Tight Private Circuits: Achieving Probing Security with the Least - - PowerPoint PPT Presentation

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Tight Private Circuits:
 Achieving Probing Security with the Least Refreshing

Sonia Belaid, Dahmun Goudarzi, and Matthieu Rivain dahmun.goudarzi@pqshield.com

Side-Channel Attacks and Higher-Order Masking

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in1 in2 in3 ⊕ ⊗ ⊗ ⊕

For each in, in = m1 + m2 + … + md

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• Higher-Order Masking:

Implementation Transformation

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x y z

⊕ ⊗ ⊗ ⊕

[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R

⊕ [ ]

[x] d-sharing addition gadget : :

⊗ [ ] R

multiplication gadget refresh gadget : :

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Secure Gadgets

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[x] [y] [z] = [x + y] ⊕ [ ] R ⊗ [ ] [x] [y] [z] = [x ⋅ y] [x] [z] = [x]

Addition gadget Multiplication gadget Refresh gadget

• Addition gadget:
• Refresh gadget:

[x] + [y] = (x1 + y1, x2 + y2, …, xd + yd) Ref([x]) = [x] [1]

⊗ [ ]

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ISW Multiplication

• Ishai, Sahai, Wager. Private Circuits: Securing Hardware against Probing Attacks (Crypto 2003)

• Compute the cross products
• Pack the cross products to get output shares
• Interleave of fresh randomness in the packing


to avoir security flaws d2 d

      c1 c2 . . . cd       =       a1b1 a1b2 . . . a1bd a2b1 a2b2 . . . . . . . . . . . . ... . . . adb1 adb2 . . . adbd       +       r1,2 . . . r1,d r1,2 . . . . . . ... rd,d−1 r1,d rd,d−1      

Probing Security

6 6 memory computation leakage

I1

x1 y1 z1 f1(x1, y1)

I2

x2 y2 z2 f2(x2, y2)

…

Is

xs ys zs fs(xs, ys)

• The adversary can probe instructions
• t

Ii f(xi, yi) = (xi, yi) for t instrucRons f(xi, yi) = ⊥ for other instrucRons ISW scheme is t-probing secure for t < d

Composition Flaws

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⊕ [ ] ⊗ [ ] [x] [y] ⊗ [ ]

• placed on
• placed on
• d = 3

P1 x1 ⋅ (x3 + y3) P2 x2 ⋅ y3

f (x1, x3 + y3) = (x1, x3 + y3) f (x2, y3) = (x2, y3)

Composition Flaws

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⊕ [ ] ⊗ [ ] [x] [y] ⊗ [ ]

• placed on
• placed on
• d = 3

P1 x1 ⋅ (x3 + y3) P2 x2 ⋅ y3

f (x1, x3 + y3) = (x1, x3 + y3) f (x2, y3) = (x2, y3)

x1 x2 (x3 + y3) + y3 = x3

Refresh Gadget

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⊕ [ ] ⊗ [ ] [x] [y] ⊗ [ ]

R

• Add refresh gadget at carefully chosen position
• Drawback: randomness overhead
• Challenge: where and how many needed?
• Solution: formal verification tools !

Limitation of Existing Tools

• Adds a refresh but circuit was already probing secure
• Complexity highly impacted by the masking order

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⊕ [ ] ⊗ [ ] [x] [y] R ⊕ [ ] ⊗ [ ] [x] [y]

existing tools

New Proposal

10 t-probing security game-based definition simplified problem formal method

• Determine exactly whether a circuit is probing secure for any order

Game Based Probing Security

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𝒝

(𝒬, x1, …, xn) (𝒬, x1, …, xn)

[x1] ← Enc(x1) [x2] ← Enc(x2) [xn] ← Enc(xn) ⋮

C([x1], [x2], …, [xn])𝒬

𝒯(𝒬)

𝒝

(v1, v2, …, vt) (v1, v2, …, vt)

Real Ideal 1. 1. 2.

Game 0 to Game 1

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𝒝

(𝒬, x1, …, xn) (𝒬, x1, …, xn) [x1] ← Enc(x1) [x2] ← Enc(x2) [xn] ← Enc(xn) ⋮

C([x1], [x2], …, [xn])𝒬

𝒯(𝒬)

𝒝

(v1, v2, …, vt) (v1, v2, …, vt) (𝒬′, x1, …, xn) (𝒬′, x1, …, xn)

C([x1], [x2], …, [xn])𝒬′

(v1, v2, …, vq) (v1, v2, …, vq) 𝒯(𝒬′)

Real Ideal 1. 2. 1.

