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Out of my head, I propose this EC-ElGamal variation on a standard 256-bit Elliptic Curve, using hash-and-XOR for the encryption part (nearly halving ciphertext size over textbook EC-ElGamal), and a simple work/space trade-off to further squeeze into a 32-byte ciphertext a message $m$ of $b$ bits, with small fixed $b$ (e.g. $b=8$).
<ul>
<li>Use a standard Elliptic Curve over field $\mathbb F_p$ with generator $G$ of order $n$, with $p$ and $n$ 256-bit primes and (conjectured) 128-bit security. <a href="https://www.secg.org/sec2-v2.pdf#subsubsection.2.4.2" rel="nofollow noreferrer">secp256r1</a> (aka NIST P-256) and <a href="https://www.secg.org/sec2-v2.pdf#subsubsection.2.4.1" rel="nofollow noreferrer">secp256k1</a> qualify. I refer to <a href="https://www.secg.org/sec1-v2.pdf#section.2" rel="nofollow noreferrer">Sec1v2</a> for ECC math and notation.</li>
<li>Key generation:
<ul>
<li>Choose private key $d$ randomly in the interval $[1,n)$</li>
<li>Compute public key $Q\gets d\,G$</li>
</ul>
</li>
<li>Encryption of $b$-bit plaintext $m$
<ul>
<li>Choose $e$ randomly in the interval $[1,2^{32})$</li>
<li>Compute $F\gets e\,G$</li>
<li>Choose $k$ randomly in the interval $[1,n)$</li>
<li>Compute $R\gets k\,G$</li>
<li>While $R_x\not\equiv0\pmod{2^b}$
<ul>
<li>Compute $R\gets R+F$ and $k\gets k+e$, which maintains $R=k\,G$</li>
<li>If $k\ge n$ the RNG that chose $k$ is most likely broken; abort or restart encryption</li>
</ul>
</li>
<li>If $R_y$ is odd, set $R_y\gets p-R_y$ and $k\gets n-k$, which maintains $R=k\,G$</li>
<li>Compute $S\gets k\,Q$ (the shared secret point)</li>
<li>Compute $u\gets \operatorname{SHA-256}(S_x)\bmod2^b$, where $S_x$ is expressed as a 32-byte bytestring</li>
<li>Compute $v\gets u\oplus m$</li>
<li>Output ciphertext of 256-bit, consisting of $256-b$ bits for $R_x$ with the low-order $b$ bits removed (they are zero by construction), and $b$ bits for $v$</li>
</ul>
</li>
<li>Decryption:
<ul>
<li>From the ciphertext extract $R_x$ (inserting $b$ low-order bits at zero) and $v$</li>
<li>Compute an $R_y$ matching $R_x$ (that's a standard technique to undo point compression, see <a href="https://www.secg.org/sec1-v2.pdf#subsubsection.2.3.4" rel="nofollow noreferrer">Sec1v2 §2.3.4</a>, step 2.4.1); if that fails, abort decryption.</li>
<li>If $R_y$ is odd, set $R_y\gets p-R_y$</li>
<li>Compute $S\gets d\,R$ (the shared secret point)</li>
<li>Compute $u\gets \operatorname{SHA-256}(S_x)\bmod2^b$, where $S_x$ is expressed as a 32-byte bytestring</li>
<li>Compute and output the deciphered plaintext $m\gets u\oplus v$</li>
</ul>
</li>
</ul>
I conjecture <a href="https://en.wikipedia.org/wiki/Ciphertext_indistinguishability#Indistinguishability_under_chosen_ciphertext_attack/adaptive_chosen_ciphertext_attack_(IND-CCA1,_IND-CCA2)" rel="nofollow noreferrer">IND-CCA1</a> (thus <a href="https://en.wikipedia.org/wiki/Ciphertext_indistinguishability#Indistinguishability_under_chosen-plaintext_attack_(IND-CPA)" rel="nofollow noreferrer">IND-CPA</a>) security to 128-bit level, under plausible assumptions (hardness of a variant of the decisional Diffie-Hellman problem for the Elliptic Curve, and SHA-256 modeled as a random oracle). There is a trivial attack against IND-CCA2 (thus access to a decryption oracle compromises the confidentiality of past messages even if the decryption oracle has blacklisted the original ciphertexts; this is a non-issue in practice).
Beware that the cipher is trivially malleable. This allows an adversary to flip desired bits in the plaintext by altering these bits in the ciphertext. This can be a practical issue. Some level of mitigation of that is possible by replacing $u\oplus\ldots$ by <a href="https://en.wikipedia.org/wiki/Format-preserving_encryption" rel="nofollow noreferrer">Format Preserving Encryption</a> with a wider $u$ as key; or/and more work or/and space.
The while loop at encryption is a time/space trade-off, searching $k$ by enumeration so that it's $x$ coordinate $R_x$ has $b$ low-order bits at zero that need not be transmitted. The search is optimized to require $\approx2^b$ point additions, but for $b$ starting about $8$ this is going to become a pain. Use of secret increment $e$ and matching $F=e\,G$ is a nice-to-have making the final $k$ more uniform (see note), but I know no attack if we take $e=1$ and thus $F=G$. Secret increment further could help mitigate side-channel attacks.
If we want to avoid some or all of that time/space trade-off, we can
<ul>
<li>trivially, increase the ciphertext size.</li>
<li>or/and venture into generating a custom curve, with bit size of $n$ and $p$ reduced by a few bits from 256. For each 1 bit of security we give up, we gain about 2 for $n$ and $p$, thus on ciphertext bits, or 4 times less guesswork in the while loop. And for secp256r1 the best known attacks arguably require $&gt;2^{140}$ bit operations (based on a conservative extrapolation of a similar <a href="https://ed25519.cr.yp.to/ed25519-20110926.pdf#page=2" rel="nofollow noreferrer">claim</a> of $&gt;2^{140}$ for Ed25519).</li>
</ul>
<hr />
Note: In textbook Diffie-Hellman or ElGamal, we choose $k$ uniformly in $[0,n)$. Our variant restricts to $k$ with $k\,G$ having an $x$ coordinate with it's low-order $b$ bits at zero. By a statistical argument, there are about $n/2^b$ such $k$. We'd want to choose $k$ uniformly in this subset of $[0,n)$. The simplest would be repeating for random $k$, but that has prohibitive computational cost for the point multiplication $R\gets k\,G$. We save on this by moving from one candidate $k$ to the next by a fixed increment $e$, allowing to update $R$ by a single point addition $R\gets R+F$.
If we used fixed $e=1$, then the $k$ selected by the while loop would have a distinguishable property (for one knowing $k$): the probability that a $k$ is selected is proportional to $j$ computable from $k$ as the smallest $j&gt;0$ such that $k-j$ is in the subset (or $k+1$ if there's no such $j$, which occurs for a singe $k$ in the subset). The problem is similar to the non-uniform choice of prime obtained by picking a random starting point and searching for the next prime.
Choosing any public fixed $e$ does not solve the issue, because the defintion of $j$ can be adapted to $k-e\,j$ in the subset. However, choosing a random secret $e$ for each choice of $k$ does. A similar technique is used in random prime generators for RSA. I have no reference, but an argument is that an adversaries not knowing $e$ do not know what $e$ to use for a test; they can test for all $e$ and combine the outcomes, but then their test is overwhelmed with noise.
I picked the upper limit of the interval $[1,2^{32})$ for $e$ so that the computation $F\gets e\,G$ has marginal expected cost compared to $R\gets k\,G$. I am confident (without evidence or reference) this is more than enough to make the bias in the selection of $k$ practically undetectable even for hypothetical adversaries knowing $k$; and actual adversaries could only get $k$ thru a side channel.
<hr />
Final thought: we could damp any fear that the criteria $R_x\equiv0\pmod{2^b}$ creates a weakness by adjusting that to: the low $b$ bit of $R_x$ are some function (having about uniform distribution) of the other bits of $R_x$. A $b$-bit CRC would do. The receiver can reconstruct the necessary low-order bits of $R_x$ by applying this function.

