Questions and Answers
What is the complexity of brute force factorization?
O((2n/2) / (n/2)) where n is the number of bits
What does the equation (x+y)(x-y)=x²-y² represent in Fermat’s method?
The difference of squares
Which step involves setting d to d + t in Fermat's method?
In Pollard Rho Factorization Algorithm, what polynomial is commonly used?
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The Pollard Rho Algorithm can guarantee finding a non-trivial factor of n.
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What is B-powersmooth in relation to Pollard's p-1 method?
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For N=3675, what are the final factors returned by the factorization process?
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What should be the initial value of a and b in the Pollard Rho Factorization algorithm?
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What type of numbers does the Pollard Rho algorithm aim to factor?
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Study Notes
Factorization Methods
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Brute Force Factorization
- Attempts every possible solution, focusing primarily on prime numbers.
- Avoids unnecessary division (e.g., skipping even divisions if not divisible by 2).
- Complexity: O((2^n/2) / (n/2)), where n is the number of bits.
- Advantageous due to guaranteed success; operates without randomness.
Fermat’s Differences of Squares
- Based on the identity: (x+y)(x-y) = x² - y².
- Process starts with an integer N, aiming to express it in the form: (x+y)(x-y)=x²-N.
- Key steps:
- Set x = ceil(sqrt(N)) and initiate t = 2x + 1.
- Calculate d = x² - N; if N is not a perfect square, d should be positive.
- Repeat increasing d by t, while adjusting t until d is a perfect square.
- Final factors derived from x+squareroot(d) and x-squareroot(d).
Example: Factorization Using Fermat’s Method
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Example 1: N=88
- x starts at 10, d evaluated through adjustments until d becomes a square.
- Resulting factors: 22 and 4.
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Example 2: N=3675
- Starting with x at 61, similar adjustments lead to factors of 75 and 49.
Pollard Rho Factorization Algorithm
- Efficiently factors composite numbers in the form n = pq.
- Utilizes polynomial g(x) = (x² + 1) for generating a pseudorandom sequence.
- Describes a structured cycling of sequences which eventually helps identify factors.
Steps for Pollard Rho Algorithm
- Initialize both a and b at 2, with d starting at 1.
- Define the polynomial modulo n, g(x) = (x² + 1) mod n.
- Continuously compute values and check conditions for gcd until d is non-trivial.
- If unsuccessful, attempt with alternate starting values or functions.
Example: Pollard Rho
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Example: n=8051
- Sequence progresses with a and b until a non-trivial gcd is identified (97).
- Outcome reveals factors: 97 and 83 (since 8051 = 97 * 83).
Homework Factorization
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Tasks:
- Factor n=7171 and n=455459 using the Pollard Rho method, tracking changes in a, b, and gcd outcomes.
- For n=245, identifying the process leads to gcd revealing 5 as a factor.
- Factor n=2781 eventually leads to discovering 927 and 3.
Pollard’s p-1 Method
- Focuses on the likelihood that a factor is B-powersmooth.
- Acknowledges that every power dk dividing factors of p-1 remains bounded by B.
- Historical context: Developed by John Pollard in 1974 for extracting B-powersmooth factors effectively.
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Description
Explore the fundamentals of factorization techniques, including brute force and Fermat’s differences of squares. Understand how these methods provide reliable solutions for various mathematical problems while delving into their complexities and advantages. This quiz will guide you through the essential concepts and applications of these algorithms.
