On the Adaptive Security of Free-XOR-Based Garbling Schemes in the Plain Model

A Garbling Scheme is a fundamental cryptographic primitive, with numerous theoretical and practical applications. Since its inception by Yao (FOCS’82, ’86), optimizing the communication and computation complexities of securely garbling circuits has been an area of active research. One such optimization, and perhaps the most fundamental, is the ‘Free-XOR’ technique (Kolesnikov and Schneider, ICALP’08) which allows XOR gates in a function garbling to not require representation, and therefore communication. Since then, several works have designed and analysed the security of schemes that adopt the Free-XOR optimisation. In particular: (1) Applebaum (JoC’16) proved that this can be securely instantiated assuming symmetric-key encryption satisfying a notion called RK-KDM security; and (2) Zahur, Rosulek and Evans (Eurocrypt’15) proposed the so-called ‘Half Gates’ scheme, and proved that it can be instantiated assuming hash functions satisfying a notion called CCR security. Although both schemes have been proven selectively secure, prior work leaves it open to analyze whether they satisfy a stronger security notion – adaptive security – in the plain model. In this work, we formally show that the selective security of these two schemes cannot be lifted to adaptive security under the same assumptions. To establish these barriers, we adopt techniques from the work of Kamath et al. (Crypto’21), who proved similar negative results for Yao’s garbling. We use that as a starting point and introduce new techniques tailored towards addressing Free-XOR-based schemes.