I am having trouble coming up with a use case for RSA or DSA. It appears that ECC is better in every way.
Is this true?
I am looking for cases where RSA/DSA is superior to ECC, not where it is used for historical reasons.
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3$\begingroup$ Tempted to close as dupe of this question. Is there anything that is not answered in the answers there? If so, you should edit to clarify. $\endgroup$– otusFeb 24, 2016 at 8:10
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$\begingroup$ By themselves, elliptic curves are useless. It is when certain algorithms are done over elliptic curves that they become useful. ECC is not a replacement for DSA, but ECDSA is. $\endgroup$– MelabFeb 26, 2016 at 6:53
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2 Answers
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There are three use cases where RSA beats common ECC algorithms, such as ECDSA:
Signature with verification frequent or/and by low-power devices or/and where the verification code needs to be small. The verification cost of $n$-bit RSA with usual public exponents is $O(n^2)$, but the verification cost of ECC-based signatures is $O(n^3)$ (using usual algorithms). Together with simpler math, that's why RSA can be way over 10 times faster for signature verification at usual security levels, even though it must use a larger $n$ for equivalent security level. RSA verification also requires significantly less code than ECC computations, which makes it attractive, for example, in ROM code where space is expensive and a bug can't be patched.
Similarly, encryption by low-power devices or/and with decryption comparatively rare.
Need to minimize the size overhead of adding a signature; using signature with message recovery, that can be 34 bytes for RSA (using SHA-256 hash, ISO 9796-2 mode 3 or the deprecated mode 1, for messages at least 222 bytes before signature at the 2048-bit security level), versus 64 bytes for ECDSA for comparable security.
RSA is thus a good choice (and indeed still the dominant one, I believe) for signing public-key certificates; beside inertia, in the internet domain that's mostly for reason 1 (certificates are verified often), but in the Smart Card and payment industry reason 3 adds up.
Additional arguments for RSA (vs ECC) are
- RSA was first there / is the most time-proven, and became an industry standard.
- Simplicity. RSA signature verification is much easier to code, and get right, than ECDSA signature verification.
- RSA has long been clearly patent-free. But that now applies to ECDSA as well, and arguably applied all along and indisputably since August 2014, at least for curves in a prime field; e.g. ECDSA on secp256r1 and EdDSA on Curve25519 are patent unencumbered.
- Perhaps (but see these comments) slightly more quantum resilience; that is, at comparable level of resistance versus non-quantum attacks, RSA arguably would fall after ECC if it ever emerged quantum computers usable for cryptanalysis; see this other answer and section 5.4 of its source; note that Koblitz and Menezes are not making any strong statement, rather, their intro is (emphasis mine):
We next examine some conjectures about the NSA’s motives in its PQC announcement (..)
The NSA believes that RSA-3072 is much more quantum-resistant than ECC-256 and even ECC-384. (..)
Note: this answer does not touch use cases where ECC is preferable, or its virtues.
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3$\begingroup$ Case 2 could occur in a sensor grid scenario where the encryption is on embedded systems but decryption is on a much more powerful server. $\endgroup$– DemiFeb 24, 2016 at 18:26
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$\begingroup$ OTOH, using RSA signature and key generation on smart card devices really can bog things down, and using the more accpted / proliferate RSA PKCS#1 extends the signature size to less acceptable levels. $\endgroup$– Maarten Bodewes ♦Feb 28, 2016 at 20:02
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3$\begingroup$ I think citing improved "quantum resilience" is a bit irresponsible, since both algorithms should be considered utterly broken in the presence of scalable quantum computing. While there may be a gap, the authors themselves suggest "It is not likely that the gap between quantum cryptanalysis of a 384-bit key and a 3072-bit key will be great enough to serve as a basis for a cryptographic strategy." This should be interpreted as quantum resilience NOT being a reason to use RSA over ECC algorithms. $\endgroup$– bkjvbxOct 18, 2016 at 9:40
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1$\begingroup$ @bkjvbx I interpret this as “yes, both will fall to a quantum computer, but ECDSA will fall first, which might give a bit more warning to switch.” Of course, one should be switching to post-quantum schemes ASAP, but in some cases (such as code signing), all that matters is that I have switched to a post-quantum scheme before the original scheme is broken. As long as RSA holds out long enough for a successor to be deployed, software signatures based on it are safe. $\endgroup$– DemiDec 11, 2021 at 6:46
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$\begingroup$ Note that an EC Schnorr signature only requires 48 bytes, since it is unnecessary for the hash to produce a challenge greater than 128 bits. Signing keypair is (x, xG). Signature is (c,r), VER is c =?= hash_to_128_bits(rG + cX || m), SIG is choose random k, r = c - kx. c = hash_to_128_bits(kG || m). $\endgroup$– knacccDec 11, 2021 at 11:24
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If practical quantum computers become a reality, the larger bitlengths of RSA keys would make them more quantum-resistant than their ECC counterparts. See section 5.4 of this Koblitz & Menezes paper
