y
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
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
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
qc
Ideal world
y
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
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
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
Yannick Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
Constructing TBCs in the RPM
y
ASK 2015
24 / 36
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
Yannick Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
Constructing TBCs in the RPM
y
ASK 2015
24 / 36
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
k1 ⊗ t
k2 ⊗ t
P1
P2
y
u1 v1
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:
x
Yannick Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
u1 v1
u2 v2
Constructing TBCs in the RPM
y
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
(t, x )
u1 v1
u2 v2
Yannick Seurin
Constructing TBCs in the RPM
y
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
(t, x )
u1 v1
u2 v2
Yannick Seurin
Constructing TBCs in the RPM
y
proba ≤
qc qp2 22n
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
(t, x )
u1 v1
u2 v2
Yannick Seurin
Constructing TBCs in the RPM
y
proba ≤
qc qp2 22n
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
(t, x )
u1 v1
u2 v2
y
proba ≤
qc qp2 22n
(t, x )
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
(t, x )
u1 v1
u2 v2
y
proba ≤
qc qp2 22n
(t, x ) (t 0, 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
(t, x )
u1 v1
u2 v2
(t, x ) (t 0, x 0)
Yannick Seurin
Constructing TBCs in the RPM
y
proba ≤
qc qp2 22n
proba ≤
qc2 22n
ASK 2015
24 / 36
Tweakable BC
Tweakable EM
Birthday Security
BBB Security
Conclusion
The Ten “Bad Collision” Cases P1
(t, x )
(t, x )
u1
v1
u1
P2
v2
u2
(t, y )
v1
(t, x )
(t, y )
(t 0 , x 0 )
(t 00 , y 00 )
u2
v2
(t, y )
(t, x )
(t, y )
(t 0 , x 0 )
(t 0 , y 0 )
(t, x )
u1
(t, y )
(t, x )
(t 0 , y 0 )
(t 0 , x 0 )
v2
(t, y )
(t, x )
u1
v1
u2
v2
(t, y )
(t 0 , x 0 )
u10
v10
u20
v20
(t 0 , y 0 )
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 QU1
QV2
• assuming there are no
P2
U1
V1
f U 1
f V 1
f U 2
f V 2
U2
V2
bad collisions, show that the answers of the TEM construction are close to answers of a random tweakable permutation • for each query, there is
QX
U10
V10
U20
V20
QY
U100
V100
U200
V200
a “fresh” value of P1 or P2 which randomizes the output
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 QU1
QV2
• assuming there are no
P2
U1
V1
f U 1
f V 1
f U 2
f V 2
U2
V2
bad collisions, show that the answers of the TEM construction are close to answers of a random tweakable permutation • for each query, there is
QX
U10
V10
U20
V20
QY
U100
V100
U200
V200
a “fresh” value of P1 or P2 which randomizes the output
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
P1
P2
Pr
x
y
• r rounds, r even, with independent keys k1 , . . . , kr secure up to (r /2)n
rn
∼ 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
P1
P2
Pr
x
y
• r rounds, r even, with independent keys k1 , . . . , kr secure up to (r /2)n
rn
∼ 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
P1
P2
Pr
x
y
• r rounds, r even, with independent keys k1 , . . . , kr secure up to (r /2)n
rn
∼ 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
P1
P2
Pr
x
y
• r rounds, r even, with independent keys k1 , . . . , kr secure up to (r /2)n
rn
∼ 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 x
k2 ⊕ t P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t 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 x
k2 ⊕ t P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t 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 x
k2 ⊕ t P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t 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
x
P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t P4
y
2. nonlinear TKS
x
k1 ⊗ t
k2 ⊗ t
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
x
P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t P4
y
2. nonlinear TKS
x
k1 ⊗ t
k2 ⊗ t
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
x
P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t P4
y
2. nonlinear TKS
x
k1 ⊗ t
k2 ⊗ t
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
x
P1
k1 ⊕ t P2
k2 ⊕ t P3
k1 ⊕ t P4
y
2. nonlinear TKS
x
k1 ⊗ t
k2 ⊗ t
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: g
t k
g
f
x
g
f P1
f P2
Pr
y
• sufficient conditions on f and g to have provable
beyond-birthday-bound security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
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: g
t k
