Encryptions

We introduced private-key encrytion earlier, but public-key encryption is deferred to this section. It is because the hardness assumption: private-key encryption and other primitives can be based on the existence of OWFs, but we need other assumptions to obtain PKE.

Learning with Errors, LWE

In this section, we change the representation of \(\Z_q\) so that

\[\Z_q \equiv \set{ -\floor{\frac{q-1}{2}}, ..., 0, ..., \floor{\frac{q}{2}}}.\]

The modular arithmetic is the same. We also say that an element in \(\Z_q\) is small if it is close to 0.

We consider matrix \(A \in \Z_q^{m \times n}\) and vector \(\vec{s} \in \Z_q^n\) such that \(m \gg n\). Let \(\vec{t}' := A \cdot \vec{s} \in \Z_q^m\). Using standard linear algebra, it is efficient to solve \(A \cdot \vec{x} = \vec{t}'\) (when \(A\) is full rank).

The LWE problem considers the following variant: \(A\) and \(\vec{s}\) are chosen as before, but an additional error vector \(\vec{e} \in \Z_q^m\) is sampled such that \(\norm{\vec{e}}\) is small (we will have each coordinate of \(\vec e\) small in \(\Z_q\)). Then, the goal is to find \(\vec{x}, \vec{y}\) such that

\[A \cdot \vec{x} + \vec{y} = \vec{t},\]

where \(\vec{t} := A \cdot \vec{s} + \vec{e}\), \(\vec{x} \in \Z_q^n\), and \(\vec{y} \in \Z_q^m\) is small. The hardness of this problem, Learning with Errors, is parameterized by \(q, m, n\), and the bound of small. When parameters are chosen appropriately, this problem is believed to be hard even for quantum algorithms (the belief is established by reductions from lattice problems which is beyond the scope).

For our purpose, it is easier to use the decision variant of LWE, formalized next.

Definition: Learning with Errors (LWE) Assumption

We say the decisional \(\LWE_{m,q,\psi}\) problem is quantum-hard if for all quantum polynomial-time distinguisher \(D\) there is a negligible function \(\eps\) such that for all \(n\in\N\),
\[\begin{align*} \Pr[A \gets \Z^{m\times n}_q; \vec{s} \gets \psi^n ; \vec{e} \gets \psi^m : D(A, A \cdot \vec{s} + \vec{e}) = 1] \\ - \Pr[A \gets \Z^{m\times n})_q ; \vec{t} \gets \Z^m_q : D(A,\vec{t}) = 1] \le \eps(n), \end{align*}\]
where \(m, q \in \N\) are functions of \(n\), and \(\psi\) is an efficiently samplable distribution over \(\Z_q\).

Discuss:

Again, the hardness depend on the parameters \(m,q,\psi\). For example, when \(\psi\) is uniform over \(\Z_q\), the two distributions are identical (but useless).
Notice that the above definition also samples \(\vec{s}\) from \(\psi\).
The two distributions differ in entropy: \((A,\vec{t})\) consists of \(mn + m\) uniform elements from \(\Z_q\), but \((A, A \cdot \vec{s} + \vec{e})\) is sampled from \(mn\) uniform elements plus \((m+n)\) elements from \(\psi\).

An useful set of parameter:

\(m(n) = \poly(n) \ge n^2\), the smaller the harder LWE
\(q(n) = n^{\poly(n)}\) so that we can write an element in \(\poly(n)\) bits
\(\psi\) a distribution such that \(\exists B = \poly(n)\), \(\psi\) outputs \(|a| \le B\) except with negligible probability (think \(\psi\) as a uniform dist); the larger \(B\) compared to \(q\) the harder LWE

NIST candidate, CRYSTALS-Kyber:

\(n = 256 k\) for \(k=3,4,5\)
\(q = 3329\) so it can be viewed as \(O(n)\)
\(\psi\) binomial distribution, 3 or 5 fair trials (so that \(\card{e} \le 5\))

Ref: [KL, 14.3]. Remark: Chen [Chen24] recently posted a quantum algorithm that solves LWE for a large range of parameters in polynomial time. It is not yet fully reviewed, but it would be a huge breakthrough if it holds.

