Bit-Decomposition Protocol

Next, suppose that a̸= b. This implies the existence of a pair of most significant bits⃗a_kand⃗b_k, where⃗a_k ̸=⃗b_k. Fori > k, we have⃗t_i, ⃗q_i < ^λ₂+1 andd⃗_i =⃗e_i = 0, since⃗s_i = 0.

For i < k, we have ⃗s_i, ⃗s_i+1 ≥ 1, since a_k ̸= b_k, which implies that t_i > ^λ₂ ⇐⇒ β = ^λ₂ and q_i > ^λ₂ ⇐⇒ β = ^λ₂. This leads to the fact that d⃗_i =⃗e_i, for i < k. Based on this, c=d⃗_k⊕⃗e_k if k ≤ℓ−2 and c=d⃗_k, otherwise. But, since⃗s_k = 1 and⃗a_k ̸=⃗b_k, ifk≤ℓ−2, we will have⃗s_k+1 = 0, which leads to⃗e_k = 0. Thus,c=d⃗_k for 0≤k≤ℓ−1, ifa̸=b.

Suppose that a < b. Since a ̸= b, we have c = d⃗k. Because b > a, we have

⃗bk = 1, ⃗ak = 0 and⃗sk= 1, implying that d⃗k = 1. This means that c= 1, if a < b.

Suppose that a > b. Since a ̸= b, we have c = d⃗_k. Because b < a, we have

⃗bk = 0, ⃗a_k = 1 and⃗s_k= 1, implying that d⃗_k = 0. This means that c= 0, if a > b.

This demonstrates that the described protocol will output 1, if a < b and 0, otherwise.

Security: The same rationale used to prove the security of ΠEEQN can also be used to prove the security of ΠBLTℓ.

Theorem 6. Protocool Π^PBLTℓ is correct and securely implements the functionality F_BLTℓ against semi-honest adversaries in the commodity-based model.

Proof. Correctness and Security: The same arguments used to prove the correctness and security of ΠBLTℓ can be used to prove the correctness and security of Π^PBLTℓ.

Functionality: (J⃗bK

A 2,J⃗bK

B

2)← F_BDℓ(JβK

A 2^ℓ)(JβK

B 2^ℓ)

F_BDℓ runs with the parties Alice and Bob, and is parametrized by ℓ≥2.

• Input: Upon receiving a message from a party with its share of Jβ⃗K2^ℓ, check if its share is contained inZ2^ℓ. If it’s not, then abort. Otherwise, record the share, ignore any subsequent message from that party and inform the other parties about the receival.

• Output: Upon receiving both parties shares, reconstruct β. After reconstruc-tion, compute the binary expansionbℓ−1bℓ−2. . . b₀ of β and return to Alice and Bob there respective shares of Jbℓ−1K2,Jbℓ−2K2. . .Jb₀K2.

With this definition in mind, we now present two protocols which efficiently imple-ment it. The general idea behind both protocols is the same, which is to take the binary expansionsu, v ∈ Z^ℓ2 of JβK

B

2^ℓ and −JβK

A

2^ℓ, respectively, and perform binary addition over

⃗u and ⃗v, but ignoring the last carry bit generated when computing the binary addition, which causes the binary addition result to be the same as that of computing binary ad-dition modulo 2^ℓ. Since the output of the binary addition modulo 2^ℓ, performed by the protocol, is a sequence of shared bits andβ =JβK

B 2^ℓ−JβK

A

2^ℓ (mod 2^ℓ), the output of the addition is the desired output for the bit-decomposition protocol.