Imagine that one needs to periodically encrypt very short messages (i.e., a boolean Yes/No, a single byte, or 3-4 bytes in the worst case). We assume that there is no session, and we just need to encrypt under a receiver's public key (i.e., posting an encrypted byte to the blockchain).
I'm aware of ElGamal, Cramer-Shoup, RSA, ECIES encryption etc. and I'm looking for the algorithm with the shortest output when encrypting tiny amount of information.
I want to maintain at least 128bits of security, thus wondering if an ElGamal variant, optimized for short messages, exists. We know that ElGamal over elliptic curves outputs 64 bytes ciphertexts (assuming we encrypt a message with length up to 32 bytes when using a 256bit elliptic curve).
Is there an asymmetric scheme in the literature that could output less bytes than this? ideally 30-32 bytes?

Asymmetric encryption scheme with the shortest output for encrypting 1 byte of information

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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功能1上新10个字符

功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符功能2描述100个字符。

功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符功能2上新100个字符。

功能5描述100个字符功能5描述100个字符功能5描述100个字符功能5描述100个字符功能5描述100个字符功能5描述100个字符

功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符功能5上新100个字符

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假设需要定期加密非常短的消息(例如，布尔值是/否，单个字节，或者在最坏的情况下是3-4字节)。我们假设没有会话，我们只需要在接收者的公钥下加密(即将加密的字节发送到块链)。我知道ElGamal，Cramer，RSA，ECIES加密等，我正在寻找算法的最短输出时，加密少量的信息。我想维护至少128位的安全性，因此想知道是否存在针对短消息进行优化的ElGamal变体。我们知道，椭圆曲线上的ElGama

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