Corollary:

Suppose that \(\gcd(q,2)=1\). If \((A, A \cdot \vec s + \vec e) \approx (A, \vec t)\), then \((A, A \cdot \vec s + 2\cdot \vec e) \approx (A, \vec t)\).

Proof idea: because \(q\) and \(2\) coprime, \(A\) and \(2A\) are identically distributed (and \(\vec t\) as well).

An Encryption Based on LWE

If we have the following decisional \(\LWE_{m,q,\psi}\):

\[\set{A, A\cdot \vec{s} + \vec{e}} \approx \set{A, \vec{t}}\]

Then, given \(d \in \N\) such that \(\gcd(d,q) = 1\), we also have

\[\set{A, A\cdot \vec{s} + d\vec{e}} \approx \set{A, \vec{t}}\]

by a simple reduction that multiply the former indisintinguishability by \(d\) (because \(A\) and \(\vec{t}\) are both uniform over \(\Z_q\)). Hence, we can construct a secret-key encryption.

Construction: Secret Key Encryption from LWE

Parameters: \(m, q \in \N\) as a function of \(n\), \(\psi\) a distribution over \(\Z_q\).
\(\Gen(1^n)\): output \(k := \vec{s} \gets \psi^n\)
\(\Enc_k(m)\): for binary message \(m\in\bit\), sample \(\vec{a} \gets \Z_q^n\) and \(e \gets \phi\), output \(c:=(\vec{a}, t = \vec{a} \cdot \vec{s} + 2e + m)\) (where the arithmetic is in \(\Z_q\)).
\(\Dec_k(c)\): output
\[m' := (t - \vec{a}\cdot \vec{s} \mod 2)\]

The correctness is direct. The (secret-key) CPA security can be proved by a reduction \(R\) such that when ever the adversary \(A\) of the encryption asks for an encryption, \(R\) takes the next row from its LWE input and adds \(m\). The details are skipped here.

Homomorphic Encryption

Definition: Homomorphic encryption.

A (public or secret key) encryption scheme \((\Gen,\Enc,\Dec)\) is said to be homomorphic if the scheme provides efficient \((\Add, \Mul)\) operations that satisfies the syntax and correctness below.
Addition: For any messages \(m_0, m_1 \in \Z_2\),
\[\Pr_k[(c_i \gets \Enc_k(m_i))_{i\in\bit}; \Dec_k(\Add(c_0, c_1)) = m_0+m_1] = 1\]
Multiplication: For any messages \(m_0, m_1 \in \Z_2\),
\[\Pr_k[(c_i \gets \Enc_k(m_i))_{i\in\bit}; \Dec_k(\Mul(c_0, c_1)) = m_0 \cdot m_1] = 1\]
Multiply by constant: For any messages \(m_0, m_1 \in \Z_2\),
\[\Pr_k[c_0 \gets \Enc_k(m_0); \Dec_k(\Mul(c_0, m_1)) = m_0 \cdot m_1] = 1\]
Here, the message space and the arithmetic operations \((+,\cdot)\) are in \(\Z_2\), but one may define similarly for other algebra.

[Ref: Barak@Prinecton]

Discuss:

The homomorphic operations are requiring correctness (not security).
We sometimes relax the correctness to “except for negligible probability” due to technical construction
The above definition requires only for “single hop” homomorphic operation, which means that \(\Add\) and \(\Mul\) work for ciphertexts freshly encrypted by \(\Enc\)
We may define \(t\)-hop operations so that \(\Add\) or \(\Mul\) work also for ciphertexts that is output by \((t-1)\)-hop \(\Add\) or \(\Mul\)
Ideally, we want unlimited-hop operations. Since \((+,\cdot)\) in \(Z_2\) implement logical (XOR, AND), that enables any boolean-circuit computation on ciphertexts. This is called Fully Homomorphic Encryption, FHE.
Less ideally, if a scheme achieves unlimited-hop \(\Add\) and multiply by constant, it is called additive homomorphic encryption (or linear homomorphic), which is still useful.
Notice that \(\Add\) and \(\Mul\) need no key, which is essential (otherwise, given key, it would be trivial to perform the operations)
It is trivial and cheating to output \((\text{add}, c_0, c_1)\) as the output of \(\Add\) (and \(\Mul\) resp.), which belows up the size of the ciphertext and leaves the actual arithmetic to \(\Dec\). However, in the above 1-hop Add and Mul definition, we have no way to require it (because one may always pad \(c_0\) with unused bits). For more-than-constant hops, we require the ciphertext size to be small even after homomorphic evaluations.