In order to compute the binary addition modulo 2^ℓ, we first compute the carry bit vector⃗c, the vector containing the carry bits generated when computing the binary addition of⃗u and ⃗v, and then compute⃗b_i =⃗u_i⊕⃗v_i⊕⃗c_i, for 0≤ i≤ℓ−1, which will be the vector containing the binary expansion of β. By writing down the expressions that define the values of the least significant bits of⃗c we can easily get to the expression that defines⃗c as a whole. The expressions that define the four least significant bits of⃗c are:

⃗c₀ = 0

⃗c₁ =⃗u₀∧⃗v₀

⃗c₂ = (⃗u₁∧⃗v₁)⊕((⃗u₁⊕⃗v₁)∧(⃗u₀∧⃗v₀))

⃗c₃ = (⃗u₂ ∧⃗v₂)⊕((⃗u₂ ⊕⃗v₂)∧(⃗u₁∧⃗v₁))⊕((⃗u₂ ⊕⃗v₂)∧(⃗u₁⊕⃗v₁)∧(⃗u₀∧⃗v₀))

As previously stated, we can analyse these expressions and workout the following set of Boolean equations that define the carry bit vector⃗c.

⃗c₀ = 0 and⃗c_i =^Mi−1

j=0

⃗t_i,j, for 1 ≤i≤ℓ−1

⃗t_i,j =⃗g_j ∧

i−1

^

k=j+1

⃗

x_k, for 0≤j < i≤ℓ−1

⃗g_i =⃗u_i∧⃗v_i, for 0 ≤i≤ℓ−1

⃗

x_i =⃗u_i⊕⃗v_i, for 0 ≤i≤ℓ−1

Both these protocols just privately compute the vector⃗c, by using SOTs to evaluate the previously described Boolean equations, and then finish by computing the vector⃗b. Note that the only difference between the the two protocols is found in their 4th step. But the values of⃗t_i,j, computed in the protocols 4th step, will be the same in both protocols, since the protocols differ only in how the values are computed. More specifically, ΠBDuses SOTs to compute the values of⃗t_i,j, while Π^′BD uses the functionalityFEEQ_ℓ.

Protocol: (J⃗bK

A 2,J⃗bK

B

2)←ΠBDℓ(JβK

A 2^ℓ)(JβK

B 2^ℓ)

Let ⃗v ∈ Z^ℓ2 and ⃗u ∈ Z^ℓ2 be the binary expansions of (−JβK

A

2^ℓ (mod 2^ℓ)) and JβK

B

2^ℓ, respectively.

1. Execute J⃗g_iK^ℓ ← F_SOT³

ℓ((0,0,1),J⃗u_i+⃗v_iK³), for 0 ≤i≤ℓ−1 (⃗g_i =⃗u_i∧⃗v_i).

2. Execute J⃗x_iK^ℓ ← F_SOT²

ℓ((0,1),J⃗u_i+⃗v_iK²), for 0 ≤i≤ℓ−1. (⃗x_i =⃗u_i⊕⃗v_i)

3. Locally compute J⃗h_i,jKℓ ← J⃗g_jKℓ+^Pi−1k=j+1J⃗x_kKℓ, for 0 ≤ j < i ≤ ℓ−1. (⃗h_i,j = i−j ⇐⇒ ⃗gj∧^Vi−1_k=j+1⃗xk)

4. Perform J⃗t_i,jK² ← F_SOT^ℓ

2(Oneℓ(i−j),J⃗hi,jK^ℓ), for 0≤j < i≤ℓ−1. (⃗t_i,j = 1 ⇐⇒

⃗hi,j =i−j ⇐⇒ ⃗g_j ∧^Vi−1_k=j+1⃗x_k;⃗t_i,j =⃗g_j∧^Vi−1_k=j+1⃗x_k)

5. Let ⃗c₀ = 0. Locally compute Jc_iK² = ^Li−1j=0J⃗t_i,jK², for 1 ≤ i ≤ ℓ −1. (⃗c_i =

Li−1 j=0⃗t_i,j)

6. Locally computeJb_iK² =Ju_iK²+Jv_iK²+Jc_iK², for 0≤i≤ℓ−1. (⃗b_i =⃗u_i⊕⃗v_i⊕⃗c_i)

Protocol: (J⃗bK

A 2,J⃗bK

B

2)←Π^′BDℓ(JβK

A 2^ℓ)(JβK

B 2^ℓ)

Let ⃗v ∈ Z^ℓ2 and ⃗u ∈ Z^ℓ2 be the binary expansions of (−JβK

A

2^ℓ (mod 2^ℓ)) and JβK

B

2^ℓ, respectively.

