Factoring By Digital Coding

April 1, 2014

A few days ago, an announcement was made in /r/crypto of a new factoring algorithm that runs in time O(log^4 n). A paper describes the algorithm. And even better, coding for the new algorithm is simple:

1)Set mn.
2) Express m as a list of its decimal digits.
3) Express each decimal digit as a list of binary bits.
4) Append the individual lists of binary bits.
5) Convert the resulting list of binary bits into a decimal number. Call it d.
6) If the greatest common divisor of n and d is between 1 and n, it is a factor of n. Report the factor and halt.
7) Set md and go to Step 1.

Steps 1 through 4 form the digital code of a number. For example, 88837 forms the bit-list 1 0 0 0, 1 0 0 0, 1 0 0 0, 1 1, 1 1 1, where we used commas to delineate the decimal digits. That bit-list is 69919 in decimal, which is the digital code of 88837. To find a factor of 88837, compute successive digital codes until a gcd is greater than 1; the chain runs 88837, 69919, 54073, 2847, 2599, 5529, 2921, 333, at which point gcd(88837, 333) = 37 and we have found a factor: 88837 = 74 × 37. Factoring 96577 is even faster: the digital code of 96577 is 40319, and gcd(96577, 40319) = 23, which is a factor of 96577 = 13 × 17 × 19 × 23. But trying to factor 65059 = 17 × 43 × 89 causes an infinite loop. The paper gives a heuristic solution to that problem, but we’ll stop there and accept that sometimes a factorization fails.

Your task is to write a function that factors by digital coding. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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4 Responses to “Factoring By Digital Coding”

  1. Lucas A. Brown said

    I use the cycle-finding method from Pollard’s Rho algorithm to check whether we’re stuck in a loop.

    def digital_coding(n):
        if isprime(n): return n
        x = y = n
        while True:
            x = eval(''.join(bin(int(digit))[2:] for digit in str(x)))
            y = eval(''.join(bin(int(digit))[2:] for digit in str(y)))
            y = eval(''.join(bin(int(digit))[2:] for digit in str(y)))
            f = gcd(x, n)
            g = gcd(y, n)
            if 1 < f < n: return f
            if 1 < g < n: return g
            if x == y: return None
    
  2. Lucas A. Brown said

    While fast, this has a disturbingly high failure rate: 338,793 out of the first million natural numbers fail to produce a factor by this method.

    fail = 0
    for n in count(2):
        if n % 10**5 == 0: print fail, n, float(fail) / float(n)
        f = digital_coding(n)
        if f is None: fail += 1
    

    I suppose it might be useful as a prefilter when attempting to factor large numbers with no small factors — it’s a computationally cheap procedure, so it wouldn’t introduce much overhead and has a chance of dramatically speeding up the task by skipping the heavy artillery such as the ECM or MPQS if it succeeds.

  3. programmingpraxis said

    @Lucas: Did you see the last paragraph on the second page of the exercise?

  4. Situxuming said

    The following is my Mathematica code:

    1. Firstly define a function that create the digital code of a specific number:

    dc[n_] := 
     FromDigits[Flatten[IntegerDigits[#, 2] & /@ IntegerDigits[n]], 2]
    

    2. Then a function that computes successive digital codes until a gcd is greater than 1 is defined as follows:

    gcd[n_]:=NestWhileList[dc, n, ! (1 < GCD[#, n] < n) &]
    

    3. Finally, define a function that returns the expected results:

    result[n_]:=Module[{tmp=gcd[n]},Flatten[{Rest[tmp],GCD[tmp[[-1]],n]}]]
    

    In Mathematica, we get:

    In[1]:=result[88837]
    
    Out[1]={69919, 54073, 2847, 2599, 5529, 2921, 333, 37}
    
    In[2]:=result[96577]
    
    Out[2]={40319, 23}
    

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