Alternatively, we define homomorphic encryption with respect to a class of circuits \(\cC\).

Definition: Homomorphic encryption for class \(\cC\).

Let \(\cC\) be a class of circuits. We say the encryption scheme \((\Gen,\Enc,\Dec, \Eval)\) is homomorphic for \(\cC\) if for any \(C \in \cC\), for any \(m_1,...,m_\ell \in \bit\) where \(\ell\) is the input size of \(C\), let \(k \gets \Gen(1^n)\) be the encryption key,
\[\Pr_k[(c_i \gets \Enc_k(m_i))_{i\in[\ell]}; \Dec_k(\Eval(C, c_1,..., c_\ell)) = C(m_1,...,m_\ell)] = 1.\]
Moreover, we require the output size of \(\Eval\) to be compact, that is, bounded by \(\ell' \cdot |\Enc_k(m_i)|\).

The above secret-key encryption based on LWE has a direct homomorphic addition. To see why, consider two ciphertexts that is encrypted using the same key \(\vec{s}\),

\[c_0:=(\vec{a}_0, \vec{a}_0 \cdot \vec{s} + 2e_0 + m_0), c_1:=(\vec{a}_1, \vec{a}_1 \cdot \vec{s} + 2e_1 + m_1)\]

The homomorphic \(\Add(c_0, c_1)\) is defined to be the coordinate-wise addition:

\[\begin{align*} \Add(c_0, c_1) := & (\vec{a}_0+\vec{a}_1, (\vec{a}_0 \cdot \vec{s} + 2e_0 + m_0) + (\vec{a}_1 \cdot \vec{s} + 2e_1 + m_1))\\ = & (\vec{a}', \vec{a}'\cdot \vec{s} + 2e' + (m_0+m_1)) \end{align*}\]

where \(\vec{a}' = \vec{a}_0+\vec{a}_1\) and \(e' = e_0+e_1\). The multiplication by constant is also coordinate-wise multiply by plaintext \(m_1\). Notice that the correctness holds as long as \(e' \le q / 4\) before modulo \(q\). Hence, we can perform \(O(q/B)\) operations (of addition or multiply by constant) and then still obtain the correct decrytion, where \(B, q\) are the LWE parameters.

Multiplicative homomorphism is more involved. We show an application of additive homomorphic encryption first.

Definition: Public-key encryption.

\((\Gen,\Enc,\Dec)\) is said to be a public-key encryption scheme if the following syntax, correctness, and security holds.
\(\Gen\) is a PPT algorithm, \((\pk, \sk) \gets \Gen(1^n)\)
\(\Enc\) is a PPT algorithm, for all \(\pk\) and all \(m \in \bit\), \(c \gets \Enc_\pk(m)\)
\(\Dec\) is a deterministic algorithm, for all \(\sk\) and \(c\), \(m \gets \Dec_\sk(c)\) such that \(m \in \bit \cup \bot\).
Correctness: For all \(n \in \N, m \in \bit\),
\[\Pr[(\pk, \sk) \gets \Gen(1^n) : \Dec_\sk(\Enc_\pk(m)) = m] = 1.\]
Security: For all NUPPT \(D\), there exists a negligible function \(\eps(\cdot)\) such that for all \(n\in\N\), \(m_0, m_1 \in \bit\), \(D\) distinguishes between the following distributions with probability at most \(\eps(n)\):
\[\set{(\pk, \sk)\gets \Gen(1^n) : (\pk, \Enc_\pk(m_0))}_n\]
\[\set{(\pk, \sk)\gets \Gen(1^n) : (\pk, \Enc_\pk(m_1))}_n\]

With this definitions, there are some immediate impossibility results:

Perfect secrecy is impossible: given \(\pk\), the adversary can try to encrypt all messages with all randomness.
Deterministic encryption is also impossible: the adversary can try to encrypt the same message and get the same ciphertext.