1. Execute J⃗g_iK^ℓ ← F_SOT³

ℓ((0,0,1),J⃗u_i+⃗v_iK³), for 0 ≤i≤ℓ−1. (⃗g_i =⃗u_i∧⃗v_i) 2. Execute J⃗x_iK^ℓ ← F_SOT²

ℓ((0,1),J⃗u_i+⃗v_iK²), for 0 ≤i≤ℓ−1. (⃗x_i =⃗u_i⊕⃗v_i)

3. Locally compute J⃗h_i,jK^ℓ ← J⃗g_jK^ℓ+^Pi−1_k=j+1J⃗x_kK^ℓ, for 0 ≤ j < i ≤ ℓ−1. (⃗h_i,j = i−j ⇐⇒ ⃗g_j∧^Vi−1_k=j+1⃗x_k)

4. Perform J⃗t_i,jK2 ← FEEQ_ℓ(J⃗hi,jK^ℓ,Ji−jK^ℓ), for 0 ≤ j < i ≤ ℓ−1. (⃗t_i,j = 1 ⇐⇒

⃗hi,j =i−j ⇐⇒ ⃗gj ∧^Vi−1_k=j+1⃗xk;⃗t_i,j =⃗gj∧^Vi−1_k=j+1⃗xk)

5. Let ⃗c₀ = 0. Locally compute Jc_iK² = ^Li−1j=0J⃗t_i,jK², for 1 ≤ i ≤ ℓ −1. (⃗c_i =

Li−1 j=0⃗t_i,j)

6. Locally computeJb_iK² =Ju_iK²+Jv_iK²+Jc_iK², for 0≤i≤ℓ−1. (⃗b_i =⃗u_i⊕⃗v_i⊕⃗c_i)

Below we added two examples of executions for the two bit-decomposition protocols to help the reader better understand how the protocols work.

Example: (J⃗bK

A 2,J⃗bK

B

2)←ΠBD3(5)(5), where⃗b= (0,0,0) Let ℓ = 3, JβK

B

8 = 5 and JβK

A

8 = 5. This implies that ⃗v = (0,1,1) and

⃗u= (1,0,1).

1. Alice and Bob execute the following 3 SOTs:

• J⃗g₀K³ ← F_SOT³

3((0,0,1),J2K3). (⃗g₀ = 1)

• J⃗g₁K³ ← F_SOT³

3((0,0,1),J1K³). (⃗g₁ = 0)

• J⃗g₂K³ ← F_SOT³

3((0,0,1),J1K3). (⃗g₂ = 0) 2. Alice and Bob execute the following 3 SOTs:

• J⃗x0K³ ← F_SOT²

3((0,1),J0K2). (⃗x0 = 0)

• J⃗x₁K³ ← F_SOT²

3((0,1),J1K²). (⃗x₁ = 1)

• J⃗x₂K³ ← F_SOT²

3((0,1),J1K2). (⃗x₂ = 1)

3. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗h1,0K3 =J⃗g₀K3+^P0k=1J⃗xkK3 =J⃗g₀K3 =J1K3

• J⃗h_2,0K³ =J⃗g₀K³+^P1k=1J⃗x_kK³ =J⃗g₀K³+J⃗x₁K³ =J2K3

• J⃗h2,1K³ =J⃗g1K³+^P1k=2J⃗xkK³ =J⃗g1K³ =J0K3

4. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗t_1,0K2 ← F_SOT³

2((0,1,0),J⃗h_1,0K3). (⃗t_1,0 = 1)

• J⃗t_2,0K² ← F_SOT³

2((0,0,1),J⃗h_2,0K³). (⃗t_2,0 = 1)

• J⃗t_2,1K² ← F_SOT³

2((0,1,0),J⃗h_2,1K³). (⃗t_2,1 = 0)

5. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗c₀K² =J0K2

• J⃗c₁K² =^P0j=0J⃗t_1,jK² =J⃗t_1,0K² =J1K2

• J⃗c₂K² =^P1j=0J⃗t_2,jK² =J⃗t_2,0K²+J⃗t_2,1K² =J1K2

6. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗b₀K2 =J⃗u₀K2+J⃗v₀K2+J⃗c₀K2 =J1K2+J1K2+J0K2 =J0K2

