## 1 Introduction  
The purpose of cryptography is to protect the secret message that is transmitted between the legitimate sender and the receiver from unauthorized reading or modification of the message. The task of cryptographers is to develop secure and reliable cryptographic protocols. Classical cryptography techniques such as symmetric cryptography and asymmetric cryptography are widely used. However, since classical cryptography is based on the complexity of computation, it is facing more and more challenges due to the development of supercomputers and, potentially, quantum computers. The development of quantum cryptography provides a solution that is based on the Heisenberg Uncertainty Principle and No-Cloning Theorem. In 1984, [PERSON] and [PERSON] proposed the first quantum key distribution (QKD) protocol, which is known as the BB84 protocol [1]. In this protocol, two legitimate users can establish a secure channel by using quantum resources to generate an unconditionally secure key. After 1984, several variants [2; 3; 4] of BB84 protocols concentrated around QKD were proposed. In 1999, [PERSON] and [PERSON] proposed a protocol for quantum secret sharing (QSS) [5]. In the same year, a protocol for deterministic secure quantum communication (DSQC) was proposed by [PERSON] and [PERSON] [6]. In a DSQC protocol, one bit of additional classical information transmission is required for each qubit; otherwise the receiver cannot read the correct secret message. Secure direct quantum communication does not require exchange of classical information for encryption or decryption of the message. A protocol that does not require such classical information exchange is named quantum secure direct communication (QSDC) [7, 8]. In all QSDC and DSQC protocols, the secret message can be only transmitted one-way \"from [PERSON] to [PERSON]\". In the year of 2004, the first quantum dialogue protocol was proposed by [PERSON] using Bell states [9], which enables the bidirectional quantum communication that [PERSON] and [PERSON] can transmit and receive their messages simultaneously. This protocol is a modification of the Ping-Pong protocol which started with an initial state [7]. In 2006, [PERSON] proposed the three-stage quantum cryptography protocol [10]. In this protocol, there is no need to have prior generation of keys and it does not need an initial state. Besides, multi-photons can be used to increase the stability and security of the transmission [11]. QSDC has been also extended to continuous-variable systems [12, 13]. Most quantum protocols assume a noiseless channel. But in real applications and implementations, noise should be considered as it will have a critical impact on the performance of the transmission. The concept of collective noise on quantum cryptography was explained by [PERSON] and [PERSON] [14]. Several collective noise analyses have been performed on previous protocols. Authors of [15] proposed an efficient way for quantum key distribution over collective noise. An economical setup for faithful entanglement sharing against collective noise was presented in [16]. Paper [17] analyzed the security of the \"Ping-Pong\" protocol in a noisy environment. Two quantum dialogue protocols were proposed in [18], each of which is robust against one of the two kinds of collective noise: Collective-dephasing noise and collective-rotation noise. Furthermore, continuous-variable quantum cryptography with two-way quantum communication has shown to be very robust to the presence of noise [19, 20, 21]. In this paper, we analyze the effect of collective-rotation noise in a multi-photon system under the three-stage protocol. This work distinguishes itself from existing work in the following ways: The three-stage protocol is an interesting quantum protocol as it can be used either as QKD, or as QSDC. It maps information onto non-orthogonal polarization states of photons. In the simplest form, all communications in the three-stage protocol are performed using quantum channels.