Paper 2019/364
Everybody's a Target: Scalability in PublicKey Encryption
Benedikt Auerbach, Federico Giacon, and Eike Kiltz
Abstract
For $1\leq m \leq n$, we consider a natural $m$outof$n$ multiinstance scenario for a publickey encryption (PKE) scheme. An adversary, given $n$ independent instances of PKE, wins if he breaks at least $m$ out of the $n$ instances. In this work, we are interested in the scaling factor of PKE schemes, $\mathrm{SF}$, which measures how well the difficulty of breaking $m$ out of the $n$ instances scales in $m$. That is, a scaling factor $\mathrm{SF}=\ell$ indicates that breaking $m$ out of $n$ instances is at least $\ell$ times more difficult than breaking one single instance. A PKE scheme with small scaling factor hence provides an ideal target for mass surveillance. In fact, the Logjam attack (CCS 2015) implicitly exploited, among other things, an almost constant scaling factor of ElGamal over finite fields (with shared group parameters). For Hashed ElGamal over elliptic curves, we use the generic group model to argue that the scaling factor depends on the scheme's granularity. In low granularity, meaning each public key contains its independent group parameter, the scheme has optimal scaling factor $\mathrm{SF}=m$; In medium and high granularity, meaning all public keys share the same group parameter, the scheme still has a reasonable scaling factor $\mathrm{SF}=\sqrt{m}$. Our findings underline that instantiating ElGamal over elliptic curves should be preferred to finite fields in a multiinstance scenario. As our main technical contribution, we derive new genericgroup lower bounds of $\Omega(\sqrt{m p})$ on the difficulty of solving both the $m$outof$n$ Gap Discrete Logarithm and the $m$outof$n$ Gap Computational DiffieHellman problem over groups of prime order $p$, extending a recent result by Yun (EUROCRYPT 2015). We establish the lower bound by studying the hardness of a related computational problem which we call the searchbyhypersurface problem.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 A major revision of an IACR publication in EUROCRYPT 2020
 Keywords
 publickey cryptographykey encapsulationmass surveillanceellipticcurve cryptography (ECC)generic group model
 Contact author(s)

benedikt auerbach @ ist ac at
federico giacon @ rub de
eike kiltz @ rub de  History
 20200508: last of 2 revisions
 20190410: received
 See all versions
 Short URL
 https://ia.cr/2019/364
 License

CC BY
BibTeX
@misc{cryptoeprint:2019/364, author = {Benedikt Auerbach and Federico Giacon and Eike Kiltz}, title = {Everybody's a Target: Scalability in PublicKey Encryption}, howpublished = {Cryptology {ePrint} Archive, Paper 2019/364}, year = {2019}, url = {https://eprint.iacr.org/2019/364} }