I have research and literature review question about a research finding. But first to explain a simple name associated with the remainder in modulus division.
In division between two numbers, where the divisor is not a factor of the dividend there will be a remainder. If you subtract the remainder from the divisor you get the value which I call the ‘short’. I call it the short because it is the missing part that will make an additional unit of the divisor. For example, 17 mod 7 = 3. Let’s call the divisor ‘block’. So then, from 17 you can make 2 blocks of 7 and then you left with a partial block of 3 units, the remainder. To make an additional block you need 4 more units. So that means you short by 4. So that is how I call it the short. The short is always divisor – remainder.
In my research I found a math configuration whereby the short from modulus division equals the remainder in another division configuration which uses the factors of the dividend. Always easy to explain with an example So I will get into it now:
n is the dividend. It is the product of 67 x 313 x 277 x 499 = 2898674533. The divisor (called r) can be arbitrarily chosen, satisfying three conditions: • It must be below the square root of n • It must be below f1 • It must not be a factor of n
I haven’t told you what f1 is. f1 is a number that can only be produced from the factors of n. In this example f1 will be 67 x 313 = 20971. Since f1 has two prime factors the 67 will be called f1a and the 313 will be called f1b.
There is an f2. It is the remaining factors of n plus 1. So then f2 = (277 x 499) + 1 = 138224. Similar to f1 there is an f2a which the 277 and f2b is the 499.
Part 1:
The r value chosen is 5764. Thus, n mod r = 2898674533 mod 5764 = 5045. The short is simply divisor – remainder = 5764 – 5045 = 719.
Part 2:
Start with the following quotient division: n \ r = 2898674533 \ 5764 = 502892. This value is called q.
Note: I used back slash to signify quotient division. Forward slash is for full decimal division.
(q - f2) + 2 = (502892 – 138224) + 2 = 364670. This value is call Lr.Lr is just an abbreviation for a research name I made up called ‘Long row’. The dividend is Lr x f1 = 364670 x 20971 =7647473599. This value is called d.
The divisor is q + 1. Thus, d mod (q+1) = 7647473599 mod 502893 = 719.
This remainder, short invariance interests me because I can generate the same remainder, short set even after altering f1 and f2.
As an example a factor from f1 will go to f2-1 and factor from f2-1 will go to f1. Previously f1 was 67 x 313 and f2-1 was 277 x 499. The 313 and 277 will swap so that the new f1 is 67 x 277 and the new f2-1 is 313 x 499.
f1 = 18559 [67 x 277]
f2 = 156188 [(313 x 499) + 1]
Everything in part 1 remains the same.So you get remainder of 5045 and short of 719 because it is the same n mod the same 5764.
In part 2 q also remains the same but Lr and d are different because f1 and f2 have been altered.Lr is 346705 and d = 346705 x 18559 = 6434498095. Now also d mod (q+1) = 6434498095 mod 502893 = 719.
So the question I have to ask is where in existing literature can I find similar research findings. If someone knows please point it out to me or explain it.
After this question I want to explain why the name Long row. Consider the following sequence table with 3 columns. The first column is an index column that counts from 1 to divisor d. the second column multiplies a constant value c to each incremented index value. The third column modulus divides the dividend (from column 2) by the divisor d. Here below are a few rows of the table based on the following input values:
c = 13
d = 17
in row 1 you compute 13 mod 17
in row 2 you compute 26 mod 17
in row 3 you compute 39 mod 17
and so on
I want you to imagine another related table whereby at each incremented row the divisor increases by 1. Thus:
in row 1 you compute 13 mod 17
in row 2 you compute 26 mod 18
in row 3 you compute 39 mod 19
and so on
This understanding of this new modified sequence table underpins the remainder, short invariance. The short from trial division [r – (n mod r)] equals the remainder that is found in the ‘Long row’ of the above defined modified sequence table which is based on incremented divisors.
My research is directed at large integer factorization. I am investigating different setup configurations. In the example that I started with think of the first f1 of 67 x 313 as known factors. Treat the factors of f2-1 as unknown but we do know what the f2-1 is. There is enough information to compute part 1 and part 2 and get the remainder, short invariance of 719. I don’t know what the factors of f2-1 are but I do know that having swapped factors between f1 and f2-1, thereby creating new a f1 and f2, that the remainder, short invariance will be fixed at 719. I know what the divisor is I just don’t know what Lr and the dividend d are.
Are there ways that I can pick my input factors and divisor r in such a configuration that will reduce the time (loop iterations)in searching for the unknown Lr and unknown d. Maybe even better that iterative searching, a linear formulation that will produce either unknown Lr or unknown d in polynomial time. That would mean Game Over for RSA 2048.
I self publish at www.scribd.com/ziadcassim.
