Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweaking Even-Mansour Ciphers Benoît Cogliati1
Rodolphe Lampe1 1 Versailles
Yannick Seurin2
University, France
2 ANSSI,
France
August 17, 2015 — CRYPTO 2015
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
1 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Outline
Background: Tweakable Block Ciphers Our Contribution Overview of the Proof for Two Rounds Longer Cascades Conclusion and Perspectives
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
2 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
3 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
3 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
3 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
3 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
3 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Generic Constructions of TBCs • 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) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
4 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Generic Constructions of TBCs • 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) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
4 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Generic Constructions of TBCs • 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) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
4 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Generic Constructions of TBCs • 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) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
4 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Generic Constructions of TBCs • 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) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
4 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the LRW Construction
x
k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
5 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the LRW Construction
x
k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
5 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the LRW Construction
x
k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
5 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the LRW Construction
x
k10 ⊗ t
k20 ⊗ t
kr0 ⊗ t
Ek 1
Ek2
Ek r
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
5 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Outline
Background: Tweakable Block Ciphers Our Contribution Overview of the Proof for Two Rounds Longer Cascades Conclusion and Perspectives
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
6 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) k f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) k f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) k f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) k f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) k f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) (k, t) f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Tweakable Even-Mansour Constructions 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 work: SPN ciphers (more gen. key-alternating ciphers) (k, t) f0 x
f1 P1
fr P2
Pr
y
• analysis in the Random Permutation Model
⇒ “tweakable” Even-Mansour construction(s) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
7 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
8 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
8 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
8 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
8 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Previous Result k⊕t x
k⊕t P1
k⊕t P2
k⊕t y
P3
• provably secure in the RPM up to ∼ 2n/2 queries [CS15, FP15] 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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
9 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Previous Result k⊕t x
k⊕t P1
k⊕t P2
k⊕t y
P3
• provably secure in the RPM up to ∼ 2n/2 queries [CS15, FP15] 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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
9 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Previous Result k⊕t x
k⊕t P1
k⊕t P2
k⊕t y
P3
• provably secure in the RPM up to ∼ 2n/2 queries [CS15, FP15] 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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
9 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Previous Result k⊕t x
k⊕t P1
k⊕t P2
k⊕t y
P3
• provably secure in the RPM up to ∼ 2n/2 queries [CS15, FP15] 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
Question How can we obtain a construction with security beyond the birthday-bound?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
9 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Back to LRW • instantiate E with the 1-round Even-Mansour construction
(1-round) Tweakable Even-Mansour (TEM) construction k ⊗t x
k ⊗t y
P
• provably secure in the RPM up to ∼ 2n/2 queries:
Adv(qc , qp ) ≤
qc2 2qc qp + . 2n 2n
• t 6= 0 ⇒ k 0 is superfluous (k ⊗ t unif. random for any t 6= 0) Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
10 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the TEM Construction • k1 , k2 independent n-bit keys
x
k1 ⊗ t
k2 ⊗ t
P1
P2
y
• our main result: secure up to ∼ 22n/3 queries in the RPM: 3/2
34qc Adv(qc , qp ) ≤ 2n
Cogliati, Lampe, Seurin
√ 30 qc qp + . 2n
Tweaking Even-Mansour Ciphers
CRYPTO 2015
11 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Cascading the TEM Construction • k1 , k2 independent n-bit keys
x
k1 ⊗ t
k2 ⊗ t
P1
P2
y
• our main result: secure up to ∼ 22n/3 queries in the RPM: 3/2
34qc Adv(qc , qp ) ≤ 2n
Cogliati, Lampe, Seurin
√ 30 qc qp + . 2n
Tweaking Even-Mansour Ciphers
CRYPTO 2015
11 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Outline
Background: Tweakable Block Ciphers Our Contribution Overview of the Proof for Two Rounds Longer Cascades Conclusion and Perspectives
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
12 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Formalization of the Security Experiment Real world
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
Ideal world
y
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
qp
qc
qp
qc
0/1
0/1
• real world: TEM construction with random keys k1 , . . . , kr e0 independent • ideal world: random tweakable permutation P
from P1 , . . . , Pr • RPM: D has oracle access to P1 , . . . , Pr in both worlds Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
13 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Formalization of the Security Experiment Real world
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
Ideal world
y
P1 , . . . , Pr
e P 0
P1 , . . . , Pr
qp
qc
qp
qc
0/1
0/1
• real world: TEM construction with random keys k1 , . . . , kr e0 independent • ideal world: random tweakable permutation P
from P1 , . . . , Pr • RPM: D has oracle access to P1 , . . . , Pr in both worlds Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
13 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
