is achievable. that the decoded messages are not equal to the transmitted messages; that is Proof: The proof is available in the Appendix. Reference [5] derived an achievable rate region with Gaus- sian codebooks and power control for the degraded Gaus- sian secure multiple-access-channel, when all the transmitters and receivers are equipped with a single antenna. Theorem 1, however, gives an achievability secrecy rate region for the general discrete memoryless multiple-access-channel. Our achievability rate region is also larger than the region of [5] in a special case of the Gaussian degraded case. When the transmitters and receivers are equipped with a single antenna, the total achieved S-DoF by using Gaussian codebooks is 0 . Here, we will provide a coding scheme based on integer codebooks, and show that for almost all channel gains a positive total S-DoF is achievable. The following theorem illustrates our results. The secrecy level is measured by a normalized equivocation, defined as follows: The normalized equivocation for each subset of messages $^{W}$S for S ⊆ K is The rate-equivocation tuple ( R$_{1}$, ..., R$_{K}$, d ) is said to be achievable for the Gaussian single-antenna multiple-access- channel with confidential messages, if for any ϵ > 0 , there exists a sequence of ((2 $^{nR$_{1}$}$, ..., 2 $^{nR$_{K}$}$) , n ) secret codes, such that for sufficiently large n , P ( n ) e = 1 ∏ K k $_{=1}$2 nR$_{k}$ ∑ ( w$_{1}$,...,w$_{K}$ ) ∈W 1 × .... ×W K (2) P { φ ( Y $^{n}$) = ( w$_{1}$, ..., w$_{K}$ ) | ( w$_{1}$, ..., w$_{k}$ ) is sent } . $^{Δ}$S △ = H ( $^{W}$S | Z $^{n}$) H ( $^{W}$S ) . (3) P ( n ) e ≤ ϵ, (4) $^{Δ}$S ≥ d − ϵ, ∀S ⊆ K . (5) Y = A [ K − 1 ∑ k =1 ˜ h$_{k}$ ˜ X$_{k}$ + ˜ X$_{K}$ ] + ˜ W$_{1}$ (9) Z = A K k =1 ˜ X$_{k}$ + W$_{2}$, Theorem 2: For the Gaussian single antenna multiple- access-channel, a total K − 1 K secure degrees-of-freedom can be achieved for almost all channel gains. Proof: Let us define ˜ X$_{k}$ △ = h$_{k,e}$ A X$_{k}$ and ˜ h$_{k}$ △ = h$_{k}$ $_{h$_{k,e}$}$. Without loss of generality let us assume that h$_{K}$ = 1 ; the cannel model then is equivalent as follows: and The perfect secrecy rate tuple ( R$_{1}$, ..., R$_{K}$ ) is said to be achievable when d = 1 . When all the transmitted messages are perfectly secure, we have In [5] it is shown that when all of the K messages are perfectly secure, then it guarantees that any subset of the messages become perfectly secure. The total Secure Degrees-of Freedom (S-DoF) of η is said to be achievable if the rate-equivocation tuple ( R$_{1}$, ..., R$_{K}$, d = 1) is achievable, and where, A $^{2}$E [ ˜ X 2 $_{k}$] ≤ ˜ P △ = h 2 $_{k,e}$P . In this model we say that the signals are aligned at the eavesdropper, according to the following definition: Definition 1: The signals ˜ X$_{1}$ , ˜ X$_{2}$ ,..., ˜ X$_{K}$ are said to be aligned at a receiver if its received signal is a rational combi- nation of them. $^{Δ}$K ≥ 1 − ϵ, (6) η = lim P →∞ ∑ K k $_{=1}$R$_{k}$ 1 2 log P (7) Note that, in n -dimensional Euclidean spaces ( n ≥ 2 ), two sig- nals are aligned when they are received in the same direction at the receiver. In general, m signals are aligned at a receiver if they span a subspace with a dimension less than m . The above definition, however, generalizes the concept of alignment f or the one-dimensional real numbers. Our coding scheme is based on integer codebooks, which means that ˜ X$_{k}$ ∈ Z for all k ∈ K . If some integer signals are aligned at a receiver, then their effect is similar to a single signal at high SNR regimes. This is due to the fact that rational numbers form a filed; therefor e the sum of constellations from Q form a constellation in Q with an enlarged cardinality. In this section, we consider the achievable S-DoF of the Gaussian single-antenna multiple-access-channel under the perfect secrecy constraint. In order to satisfy the perfect secrecy constraint, we use the random binning coding scheme to generate the codebooks. To maximize the achievable de- grees of freedom, we adopt the signal alignment scheme used in [8] to separate the signals at the intended receiver and simultaneously align the signals into a small subspace at the eavesdropper. The main results of this section are presented in the following theorems. First, we present an achievable secrecy rate region for the discrete memoryless multiple- access-channel. Theorem 1: For the perfectly secure discrete memoryless multiple-access-channel of P ( y, z | x$_{1}$, ..., x$_{K}$ ) , the region of Before we present our achievability scheme, we need to define the rational dimension of a set of real numbers. III. S-DOF OF THE SINGLE-ANTENNA MAC Definition 2: ( Rational Dimension ) The rational dimension of a set of real numbers { ˜ h$_{1}$, ˜ h$_{2}$, ..., ˜ $^{h$_{K}$}$− $_{1}$, ˜ h$_{K}$ = 1 } is M if there exists a set of real numbers { g$_{1}$, g$_{2}$, ..., g$_{M}$ } , such that each ˜ h$_{k}$ can be represented as a rational combination of g$_{i}$ s, i.e., h$_{k}$ = a$_{k,}$$_{1}$g$_{1}$ + a$_{k,}$$_{2}$g$_{2}$ + ... + a$_{k,M}$g$_{M}$ , where a$_{k,i}$ ∈ Q for all k ∈ K and i ∈ M . In fact, the rational dimension of a set of channel gains is the effective dimension seen at the corresponding receiver. In particular, { ˜ h$_{1}$, ˜ h$_{2}$, ..., ˜ h$_{K}$ } are rationally independent if the rational dimension is K , i.e., none of the ˜ h$_{k}$ can be represented as the rational combination of other numbers. for any distribution of { ( R$_{1}$, ..., R$_{K}$ ) | ∑ i ∈S R$_{i}$ ≤ I ( $^{U}$S ; Y | $^{U}$S c ) , ∀S ⊂ K , (8) k ∈K$^{R$_{k}$}$ ≤ [ I ( $^{U}$K ; Y ) − I ( $^{U}$K ; Z )] + } , P ( u$_{1}$ ) P ( u$_{2}$ ) ....P ( u$_{K}$ ) P ( x$_{1}$ | u$_{1}$ ) P ( x$_{2}$ | u$_{2}$ ) ...P ( x$_{K}$ | u$_{K}$ ) × P ( y, z | x$_{1}$, ..., x$_{K}$ ) , 2