• J⃗b₁K2 =J⃗u₁K2+J⃗v₁K2+J⃗c₁K2 =J0K2+J1K2+J1K2 =J0K2

• J⃗b₂K² =J⃗u₂K²+J⃗v₂K²+J⃗c₂K² =J1K²+J0K²+J1K² =J0K²

Example: (J⃗bK

A 2,J⃗bK

B

2)←Π^′BD3(5)(5), where⃗b= (0,0,0) Let ℓ = 3, JβK

B

8 = 5 and JβK

A

8 = 5. This implies that ⃗v = (0,1,1) and

⃗u= (1,0,1).

1. Alice and Bob execute the following 3 SOTs:

• J⃗g0K³ ← F_SOT³

3((0,0,1),J2K3). (⃗g0 = 1)

• J⃗g₁K3 ← F_SOT³

3((0,0,1),J1K3). (⃗g₁ = 0)

• J⃗g₂K3 ← F_SOT³

3((0,0,1),J1K3). (⃗g₂ = 0) 2. Alice and Bob execute the following 3 SOTs:

• J⃗x₀K³ ← F_SOT²

3((0,1),J0K²). (⃗x₀ = 0)

• J⃗x₁K³ ← F_SOT²

3((0,1),J1K2). (⃗x₁ = 1)

• J⃗x₂K³ ← F_SOT²

3((0,1),J1K2). (⃗x₂ = 1)

3. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗h1,0K³ =J⃗g₀K³+^P0k=1J⃗x_kK³ =J⃗g₀K³ =J1K3

• J⃗h2,0K³ =J⃗g0K³+^P1k=1J⃗xkK³ =J⃗g0K³+J⃗x1K³ =J2K3

• J⃗h_2,1K3 =J⃗g₁K3+^P1k=2J⃗x_kK3 =J⃗g₁K3 =J0K3

4. Alice and Bob perform FEEQ₃ 3 times in the following way:

• J⃗t_1,0K² ← F_EEQ3(J⃗h_1,0K³,J1K³). (⃗t_1,0 = 1)

• J⃗t_2,0K² ← F_EEQ3(J⃗h_2,0K³,J2K3). (⃗t_2,0 = 1)

• J⃗t_2,1K² ← F_EEQ3(J⃗h2,1K³,J1K3). (⃗t_2,1 = 0)

5. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗c₀K² =J0K2

• J⃗c₁K2 =^P0j=0J⃗t_1,jK2 =J⃗t_1,0K2 =J1K2

• J⃗c₂K2 =^P1j=0J⃗t_2,jK2 =J⃗t_2,0K2+J⃗t_2,1K2 =J1K2

6. Alice and Bob compute their respective shares of the following 3 shared values:

• J⃗b₀K² =J⃗u₀K²+J⃗v₀K²+J⃗c₀K² =J1K²+J1K²+J0K² =J0K²

• J⃗b₁K² =J⃗u₁K²+J⃗v₁K²+J⃗c₁K² =J0K²+J1K²+J1K² =J0K²

• J⃗b₂K² =J⃗u₂K²+J⃗v₂K²+J⃗c₂K² =J1K2+J0K2+J1K2 =J0K2

We now proceed to prove the security and correctness of both protocols, starting by ΠBD.

Theorem 7. Protocol ΠBD_ℓ is correct and securely implements the functionality FBD_ℓ

against semi-honest adversaries in the commodity-based model.

Proof. Correctness: Let ⃗v ∈ Z^ℓ2 and ⃗u ∈ Z^ℓ2 be the binary expansions of (−JβK

A 2^ℓ

(mod 2^ℓ)) and JβK

B

2^ℓ, respectively, and⃗c^′ ∈ Z^ℓ+12 be the carry bit vector generated when computingα=JβK