g
f
x
g
f P1
f P2
Pr
y
• sufficient conditions on f and g to have provable
beyond-birthday-bound security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
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: g
t k
g
f
x
g
f P1
f P2
Pr
y
• sufficient conditions on f and g to have provable
beyond-birthday-bound security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
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: g
t k
g
f
x
g
f P1
f P2
Pr
y
• sufficient conditions on f and g to have provable
beyond-birthday-bound security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
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
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References
References I Benoît Cogliati, Rodolphe Lampe, and Yannick Seurin. Tweaking Even-Mansour Ciphers. In Rosario Gennaro and Matthew Robshaw, editors, Advances in Cryptology - CRYPTO 2015 - Proceedings, Part I, volume 9215 of LNCS, pages 189–208. Springer, 2015. Full version available at http://eprint.iacr.org/2015/539. Paul Crowley. Mercy: A Fast Large Block Cipher for Disk Sector Encryption. In Bruce Schneier, editor, Fast Software Encryption - FSE 2000, volume 1978 of LNCS, pages 49–63. Springer, 2000. Benoît Cogliati and Yannick Seurin. On the Provable Security of the Iterated Even-Mansour Cipher against Related-Key and Chosen-Key Attacks. In Elisabeth Oswald and Marc Fischlin, editors, Advances in Cryptology - EUROCRYPT 2015 - Proceedings, Part I, volume 9056 of LNCS, pages 584–613. Springer, 2015. Full version available at http://eprint.iacr.org/2015/069. Niels Ferguson, Stefan Lucks, Bruce Schneier, Doug Whiting, Mihir Bellare, Tadayoshi Kohno, Jon Callas, and Jesse Walker. The Skein Hash Function Family. SHA3 Submission to NIST (Round 3), 2010. Yannick Seurin
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References
References II Pooya Farshim and Gordon Procter. The Related-Key Security of Iterated Even-Mansour Ciphers. In Gregor Leander, editor, Fast Software Encryption - FSE 2015, volume 9054 of LNCS, pages 342–363. Springer, 2015. Full version available at http://eprint.iacr.org/2014/953. David Goldenberg, Susan Hohenberger, Moses Liskov, Elizabeth Crump Schwartz, and Hakan Seyalioglu. On Tweaking Luby-Rackoff Blockciphers. In Kaoru Kurosawa, editor, Advances in Cryptology - ASIACRYPT 2007, volume 4833 of LNCS, pages 342–356. Springer, 2007. Jérémy Jean, Ivica Nikolic, and Thomas Peyrin. Tweaks and Keys for Block Ciphers: The TWEAKEY Framework. In Palash Sarkar and Tetsu Iwata, editors, Advances in Cryptology - ASIACRYPT 2014 - Proceedings, Part II, volume 8874 of LNCS, pages 274–288. Springer, 2014. Moses Liskov, Ronald L. Rivest, and David Wagner. Tweakable Block Ciphers. In Moti Yung, editor, Advances in Cryptology - CRYPTO 2002, volume 2442 of LNCS, pages 31–46. Springer, 2002.
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References
References III Rodolphe Lampe and Yannick Seurin. Tweakable Blockciphers with Asymptotically Optimal Security. In Shiho Moriai, editor, Fast Software Encryption - FSE 2013, volume 8424 of LNCS, pages 133–151. Springer, 2013. Will Landecker, Thomas Shrimpton, and R. Seth Terashima. Tweakable Blockciphers with Beyond Birthday-Bound Security. In Reihaneh Safavi-Naini and Ran Canetti, editors, Advances in Cryptology - CRYPTO 2012, volume 7417 of LNCS, pages 14–30. Springer, 2012. Full version available at http://eprint.iacr.org/2012/450. Bart Mennink. Optimally Secure Tweakable Blockciphers. In Gregor Leander, editor, Fast Software Encryption - FSE 2015, volume 9054 of LNCS, pages 428–448. Springer, 2015. Full version available at http://eprint.iacr.org/2015/363. Atsushi Mitsuda and Tetsu Iwata. Tweakable Pseudorandom Permutation from Generalized Feistel Structure. In Joonsang Baek, Feng Bao, Kefei Chen, and Xuejia Lai, editors, ProvSec 2008, volume 5324 of LNCS, pages 22–37. Springer, 2008. Yannick Seurin
Constructing TBCs in the RPM
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References
References IV
Kazuhiko Minematsu. Beyond-Birthday-Bound Security Based on Tweakable Block Cipher. In Orr Dunkelman, editor, Fast Software Encryption - FSE 2009, volume 5665 of LNCS, pages 308–326. Springer, 2009. Phillip Rogaway. Efficient Instantiations of Tweakable Blockciphers and Refinements to Modes OCB and PMAC. In Pil Joong Lee, editor, Advances in Cryptology - ASIACRYPT 2004, volume 3329 of LNCS, pages 16–31. Springer, 2004. Richard Schroeppel. The Hasty Pudding Cipher. AES submission to NIST, 1998.
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