Also, IND-CPA security is implied directly. Yet another difference compared to secret-key encryptions is that, the security for long messages is implied directly.

Lemma:

If \((\Gen,\Enc,\Dec)\) is secure public-key encryption, then for any polynomial length \(\ell(\cdot)\), for all \(m_0, m_1 \in \bit^{\ell(n)}\), the two distributions are indistinguishable:
\[\set{(\pk, \sk)\gets \Gen(1^n) : (\pk, \Enc_\pk(m_{0,1}), ..., \Enc_\pk(m_{0,\ell(n)}))}_n\]
\[\set{(\pk, \sk)\gets \Gen(1^n) : (\pk, \Enc_\pk(m_{1,1}), ..., \Enc_\pk(m_{1,\ell(n)}))}_n\]
where \(m_{b,i}\) denotes the \(i\)-th bit of \(m_b\).

Similarly, the adversary may adaptively choose \((m_{0,i}, m_{1,i})\) depending on the earlier ciphertexts \(\Enc_\pk(m_{b,i'}), i' \lt i\). That is called multi-message security and is also implied by single-message security.

Ref: [KL, 12.2]

Public-key Encryption from Additive Homomorphic Encryption

Theorem:

Let \((G, E, D, \Eval)\) be a secret-key homomorphic encryption scheme for the class of functions \(C_\ell := \set{f_z : z \in \bit^\ell}\), where \(\ell:=\ell(n)\) is a polynomial of the security parameter \(n\) (given to \(G\) to be chosen later), and \(f_z(x) := z \odot x\) is the inner product of \(\ell\)-bit vetors. Then, the following \((\Gen, \Enc, \Dec)\) is a public-key encryption scheme.
\(\Gen(1^n)\): let \(\sk = k \gets G(1^n)\), sample \(\ell\)-bit string \(r=(r_1,...,r_\ell) \gets \bit^\ell\) such that \(r \neq 0^\ell\), compute \(R_1 \gets E_k(r_1), ..., R_\ell \gets E_k(r_\ell)\). Output \(\sk\) and public key
\[\pk = (r_1,..., r_\ell, R_1, ..., R_\ell).\]
\(\Enc_\pk(m)\): sample \(\ell\)-bit string \(u\) uniformly at random subject to \(r \odot u = m\). Output
\[c := \Eval(f_u, R_1, ..., R_\ell).\]
\(\Dec_\sk(c)\): it simply output \(D_\sk(c)\).

The correctness follows because \(r \neq 0^\ell\) implies the existence of \(u\) and then by the correctness of \(\Eval\). The efficiency is also direct (just need to sample \(r\) and \(u\) with care). The security is sketched below.

Firstly, consider the hybrid scheme that in \(\Gen\), we encrypt \(R_i \gets \Enc_k(0)\) instead of \(\Enc_k(r_i)\). By CPA security of \((G,E,D)\), the hybrid is indistinguishable from the real \((\Gen, \Enc, \Dec)\).
Secondly in the hybrid, the ciphertext \(c\) is an encryption of 0 of the \((G,E,D)\) scheme. It could be attemping to think we were finished, but unfortunately as written, \(c\) depends on \(u\) and then \(u\) still depends on \(m\) (indeed, different encryptions of 0 might reveal something). The key is to observe that \(|c|\) is short. Let \(t:=|c|\) be the output size of \(\Eval\), and let \(\ell := 4t\). Informally, \(t\)-bit \(c\) is too short to provide sufficient information about \(4t\)-bit \(u\), and then it is information-theoretically hiding \(m\).
To be formal, we use Leftover Hash Lemma as follows. Let \(F_R(\cdot) := \Eval(f_{(\odot)}, R_1, ...,R_n)\). Then, the adversary is given \((r, F_R(u))\) and aims to find \(m = r \odot u\), where \(R\) is independent of \(r\). Rewrite the computation as
\[Ext(r, u) := (r, m = r \odot u).\]
Notice that \(r\) is uniform, \(u\) is subject to \(m\), and \(F_R(u)\) is \(t\)-bits. Observe that, subject to any fixed \(m\) (of 1 bit) and any fixed \(F_R(u)\) (of \(t=\ell/4\) bits), the min-entropy of \(u\) is at least \(\ell/2\). Because \(h_r(u) = r \odot u\) is a universal hash family, by leftover hash lemma, \(r \odot u\) is statistically close to a uniform bit by statistical difference \(2^{-\Omega(\ell/2)}\).