B

2^ℓ+ (−JβK

A

2^ℓ (mod 2^ℓ)). Based on this, we haveα =⃗c_ℓ·2^ℓ+^Pℓ−1i=0(⃗u_i⊕

⃗vi⊕⃗ci)·2ⁱ, where clearly⃗cℓ ∈ {0,1} and 0 ≤ ^Pℓ−1_i=0(⃗ui⊕⃗vi ⊕⃗ci)·2ⁱ <2^ℓ, which implies that α ≡ β ≡ ^Pℓ−1_i=0(⃗ui ⊕⃗vi ⊕⃗ci)·2ⁱ (mod 2^ℓ). This means that⃗bi = ⃗ui ⊕⃗vi ⊕⃗ci, for 0≤i≤ℓ−1, is the binary expansion ofβ. Thus, if the vector⃗ccomputed by the protocol is equal to⃗c^′, from position 0 to position ℓ−1, then based on step 6 of ΠBD, we can see that the protocol’s output would in fact be the desired one. Because of this, we proceed to prove that⃗c_i =⃗c^′_i for 0≤i≤ℓ−1.

The set of Boolean equations that define the value of⃗c^′ are the following:

⃗c^′₀ = 0 and ⃗c^′_i =

i−1

M

j=0

⃗t^′_i,j, for 1≤i≤ℓ−1

⃗t^′_i,j =g⃗^′_j ∧

i−1

^

k=j+1

x⃗^′_k, for 0≤j < i≤ℓ−1

g⃗^′_i =⃗a_i∧d⃗_i, for 0≤i≤ℓ−1 x⃗^′_i =⃗a_i⊕d⃗_i, for 0 ≤i≤ℓ−1

After quickly analyzing the protocol, we can see that ΠBDℓ computes the bit vector

⃗caccording to the following equations:

⃗c₀ = 0 and⃗c_i =^Mi−1

j=0

⃗t_i,j, for 1 ≤i≤ℓ−1

⃗t_i,j =







1, ⃗h_i,j =i−j

0, otherwise , for 0≤j < i≤ℓ−1

⃗h_i,j =⃗g_j + ^i−1X

k=j+1

⃗

x_k, for 0≤j < i≤ℓ−1

⃗

x_i =⃗a_i⊕d⃗_i, for 0≤i≤ℓ−1

⃗g_i =⃗a_i∧d⃗_i, for 0 ≤i≤ℓ−1

Looking at these equations we can see that⃗x_i, ⃗g_i ∈ {0,1} for 0≤i≤ℓ−1, which implies that 0 ≤ ⃗h_i,j ≤ i−j and ⃗h_i,j = i−j iff ⃗g_j ∧^Vi−1_k=j+1⃗x_k, for 0 ≤ j < i ≤ ℓ−1.

Based on this, we can see that⃗t_i,j =⃗gj∧^Vi−1_k=j+1⃗xkfor 0≤j < i≤ℓ−1. Thus, looking at the equation that dictates the value of⃗c, we can conclude that⃗c_i =⃗c^′_i for 0≤i≤ ℓ−1.

Therefore, we have that ΠBDℓ is correct.

Security: The rationale used in ΠEEQ^∗_N,M’s security proof can be used to prove ΠBDℓ’s security.

Theorem 8. The protocol Π^′BD_ℓ is correct and securely implements the functionality F_BDℓ against semi-honest adversaries in the commodity-based model.

Proof. Correctness: By looking at the descriptions for protocols Π^′BDℓ and ΠBDℓ, we can see that the only difference between the two is 4. So if we prove that the values of⃗t_i,j in ΠBDℓ and Π^′BDℓ respect the same equation, for 0≤j < i≤ℓ−1, from the correctness proof of ΠBDℓ, we have that Π^′BD_ℓ is also correct. Again by looking at Π^′BD_ℓ’s description and by the formal definition ofF_EEQ we can see that the values of⃗t_i, j, in the description for Π^′BD_ℓ, are defined by the following equation:

⃗t_i,j =







1, ⃗h_i,j =i−j

0, otherwise , for 0≤j < i≤ℓ−1

This equation also defines the values of⃗t_i,j in the description for ΠBDℓ, for 0≤j <

i≤ℓ−1. Thus, we can conclude that Π^′BD_ℓ is correct.

Security: The same ideas used to prove the security of ΠEEQ can be applied to prove the security of Π^′BD.

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