Game 0 to Game 1

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⊕ [ ] ⊗ [ ] [x] [y] ⊕ [ ] ⊗ [ ] [x] [y]

Game 1 to Game 2

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𝒝

[x1] ← Enc(x1) [x2] ← Enc(x2) [xn] ← Enc(xn) ⋮

𝒝

(𝒬′, x1, …, xn) (𝒬′, x1, …, xn)

C′([x1], [x2], …, [xn])𝒬′

(v1, v2, …, vq) (v1, v2, …, vq) 𝒯(𝒬′) C′ ← Flatten(C) C′ ← Flatten(C) C C 𝒯′(𝒬′)

Real Ideal 1. 1. 2. 2. 3.

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Flatten Transformation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Game 2 to Game 3

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𝒝

[x1] ← Enc(x1) [x2] ← Enc(x2) [xn] ← Enc(xn) ⋮

𝒝

(𝒬′, x1, …, xn) (𝒬′, x1, …, xn)

C′([x1], [x2], …, [xn])𝒬′

(v1, v2, …, vq) (v1, v2, …, vq) 𝒯′(𝒬′) C′ ← Flatten(C) C′ ← Flatten(C) (𝒬′′, x1, …, xn) (𝒬′′, x1, …, xn)

C′([x1], [x2], …, [xn])𝒬′′

𝒯′(𝒬′′)

Real Ideal 1. 1. 2. 2. 3.

Game 2 to Game 3

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i] [x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i]

Matrix Representation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i] [x] [y] [z] [ f ] [h] [e] [g] [i] 1 1 1

ℳj

for the j-th shares

P1 P2

Matrix Representation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i] [x] [y] [z] [ f ] [h] [e] [g] [i] 1 1 1

ℳj

for the j-th shares

P1 P2

Matrix Representation

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[x] [y] [z] ⊗ [ ] ⊕ [ ] ⊗ [ ] ⊗ [ ] ⊗ [ ] R [ f ] [h] [e] [g] [i] [x] [y] [z] [ f ] [h] [e] [g] [i] 1 1 1

ℳj

for the j-th shares

P1 P2

Game 2 to Game 3

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Im(ℳT

1) ∩ Im(ℳT 2) ∩ … ∩ Im(ℳT d) = ∅

(v1, …, vq)

uniformly distributed

Resolution Method

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⊕ [ ] ⊗ [ ] [x] [y] ⊗ [ ]

d = 3

[x] [y]

Im Im Im

∩ ∩

=

[x]

ℳ1 ℳ2 ℳ3

1 1 1 1 1

⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c1] ⋅ [c2] [c2]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c1] ⋅ [c2] [c2] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

[c2] = [c3] + [c5] !

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

[c2] = [c3] + [c5] !

𝒣 𝒱

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

[c2] = [c3] + [c5] !

𝒣 𝒱

𝒣1 𝒱1

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⊕ [ ] ⊗ [ ] [c1] [c2] [c3] ⊕ [ ] ⊗ [ ] ⊗ [ ] [c4] [c5] [c2] ⋅ [c1] [c1] [c4] ⋅ [c5], [c4] ⋅ [c3] [c5], [c3]

[c2] = [c3] + [c5] !

𝒣 𝒱

𝒣1 𝒱1 𝒣2 𝒱2

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tightProve

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Bitslice for the AES S-box

23 83 XOR gates 32 AND gates Circuit of Boyer et al.

Bitslice for the AES S-box

23 83 XOR gates 32 AND gates 83 XOR inst. 32 AND inst. Bitslice transformation

Bitslice for the AES S-box

23 83 XOR gates 32 AND gates 83 XOR inst. 32 AND inst. Bitslice transformation 83 XORs 32 ISW-and Masking transformation

d

Bitslice for the AES S-box

23 83 XOR gates 32 AND gates 83 XOR inst. 32 AND inst. Bitslice transformation 83 XORs 32 ISW-and Masking transformation

d

d-shared Boolean circuit

Application of tightProve

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d-shared Boolean circuits

Application of tightProve

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d-shared Boolean circuits

How many refresh gadgets ?
 Where to insert them ?

Tight d-private Boolean circuits

?

Application of tightProve

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d-shared Boolean circuits

Existing tools

+ 32 Refresh gadgets

1 before each mult.

Tight d-private Boolean circuits

Application of tightProve

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d-shared Boolean circuits Tight d-private Boolean circuits

tightPROVE

No refresh required !

Application of tightProve

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d-shared Boolean circuits Tight d-private Boolean circuits

tightPROVE

No refresh required ! Randomness requirement 2 /

Performance

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Randomness consumption 1500 3000 4500 6000 Masking order 2 3 4 5 6 7 8 9 10

Bitslice with tightPROVE Standard (KHL) Standard (RP) Bitslice with refresh

Thank you!

Questions?

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