TY - JOUR

T1 - On a relation between knowledge-of-exponent assumptions and the DLog vs. CDH question

AU - Kraiem, Firas

AU - Isobe, Shuji

AU - Koizumi, Eisuke

AU - Shizuya, Hiroki

N1 - Funding Information:
This work was supported in part by JSPS KAKENHI Grant Numbers 19K03612 and 19K11956.
Publisher Copyright:
Copyright © 2021 The Institute of Electronics, Information and Communication Engineers

PY - 2021/1/1

Y1 - 2021/1/1

N2 - Knowledge-of-exponent assumptions (KEAs) are a somewhat controversial but nevertheless commonly used type of cryptographic assumptions. While traditional cryptographic assumptions simply assert that certain tasks (like factoring integers or computing discrete logarithms) cannot be performed efficiently, KEAs assert that certain tasks can be performed efficiently, but only in certain ways. The controversy surrounding those assumptions is due to their non-falsifiability, which is due to the way this idea is formalised, and to the general idea that these assumptions are “strong”. Nevertheless, their relationship to existing assumptions has not received much attention thus far. In this paper, we show that the first KEA (KEA1), introduced by Damgård in 1991, implies that computing discrete logarithms is equivalent to solving the computational Diffie-Hellman (CDH) problem. Since showing this equivalence in the standard setting (i.e., without the assumption that KEA1 holds) is a longstanding open question, this indicates that KEA1 (and KEAs in general) are indeed quite strong assumptions.

AB - Knowledge-of-exponent assumptions (KEAs) are a somewhat controversial but nevertheless commonly used type of cryptographic assumptions. While traditional cryptographic assumptions simply assert that certain tasks (like factoring integers or computing discrete logarithms) cannot be performed efficiently, KEAs assert that certain tasks can be performed efficiently, but only in certain ways. The controversy surrounding those assumptions is due to their non-falsifiability, which is due to the way this idea is formalised, and to the general idea that these assumptions are “strong”. Nevertheless, their relationship to existing assumptions has not received much attention thus far. In this paper, we show that the first KEA (KEA1), introduced by Damgård in 1991, implies that computing discrete logarithms is equivalent to solving the computational Diffie-Hellman (CDH) problem. Since showing this equivalence in the standard setting (i.e., without the assumption that KEA1 holds) is a longstanding open question, this indicates that KEA1 (and KEAs in general) are indeed quite strong assumptions.

KW - Cryptographic assumptions

KW - Diffie-Hellman

KW - Discrete logarithm

KW - Knowledge-of-exponent

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U2 - 10.1587/transfun.2020CIP0002

DO - 10.1587/transfun.2020CIP0002

M3 - Article

AN - SCOPUS:85099182086

SP - 20

EP - 24

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 1

ER -