Ref: Rothblum, TCC 2011, Homomorphic Encryption: From Private-Key to Public-Key

Homomorphic Multiplication

Consider the encryption scheme for LWE. The ciphertext has format \(c = (\vec a, b)\), and the decrytion is done by \(b - \vec a^T \vec s\), where \(\vec s\) is the decryption key, and \(\mod 2\) is skipped. We can view the decryption as a function,

\[f_{\vec a, b}(\vec x) := b - \vec a^T \vec x,\]

and the decryption is to compute \(f_{\vec a, b}(\vec s)\). To obtain additive homomorphism, we are exactly performing

\[f_{\vec a', b'}(\vec x) := f_{\vec a_1, b_1}(\vec x) + f_{\vec a_2, b_2}(\vec x),\]

so that the decryption works as

\[f_{\vec a', b'}(\vec s) = f_{\vec a_1, b_1}(\vec s) + f_{\vec a_2, b_2}(\vec s).\]

For multiplication, we would like to perform similarly,

\[f_{\vec a', b'}(\vec x) := f_{\vec a_1, b_1}(\vec x) \cdot f_{\vec a_2, b_2}(\vec x).\]

This would give a correct decrytion when we compute \(f_{\vec a', b'}(\vec s)\). However, we begin with linear functions \(f_{\vec a_1, b_1}\), but we end with

\[(b_1 - \vec a_1^T \vec x)\cdot(b_2 - \vec a_2^T \vec x) = b_1 b_2 - (b_1 \vec a_2^T + b_2 \vec a_1^T) \vec x + \sum_{i,j} a_{1,i} a_{2,j} x_i x_j,\]

a quadratic function (degree 2 of \(\vec x\)). Moreover, the ciphertext size goes from \(n+1\) to ~\(n^2\). This is far from ideal, and there are many techniques to resolve this. One is known as “relinearization”, and the trick is to provide the encryptions of

\(s_i\) for \(i \in [n]\), where \(s_i\) is the \(i\)-th coordinate of \(\vec s\).
\(s_{i}\cdot s_{j}\) for \(i,j\in[n]\).

Since the encryptions (of the above values \(s_i\cdot s_j\)) are another linear function \(g_{c_{i,j}}(\vec y)\), the homomorphic multiplication can then

Compute \(f_{\vec a', b'}\) from \(f_{\vec a_1, b_1}\) and \(f_{\vec a_2, b_2}\)
In \(f_{\vec a', b'}\), substitute \(x_i x_j\) with \(g_{c_{i,j}}(\vec y)\), which yields \(f_{\vec a'', b''}(\vec y)\), a linear function of size \(n+1\).

The encryption of \(s_{i}\cdot s_{j}\) are given as public evaluation key in such constructions.

Notice This relinearization is oversimplified. Actually, the \(b_1 b_2\) term in \(f_{\vec a', b'}(\vec x)\) consists of products, such as \(\vec a_1^T \vec s \cdot 2e_2\). The naive representation of \(\vec a_1^T \vec s\) is a large integer of order \(q\), but that would incur huge error and incorrect decryption. A careful bitwise decomposition of \(b_1\) and \(b_2\) will reduce the error factor to \(\poly(n, \log q)\).

Ref:

Fully Homomorphic using Bootstrapping

The above homomorphic addition and multiplication (over \(Z_2\)) are amazing because we can continue to perform the operations on the ciphertexts repeatedly even when the ciphertexts comes from the previous homomorphic operation. That correctness / functionality hold as long as the error term \(2e\) is \(\lt q/4\) before modulo \(q\).

