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Primality testing is the process of determining whether a given number is prime, which is crucial for cryptographic applications and number theory. Efficient primality tests, such as the AKS primality test, have been developed to handle large numbers, providing deterministic results in polynomial time.
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are fundamental in number theory due to their role as the building blocks of the integers, analogous to atoms in chemistry.
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A composite number is a positive integer greater than one that has at least one positive divisor other than one and itself, meaning it can be divided evenly by numbers other than one and itself. Unlike prime numbers, which have exactly two distinct positive divisors, composite numbers have more than two divisors, indicating their non-prime nature.
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The AKS primality test is a deterministic algorithm that can determine whether a number is prime in polynomial time, making it a breakthrough in computational number theory. It is significant because it is the first primality test proven to run in polynomial time without relying on unproven hypotheses, which has important implications for cryptography and complexity theory.
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A probabilistic algorithm is a computational procedure that makes random choices as part of its logic, often leading to different outcomes on different runs for the same input. These algorithms are particularly useful for problems where deterministic solutions are inefficient or unknown, offering faster average performance or simpler implementation at the cost of some uncertainty in the result.
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A deterministic algorithm is a type of algorithm that, given a particular input, will always produce the same output with a predictable computational path. These algorithms are essential in scenarios where consistent and reliable results are critical, such as in cryptographic protocols and database transactions.
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Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p). This theorem is fundamental in number theory and has applications in fields such as cryptography, particularly in algorithms like RSA encryption.
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The Miller-Rabin Primality Test is a probabilistic algorithm used to determine if a number is a prime, and it's especially useful for testing large numbers due to its efficiency. By repeatedly checking randomness under specific conditions, it can distinguish non-primes with high accuracy, although it cannot absolutely confirm primality without a non-random factorization step.
Elliptic Curve Primality Proving (ECPP) is a sophisticated algorithm used to determine whether a number is prime, leveraging the properties of elliptic curves over finite fields. It stands out for its ability to provide a certificate of primality, which can be independently verified, making it a powerful tool in computational number theory.
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Complexity Theory is a branch of theoretical computer science that focuses on classifying computational problems according to their inherent difficulty and defining the resource limits required to solve them. It provides a framework for understanding the efficiency of algorithms and the feasibility of solving problems within practical constraints.
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Cryptography is the science of securing communication and information through the use of mathematical techniques, ensuring confidentiality, integrity, authenticity, and non-repudiation. It plays a crucial role in various applications such as secure communications, digital signatures, and cryptocurrency, protecting data from unauthorized access and tampering.
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Quadratic residues are integers that can be expressed as the square of another integer modulo some number, and they play a crucial role in number theory, particularly in solving congruences and understanding the distribution of quadratic residues and non-residues. They are fundamental in the study of quadratic reciprocity, which is a cornerstone of modern number theory and has applications in cryptography and primality testing.
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A quadratic residue modulo n is an integer that can be expressed as the square of another integer modulo n. Understanding quadratic residues is fundamental in number theory and has applications in cryptography, particularly in algorithms like RSA and primality testing.
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The Miller-Rabin test is a probabilistic algorithm used to determine if a number is a prime, providing a high degree of accuracy with multiple iterations. It is widely used in cryptographic applications due to its efficiency and ability to handle very large numbers, though it does not definitively prove primality.
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