In this paper we propose an ElGamal type encryption scheme based on the concepts of public key cryptosystem over cubic finite field extension proposed by Gong and Ham (GH). The proposed encryption scheme is an ephemeral-static, which is useful in situations like email where the recipient may not be online. The security of the proposed encryption scheme depends on the difficulty of solving 3-LFSR-DLP, 3-LFSR-DHP and 3-LFSR-DDHP. It then provides secure message transmission by having also the property of semantic security. We also adapt an efficient double exponentiation algorithm for GH construction that helps us to propose the GH-Nyberg-Rueppel-type digital signature algorithm (GH-NR-DSA) with message recovery based on the proposed scheme. Moreover, we give some countermeasures for GH-NR-DSA to resist two well known forgery attacks, namely, congruence equation attack and homomorphism attack. (C) 2013 Elsevier B.V. All rights reserved.