14 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
14 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
14 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
Cogliati, Lampe, Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
Tweaking Even-Mansour Ciphers
y
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
Cogliati, Lampe, Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
Tweaking Even-Mansour Ciphers
y
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
k1 ⊗ t
k2 ⊗ t
P1
P2
y
u1 v1
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions:
x
Cogliati, Lampe, Seurin
k1 ⊗ t
k2 ⊗ t
P1
P2
u1 v1
u2 v2
Tweaking Even-Mansour Ciphers
y
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions: k1 ⊗ t
k2 ⊗ t
x
P1
P2
(t, x )
u1 v1
u2 v2
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
y
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions: k1 ⊗ t
k2 ⊗ t
x
P1
P2
(t, x )
u1 v1
u2 v2
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
y
proba ≤
qc qp2 22n
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Bad Transcripts • one needs to avoid “two-fold” collisions: k1 ⊗ t
k2 ⊗ t
x
P1
P2
(t, x )
u1 v1
u2 v2
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
y
proba ≤
qc qp2 22n
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 )
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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)
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
y
proba ≤
qc qp2 22n
proba ≤
qc2 22n
CRYPTO 2015
15 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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 )
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
16 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
17 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
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
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
17 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Outline
Background: Tweakable Block Ciphers Our Contribution Overview of the Proof for Two Rounds Longer Cascades Conclusion and Perspectives
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
18 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Longer Cascades of the TEM Construction
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
19 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Longer Cascades of the TEM Construction
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
19 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Longer Cascades of the TEM Construction
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
19 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Longer Cascades of the TEM Construction
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
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 Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
19 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Outline
Background: Tweakable Block Ciphers Our Contribution Overview of the Proof for Two Rounds Longer Cascades Conclusion and Perspectives
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
20 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Conclusion
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
y
• we analyzed the “public permutation” variant of the LRW
construction, and proved tight 22n/3 -security for 2 rounds • similar security level as LRW, yet in an idealized model rn
• open problem 1: prove tight security up to 2 r +1 queries for r ≥ 3 • open problem 2: can we avoid non-linear mixing of the key and
the tweak and still get beyond-birthday security?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
21 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Conclusion
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
y
• we analyzed the “public permutation” variant of the LRW
construction, and proved tight 22n/3 -security for 2 rounds • similar security level as LRW, yet in an idealized model rn
• open problem 1: prove tight security up to 2 r +1 queries for r ≥ 3 • open problem 2: can we avoid non-linear mixing of the key and
the tweak and still get beyond-birthday security?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
21 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Conclusion
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
y
• we analyzed the “public permutation” variant of the LRW
construction, and proved tight 22n/3 -security for 2 rounds • similar security level as LRW, yet in an idealized model rn
• open problem 1: prove tight security up to 2 r +1 queries for r ≥ 3 • open problem 2: can we avoid non-linear mixing of the key and
the tweak and still get beyond-birthday security?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
21 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
Conclusion
x
k1 ⊗ t
k2 ⊗ t
kr ⊗ t
P1
P2
Pr
y
• we analyzed the “public permutation” variant of the LRW
construction, and proved tight 22n/3 -security for 2 rounds • similar security level as LRW, yet in an idealized model rn
• open problem 1: prove tight security up to 2 r +1 queries for r ≥ 3 • open problem 2: can we avoid non-linear mixing of the key and
the tweak and still get beyond-birthday security?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
21 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
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 security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
22 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
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 security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
22 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
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 security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
22 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
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 security in the RPM? e (k, t, x ) = E (k ⊕ t, x ) • NB: f = g linear does not work since E
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
22 / 26
Tweakable Block Ciphers
Our Contribution
Proof Overview
Longer Cascades
Conclusion
The end. . .
Thanks for your attention! Comments or questions?
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
23 / 26
References
References I 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. Pooya Farshim and Gordon Procter. The Related-Key Security of Iterated Even-Mansour Ciphers. In Fast Software Encryption - FSE 2015, 2015. To appear. Full version available at http://eprint.iacr.org/2014/953.
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
24 / 26
References
References II 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. 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.
Cogliati, Lampe, Seurin
Tweaking Even-Mansour Ciphers
CRYPTO 2015
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References
References III 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. 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. 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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