Quadratic Sieve Puns

I am not from a Math background, so it took me a lot of time to understand the simplest version of the Quadratic Sieve algorithm. Each line in any text had at least 2 things I didn't understand and had to go searching for more info. So at the end, I decided to document the algorithm in detail with explanations.

Here is a link - https://risencrypto.github.io/QuadraticSieve/

There is also a link to a Google Spreadsheet inside which contains an example Sieving.

I am sure there are mistakes - so please feel free to point it out - that's one of the main reasons I am posting it here. Same again if something is not clearly explained.

My writeup on the Quadratic Sieve algorithm - https://risencrypto.github.io/QuadraticSieve/

Many of the popular Asymmetric Cryptography algorithms are based on the difficulty of factoring very large semiprimes (semi prime is a non-prime which factors into exactly 2 primes). Large means 300 digit numbers or 600 digit numbers.

It's easy to multiply 2 large primes & find the product. But reversing it to find the 2 numbers which were multiplied to get the product is a very, very hard problem.

So in the first place, for implementing the algorithm, you have to find very large primes (which are usually found using the Miller-Rabin algorithm).

Studying the reverse (factoring large semiprimes) is important for

• for attackers trying to attack it

• you need to know how large a prime can be factored in finite time with current computing power so that you know how large a number you should use to prevent that.

The Quadratic Sieve is the second fastest algorithm for factoring large semiprimes. It’s the fastest for factoring ones which are lesser than 100 digits long.

I'm just starting to look at prime factorisation, and I've come across a reference to something called a Double Sieve. However, Google just gives me links for the Quadratic Sieve. Are they one and the same?

So I know someone asked the same question in math.stackexchange but he didn't receive an answer to the question so I am asking it again(I'm copy-pasting his code for example):

1. Say, for N = 90283, I compute bound 𝐵=𝑒(12+𝑜(1))(ln(𝑛)ln(ln𝑛√n))=44, where I take 𝑜(1)=0.22 (just a random guess, I assume). What is the best way to pick good 𝑜(1)?
2. Then I take 𝐶=⌈√N⌉=301
3. Then using Sieve of Eratosthenes I get a list of primes < 𝐵:

{2,3,5,7,11,13,17,19,23,29,31,37,41,43}

4. Then by computing Jabobi (Legendre) symbol for each value in the primes list, I pick the first quadratic nonresidues to get factor base:

{2,3,7,17,23,29,37,41}

5. Then using the Tonelli-Shanks algorithm I compute modular roots ±𝑡, where 𝑡 is a solution to 𝑡^(2)t≡ 𝑁 (mod 𝑝) with 𝑝 a prime form factor base.

6. Then I get two arrays of solutions 𝑠𝑜𝑙1 = 𝑡−𝐶 (mod 𝑝) and 𝑠𝑜𝑙2 = −𝑡−𝐶 (mod 𝑝), with p's from factor base. and also get one array of logarithms for p's 𝑙𝑝 = ln(𝑝) rounded up to some precision, say two decimal digits. What is a good precision?

sol1 ={0,0,2,13,11,26,10,28}

sol2 ={0,1,5,14,8,10,17,26}

𝑙𝑝 ={0.69,1.1,1.95,2.83,3.14,3.37,3.61,3.71}

Now as far as I understand I have to initialize 'sieving_array' initialized to 0's, say, to size = 60 elements. Also, how should I choose size of the sieving interval? Is there any formula similar to the bound?

7. Then I have to add values from 𝑙𝑝 to sieving array to positions 𝑠𝑜𝑙1[𝑗]+𝑖∗factor_base[j] and 𝑠𝑜𝑙1[𝑗]+𝑖∗ factor_base[j], where 0≤𝑖≤ size and 0≤𝑗≤|factor_base|. And for prime 𝑝=2 add 𝑙𝑝 only to positions with sol1. So I get this list (rounded to two decimal digits):

SieveArray ={1.79,1.1,2.64,1.1,1.79,1.95,1.79,1.1,3.83,3.05,8.77,3.14,3.74,3.93,3.52,1.1,3.74,3.61,1.79,3.05,0.69,1.1,1.79,1.95,1.79,1.1,9.72,1.1,5.5,0.0,6.57,7.07,0.69,3.05,4.93,0.0,1.79,3.05,0.69,4.47,3.74,0.0,1.79,1.1,2.64,1.1,1.79,8.39,4.62,1.1,0.69,3.05,1.79,0.0,10.49,4.47,0.69,4.24,3.74,0.0}

Now I am confused. The next step is(in Contini's paper) is to check each value is log(2x√N) but I don't get what x is. Also can someone please explain how to calculate the size and Sieving Interval? If anyone wants to see the paper it is here and the "qs algorithm to factor n" on page 6.

Trying to understand example of basic sieve in Wikipedia article and stuck in those lines:

• Since N is small, only 4 primes are necessary.

Why 4 primes are necessary? Why 3 is not enough?

• Now we construct our sieve V_X of Y(X) = (X + \lceil\sqrt{N}\rceil)^2 - N = (X+124)^2-15347 and begin the sieving process for each prime in the basis, choosing to sieve the first 0 ≤ X < 100 of Y(X):

Why 0 ≤ X < 100 ? not 101 or 99? How upper bound is determined?

Hey, got a crypto final on monday and I'm having a hard time with the quadratic sieve, textbook isn't doing it for me. wondering if anyone can help me out with a pdf or somethin? Also a subquestion: i understand the o-notations, like big-o small-o, big theta etc. and i understand what it means when One function is = some-o of another function. But how do i read an equation with o notation somewhere within in it, like the exponent? maybe this is whats holding me back from understanding the sieve?

If I assume that n is a semi-prime, I want to approximate the size of the factor base using primes less than or equal to B.

I understand that using the Prime Number Theorem, the number of primes less than B is asymptotically (B/ln(B)). I'm trying to understand the proportion of these primes that are moduli to which n is quadratic residue. Any guidance?

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Every week on Saturday, I will post this thread. Feel free to discuss anything related to hacking, coding, startups etc. Share your github project, show off your DIY project etc. So post anything that interests to hackers and tinkerers. Let me know if you have some suggestions or anything you want to add to OP.

The thread will be posted on every Saturday evening.

Do your worst!

They were cooked in Greece.

I'm surprised it hasn't decade.

Don't you know a good pun is its own reword?

Two muffins are in an oven, one muffin looks at the other and says "is it just me, or is it hot in here?"

Then the other muffin says "AHH, TALKING MUFFIN!!!"

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

Because she wanted to see the task manager.

But that’s comparing apples to oranges

Heard they've been doing some shady business.

but then I remembered it was ground this morning.

Edit: Thank you guys for the awards, they're much nicer than the cardboard sleeve I've been using and reassures me that my jokes aren't stale

Edit 2: I have already been made aware that Men In Black 3 has told a version of this joke before. If the joke is not new to you, please enjoy any of the single origin puns in the comments

BamBOO!

They’re on standbi

And boy are my arms legs.

A play on words.

My daughter, Chewbecca, not so much.

Pilot on me!!

Christopher Walken

Nothing, he was gladiator.

Or would that be too forward thinking?

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

When I got home, they were still there.

Hindus, on the other hand, never had any beef.

I won't be doing that today!