Question

Design an algorithm that takes as input a prime \(q\) and a bound \(m,\) and outputs a random prime \(p\) between 2 and \(m\) such that \(p \equiv 1(\bmod q) .\) Clearly, we need to assume that \(m\) is sufficiently large with respect to \(q\). Analyze your algorithm assuming Conjecture 5.22. State how large \(m\) must be with respect to \(q,\) and under these assumptions, show that your algorithm runs in time \(O\left(\ell^{4} / \operatorname{len}(\ell)+k \ell^{3}\right),\) and that its output is incorrect with probability \(O\left(4^{-k} \ell\right) .\) As usual, \(\ell:=\operatorname{len}(m)\)

Short answer

Answer: The overall time complexity of the given algorithm is O(ℓ⁴/len(ℓ) + kℓ³), where ℓ is the length of m and k is the parameter used in the Miller-Rabin primality test.

Step-by-step solution

Generate random number

Choose a random number r between 2 and m-1 (inclusive). Calculate p = r * q +1. If p is within the bound m, proceed to the next step. Otherwise, repeat this process until a valid p within the bound is found. The time complexity of this step is O(len(ℓ)), as each attempt takes constant time, and we expect to succeed in O(1) attempts because m is assumed to be large.

Primality test

Perform the Miller-Rabin primality test for the chosen number p with parameter k. The Miller-Rabin test has a time complexity of O(k*len(ℓ)²) and has an incorrect output probability of O(4^(-k)). If p passes the primality test, proceed to the next step. Otherwise, go back to step 1 and choose a new random number.

Output prime number

If p passes the primality test, output it as the prime number found. Otherwise, repeat the process from step 1 until a prime number is found. The overall time complexity of the algorithm can be calculated as follows: - Step 1: O(len(ℓ)) - Step 2: O(k*len(ℓ)²) - The algorithm is expected to find a prime number in O(len(ℓ)) iterations Considering the iterations, the overall complexity is O(ℓ⁴/len(ℓ) + kℓ³). The probability of incorrect output (due to the Miller-Rabin test) is O(4^(-k) *ℓ), as the algorithm can find a composite number that passes the primality test in any of its iterations.

Key concepts

Miller-Rabin Primality Test

The Miller-Rabin primality test is a probabilistic test used to determine whether a number is a prime. It is an important algorithm in number theory due to its efficiency and simple execution. The test relies on the properties of modular arithmetic and is executed as follows:

• First, the number you want to check, say, \(n\), is expressed in the form of \(n = 2^s \times d + 1\), where \(d\) is an odd number.
• Then, a random base \(a\) is chosen such that \(1 < a < n\).
• The algorithm first computes \(a^d \mod n\). If the result equals 1 or \(-1\), then \(n\) could be a prime.
• If neither result is obtained, the algorithm proceeds to check successive powers of 2 up to \(s-1\), recalculating \(a^{2^r \cdot d} \mod n\).
• If none of these values are \(-1\), \(n\) is definitely composite.

The test is repeated with multiple bases to reduce the likelihood of error. With a sufficiently large number of tests, the probability of misclassifying a composite number as prime becomes negligible, specifically \(O(4^{-k})\) where \(k\) is the number of random bases tested. This makes it very reliable for practical use.

Time Complexity Analysis

Time complexity is an essential part to consider when designing algorithms as it measures the time an algorithm takes to run as a function of the size of its input. For the random prime generation algorithm detailed above, the complexity comes from both generating the random number and testing for primality using Miller-Rabin.

• In Step 1, choosing a random number \(r\) between 2 and \(m-1\) and calculating \(p = r\cdot q + 1\) takes \(O(\operatorname{len}(\ell))\) time, as \(\ell = \operatorname{len}(m)\).
• During Step 2, the Miller-Rabin test for every chosen \(p\) has a time complexity of \(O(k\cdot \operatorname{len}(\ell)^2)\), because of the multiple iterations through the test discussed above.
• In expectation, since the range is large, we should hit a prime quickly which impacts the overall expected complexity.
• The complete process, including multiple iterations of these steps, leads to a total time complexity of \(O(\ell^4/\operatorname{len}(\ell) + k\ell^3)\), blending number generation and primality tests.

Analyzing the time complexity helps to predict efficiency and scalability, indicating that while the process can potentially take longer with incorrectly chosen random numbers, the structure of Miller-Rabin significantly mitigates this.

Number Theory

Number theory is the foundation on which algorithms like the random prime generation are built. It deep dives into the properties and relationships of integers, primarily focusing on primes as they are building blocks of natural numbers.

• Basic concepts like divisibility, primes, congruences, and modular arithmetic play a critical role in our algorithm.
• The modulo operation ensures that when performing tests like Miller-Rabin, results between numbers are manageable, even for large integers.
• Additionally, understanding the distribution of primes is crucial for comprehending proper algorithm design, letting us decide on how large \(m\) should be compared to \(q\) to ensure that there are sufficient primes, making our number generation valid.
• Prime number theorems often estimate the existence of primes and assist in determining the likelihood of finding primes within certain ranges, thus influencing assumptions like Conjecture 5.22 used in this problem.

With proper understanding and application of number theory concepts, complex algorithms can be developed to address problems in cryptography, encryption, and other computational fields that depend heavily on prime numbers.