The homomorphic addition incurs a small error: each \(\Add\) introduces a factor at most \(2\). Starting with \(B = \poly(n)\) and \(q = n^{\poly(n)}\), we have the budget to perform many additions.

However, the homomorphic multiplication incurs a much higher error due to the relinearization. Since the error increases exponentially in the number of operations, the \(\lt q/4\) bar is indeed a critical limitation.

To this end, Gentry (STOC 2009) introduced the idea of “bootstrapping.” That is, to view the high-error ciphertext as a constant string, the decryption algorithm plus the string as a circuit, and then the decryption key as the input of the circuit. That is, define

Circuit \(f_{\Dec, c}(x)\): it takes input \(x\) and then run and ouput \(\Dec_x(c)\).

Then, we can encrypt the decryption key \(k_1\) using another key \(k_2\), and call the result as the evaluation key \(\ek := \Enc_{k_2}(k_1)\). With that, we can homomorphically perform decryption on high-error \(c\):

\(c_2 \gets \Eval(f_{\Dec, c}, \ek = \Enc_{k_2}(k_1))\).

The above \(c_2\) can be later decrypted by \(k_2\) as

\[\Dec_{k_2}(c') = f_{\Dec,c}(k_1) = \Dec_{k_1}(c),\]

where the first equality holds if the error incurred by \(f_{\Dec,c}\) is smaller than \(O(q/B)\). Thus, we need a homomorhic encryption scheme such that

The circuit depth of \(\Dec\) is bounded by small number \(d\)
The \(\Eval\) works correctly for circuit depth more than \(d+1\) (because we want to perform at least one operation on \(c_2\))

The LWE-based encryption has a simple \(\Dec\), which computes firstly a linear operation and secondly a modulo \(q\) division, which takes circuit depth \(\poly(\log q)\). This is why we want to minimize the error of homomorphic multiplication.

With that, we can generate a chain of keys \(k_1, k_2, ..., k_\ell\) such that each key encrypts the previous one, so that we can perform many operations. This is known as “leveled homomorphic encryption.”

Alternatively, let \(k_1 = k_2 = k\), and thus \(\ek = \Enc_{k}(k)\), which is known as “bootstrapping.” Because \(\ek\) is given to the evaluator (to perform \(\Eval\)), \(\ek\) is essentially public and known to any adversaries. That introduces a significant concern in security: we do not know how to prove or disprove that the encryption is still secure given such \(\Enc_{k}(k)\), and this is called “circular security” and used as an additional assumption. Circular security gives a fully homomorphic encryption (and it is still the only known way).

Application: Private Information Retrieval

Definition: Private information retrieval (PIR)

A protocol \((\Query, \Resp, \Dec)\) is a PIR if it satisfies the following syntax, correctness, and privacy. Let \(N\) be the size of a database \(x\) such that \(|x| = N\).
\[(q,\st) \gets \Query(i)\]
\[a \gets \Resp(x, q)\]
\[b \gets \Dec(\st, a)\]
Correctness: for all \(x\), all \(i\in[N]\),
\[\Pr[(q,\st) \gets \Query(i), a \gets \Resp(x, q) : \Dec(\st, a) = x_i] = 1\]
Privacy: for all \(x\), all \(i_0, i_1 \in [N]\), for all NUPPT \(D\), the following are indistinguishable:
\[\set{(q_0,\st_0) \gets \Query(i_0) : q_0} \approx \set{(q_1,\st_1) \gets \Query(i_1) : q_1}\]
Moreover, we define the computation time and the communication to be the time and output size of \(\Query, \Resp, \Dec\).

There are many settings of PIR, for example, the database \(x\) and thus the query may be answered by multiple servers. We consider the simpler single server setting here. There is trivial solutions, but we want the protocol to be efficient. Using a somewhat homomorphic encryption, we can construct an efficient PIR with short communication and client time.

Ref: Lin, Mook, Wichs, Doubly Efficient Private Information Retrieval and Fully Homomorphic RAM Computation from Ring LWE. STOC 2023.