Binary quadratic forms

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This book explores algorithmic problems related to binary quadratic forms \( f(X, Y) = aX^2 + bXY + cY^2 \) with integer coefficients \( a, b, c \), the mathematical theories that address these issues, and their applications in cryptography. A significant portion of the theory is developed for forms with real coefficients, demonstrating that integer coefficient forms arise naturally. The evolution of number theory has been propelled by the exploration of concrete computational challenges, leading to the development of profound theories from the time of Euler and Gauss to the present. Algorithmic solutions and their properties have become a distinct area of study. The book intertwines classical strands of number theory with the presentation and analysis of both classical and modern algorithms that address these motivating problems. This algorithmic perspective aims to foster an understanding of both theory and solution methods, as well as an appreciation for the efficiency of these solutions. The computer age has significantly advanced algorithmic search capabilities, enabling the resolution of complex problems, such as Pell equations with large coefficients. Additionally, the role of number theory in public-key cryptography has heightened the importance of establishing the complexity of various computational problems, as the security of many computer systems relies on their intractability.