Blind signatures have been widely adopted to construct untraceable electronic cash systems since they are both unlinkable and unforgeable. Although unlinkability protects the privacy of customers and users, it may be abused by criminals for such purposes as to launder money or to safely get a ransom. The techniques of fair blind signatures are developed to deal with the abuse of unlinkability. In this paper we propose a user efficient fair blind signature scheme which makes it possible for a government or a judge to recover the link between a signature and the instance of the signing protocol which produces that signature when the unlinkability property is abused. Only two integers are required to form a signature in the proposed fair blind signature scheme. Furthermore, it only takes several modular multiplications for a user to obtain and verify a signature. It turns out that the scheme is suitable for situations where computation capability of users or customers is limited, such as smart cards and mobile units. Compared with existing blind signature schemes proposed in the literatures, our method reduces the computation required of users by more than 99%.