Cryptology ePrint Archive: Report 2020/1436

Removable Weak Keys for Discrete Logarithm Based Cryptography

Michael John Jacobson Jr. and Prabhat Kushwaha

Abstract: We describe a novel type of weak cryptographic private key that can exist in any discrete logarithm based public-key cryptosystem set in a group of prime order $p$ where $p-1$ has small divisors. Unlike the weak private keys based on numerical size (such as smaller private keys, or private keys lying in an interval) that will always exist in any DLP cryptosystems, our type of weak private keys occurs purely due to parameter choice of $p$, and hence, can be removed with appropriate value of $p$. Using the theory of implicit group representations, we present algorithms that can determine whether a key is weak, and if so, recover the private key from the corresponding public key. We analyze several elliptic curves proposed in the literature and in various standards, giving counts of the number of keys that can be broken with relatively small amounts of computation. Our results show that many of these curves, including some from standards, have a considerable number of such weak private keys. We also use our methods to show that none of the 14 outstanding Certicom Challenge problem instances are weak in our sense, up to a certain weakness bound.

Category / Keywords: public-key cryptography / public-key cryptography, elliptic curve cryptosystem, cryptographic protocols

Original Publication (in the same form): JOURNAL OF CRYPTOGRAPHIC ENGINEERING

Date: received 15 Nov 2020, last revised 24 Nov 2020

Contact author: prabkush at gmail com

Available format(s): PDF | BibTeX Citation

Note: This is a post-peer-review, pre-copyedit version of an article published in Journal of Cryptographic Engineering. The final authenticated version is available online at:

Version: 20201124:161006 (All versions of this report)

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