Abstract
We present Lyra, a password-based key derivation scheme based on cryptographic sponges. Lyra was designed to be strictly sequential (i.e., not easily parallelizable), providing strong security even against attackers that use multiple processing cores (e.g., custom hardware or a powerful GPU). At the same time, it is very simple to implement in software and allows legitimate users to fine-tune its memory and processing costs according to the desired level of security against brute force password guessing. We compare Lyra with similar-purpose state-of-the-art solutions, showing how our proposal provides a higher security level and overcomes limitations of existing schemes. Specifically, we show that if we fix Lyra ’s total processing time \(t\) in a legitimate platform, the cost of a memory-free attack against the algorithm is exponential, while the best-known result in the literature (namely, against the scrypt algorithm) is quadratic. In addition, for an identical same processing time, Lyra allows for a higher memory usage than its counterparts, further increasing the cost of brute force attacks.
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Acknowledgments
This work was supported by National Counsel of Technological and Scientific Development (CNPq) under grants 482342/2011-0, 473916/2013-4, under productivity research grant 303163/2009-7, as well as by the São Paulo Research Foundation (FAPESP) under grant 2011/21592-8.
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Appendices
Appendix A: PBKDF2
The Password-Based Key Derivation Function version 2 (PBKDF2) algorithm [26] was originally proposed in 2000 as part of RSA Laboratories’ PKCS#5. It is nowadays present in several security tools, such as TrueCrypt [46] and Apple’s iOS for encrypting user passwords [2], and has been formally analyzed in several circumstances [6, 48].
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Basically, PBKDF2 (see Algorithm 3) iteratively applies the underlying pseudorandom function \(Hash\) to the concatenation of \(pwd\) and a variable \(U_i\), i.e., it makes \(U_i = Hash(pwd, U_{i-1})\) for each iteration \(1 \leqslant i \leqslant {T}\). The initial value \(U_0\) corresponds to the concatenation of the user-provided \(salt\) and a variable \(l\), where \(l\) corresponds to the number of required output blocks. The \(l\)-th block of the \(k\)-long key is then computed as \(K_l = U_1 \, \oplus \,U_2 \, \oplus \,\cdots \, \oplus \,U_{{T}}\), where \(k\) is the desired key length.
PBKDF2 allows users to control its total running time by configuring the \({T}\) parameter. Since the key derivation process is strictly sequential (one cannot compute \(U_i\) without first obtaining \(U_{i-1}\)), its internal structure is not parallelizable. However, as the amount of memory used by PBKDF2 is quite small, the cost of implementing brute force attacks against it by means of multiple processing units remains reasonably low.
Appendix B: Bcrypt
Another solution that allows users to configure the key derivation’s processing time is bcrypt [38]. The scheme is based on a customized version of the 64-bit cipher algorithm Blowfish [39], called EksBlowflish (“expensive key schedule blowfish”).
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Both algorithms use the same encryption process, differing only on how they compute their subkeys and S-boxes. Bcrypt consists in initializing EksBlowfish’s subkeys and S-Boxes with the salt and password, using the so-called EksBlowfishSetup function, and then using EksBlowfish for iteratively encrypting a constant string, 64 times.
EksBlowfishSetup starts by copying the first digits of the number \(\pi \) into the subkeys and S-boxes \(S_i\) (see Algorithm 4). Then, it updates the subkeys and S-boxes by invoking \(ExpandKey(salt, pwd)\), for a 128-bit salt value. Basically, this function (1) cyclically XORs the password with the current subkeys, and then (2) iteratively blowfish-encrypts one of the halves of the salt, with the resulting ciphertext being XORed with the salt’s other half and also replacing the next two subkeys (or S-Boxes, after all subkeys are replaced). After all subkeys and S-Boxes are updated, bcrypt alternately calls \(ExpandKey(0, salt)\) and then \(ExpandKey(0, pwd)\), for \(2^{T}\) iterations. The user-defined parameter \({T}\) determines, thus, the time spent on this subkey and S-Box updating process, effectively controlling the algorithm’s total processing time.
Like PBKDF2, bcrypt allows users to parameterize only its total running time. In addition to this shortcoming, some of its characteristics can be considered (small) disadvantages when compared with PBKDF2. First, bcrypt employs a dedicated structure (EksBlowfish) rather than a conventional hash function, leading to the need of implementing a whole new cryptographic primitive and, thus, raising the algorithm’s code size. Second, EksBlowfishSetup’s internal loop grows exponentially with the \({T}\) parameter, making it harder to fine-tune bcrypt’s total execution time without a linearly growing external loop. Finally, bcrypt displays the unusual (albeit minor) restriction of being unable to handle passwords having more than 56 bytes.
Appendix C: On the algorithm’s name
The name Lyra comes from Chondrocladia lyra, a recently discovered type of sponge [16]. While most sponges are harmless, this harp-like sponge is carnivorous, using its branches to ensnare its prey, envelope it in membrane and completely digest it.
Lyra ’s memory matrix displays some similarity with this species’ external aspect, and we expect it to be at least as much aggressive against adversaries trying to attack it.
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Almeida, L.C., Andrade, E.R., Barreto, P.S.L.M. et al. Lyra: password-based key derivation with tunable memory and processing costs. J Cryptogr Eng 4, 75–89 (2014). https://doi.org/10.1007/s13389-013-0063-5
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DOI: https://doi.org/10.1007/s13389-013-0063-5