This paper presents a method to construct elliptic curves of prime order with embedding degree k = 12, which significantly improves upon previous techniques that were limited to k ≤ 6. The new curves allow for efficient implementation, as non-pairing operations require arithmetic only over F_p^4, and pairing values can be compressed to one third of their length. The authors also discuss the role of large complex multiplication (CM) discriminants in minimizing the ratio ρ = log(p)/log(r), which is crucial for reducing the size of the base field p relative to the subgroup order r. For embedding degree k = 2q where q is prime, they show that handling log(D)/log(r) ≈ (q - 3)/(q - 1) enables building curves with ρ ≈ q/(q - 1). The paper also addresses point and pairing compression techniques, demonstrating that pairing values can be compressed to one third or even one sixth of their length. Additionally, the authors explore the construction of curves over extension fields and discuss the challenges of extending their method to higher embedding degrees. The proposed method allows for a sixfold compression of pairing values, which is more efficient than what is achievable with supersingular Abelian varieties. The paper concludes by highlighting the importance of further research into extending the method for higher values of k.This paper presents a method to construct elliptic curves of prime order with embedding degree k = 12, which significantly improves upon previous techniques that were limited to k ≤ 6. The new curves allow for efficient implementation, as non-pairing operations require arithmetic only over F_p^4, and pairing values can be compressed to one third of their length. The authors also discuss the role of large complex multiplication (CM) discriminants in minimizing the ratio ρ = log(p)/log(r), which is crucial for reducing the size of the base field p relative to the subgroup order r. For embedding degree k = 2q where q is prime, they show that handling log(D)/log(r) ≈ (q - 3)/(q - 1) enables building curves with ρ ≈ q/(q - 1). The paper also addresses point and pairing compression techniques, demonstrating that pairing values can be compressed to one third or even one sixth of their length. Additionally, the authors explore the construction of curves over extension fields and discuss the challenges of extending their method to higher embedding degrees. The proposed method allows for a sixfold compression of pairing values, which is more efficient than what is achievable with supersingular Abelian varieties. The paper concludes by highlighting the importance of further research into extending the method for higher values of k.