# Tag Info

115

In the first decade of the 21th century, and counting, on a given $\text{year}$, no RSA key bigger than $(\text{year} - 2000) \cdot 32 + 512$ bits has been openly factored other than by exploitation of a flaw of the key generator (a pitfall observed in poorly implemented devices including Smart Cards). This linear estimate of academic factoring progress ...

38

I've never heard that RSA becomes less secure when the modulus grows. Obviously the strength doesn't grow as fast as the number of bits, but that only means that it grows sub-exponentially. If it keeps growing (without the growth going near zero) then there is no "trap". Check for instance here where the conclusion is that there is no exponential growth but ...

35

Let's assume for an instant that you could build a large table of all primes. Then... what ? How would you use it ? What would you look up ? If you "just" scan the table and try to divide the number to factor by each prime, then this is known as trial division; there is no need to store the primes (they can be regenerated on-the-fly; that's the division ...

35

Those appear to be based on the complexity of the General Number Field Sieve, one of the fastest (if not the fastest) classical factoring algorithms. I confirmed this in Mathematica. Here is the complexity for the GNFS (source): $$\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\ln n)^{\frac{1}{3}}(\ln \ln n)^{\frac{2}{3}}\right)$$ where $n$ is a ...

28

You might want to look at NIST SP800-57, section 5.2. As of 2011, new RSA keys generated by unclassified applications used by the U.S. Federal Government, should have a moduli of at least bit size 2048, equivalent to 112 bits of security. If you are not asking on behalf of the U.S. Federal Government, or a supplier of unclassified software applications to ...

23

I don't understand at all what this claim is on the website. The claim that RSA becomes very expensive for large $N$ is true, but to say that the gap between encryption/decryption cost and factoring goes down makes no sense at all! The function describing the running time of the best factoring algorithms is clearly asymptotically larger than $n^3$ (the time ...

20

No, it is not at all feasible to build an index of prime factors to break RSA. Even if we consider 384-bit RSA, which was in use but breakable two decades ago, the index would need to include a sizable portion of the 160 to 192-bit primes, so that the smallest factor of the modulus has a chance to be in the index. Per the Prime number theorem there are in ...

19

The generic discrete logarithm problem is this: Given a group $(G, ·)$ with generator $g$ and $y \in G$, find $x \in \mathbb N$ such that $y = g^x$. The "classic" discrete logarithm problem (the one used in "classic" DSA and ECDSA) is this with some subgroup of the (multiplicative) group of a prime field, i.e. $(\mathbb Z/p \mathbb Z)^*$: Given a ...

19

2) Has anyone claimed to make any progress with this challenge? Ah that question I can answer now... I found the solution on the 15th of April 2019 and sent it to the MIT's CSAIL on the 16th of April. Another team shall have the answer by the 11th or 12th of May (they used a FPGA). I noticed around the end of 2015 that if I used GMP I could find the ...

17

The three main general-purpose algorithms for factorization are the quadratic sieve (QS), the elliptic curve method (ECM) and the number field sieve (NFS). On Complexity The running time of these algorithms is expressed with the L-notation: $L_n[a,c]$ means that the asymptotic complexity of factoring a number $n$ is $O(e^{(c+o(1))(\log n)^a(\log \log n)})^{... 15 RSA themselves (the company) say this in their RSA Factoring Challenge FAQ: Why is the RSA Factoring Challenge no longer active? Various cryptographic challenges — including the RSA Factoring Challenge — served in the early days of commercial cryptography to measure the state of progress in practical cryptanalysis and reward researchers for the new ... 15 RSA-768 took 2000 years of 2.2Ghz single core Opteron from year 2009 [1]. DJB et al wrote in 2013 [2] that RSA-1024 would take$2^{70}$differences with$2^{24}$per machine per second in 2009, so 2 million years. Hardware improved since then, and GNFS can use GPUs, so maybe better, but about a million years I guess. Absolutely the computation can be ... 14 Yes there are other hard problems you can base asymmetric cryptography on. Lattices. The NTRU systems is based on the shortest vector problem in ideal lattices. Lattice-based cryptography is of much interest these days for two reasons: (1) unlike factorization and discrete logarithms, there isn't an efficient algorithm for breaking these problems on a (... 14 Actually, the problem is that the above quote uses the term "discrete log" in a way that's different from what you're thinking of. When someone uses the term "discrete log", they can mean two things: A discrete log in the group$Z^*_p$; that is, given$p$,$g$and$g^x \bmod p$, recover$x$A discrete log in some other group; that is, given a group$G$, a ... 14 If$p=2q+1$is a safe prime (that is,$q$is a prime as well), then$p-1=2q$has exactly two prime factors:$2$and$q=(p-1)/2$. 14 First we may want RSA primes to be something like a safe prime, ie a prime$p$where$(p-1)/2$is prime as well. Back in 1974 Pollard found an algorithm to factor moduli whereby you can factor$N=pq$if$p-1$or$q-1$are smooth, that is all prime-factors of$p-1$or$q-1$are smaller than a bound$B$. The algorithm will then factor$N$in time$\mathcal O(...

14

Proving P=NP would not necessarily give you an algorithm, because there are many different methods to prove something (i.e. Direct proof, Proof by contradiction, etc.). But it is shown that if you were to find a polynomial time algorithm to solve a NP-complete problem that you could modify that algorithm to solve all NP-problems, including the Integer ...

13

I don't believe a lower bound has ever been proven for the "fewest" number of bits needed. Coppersmith showed, however, how given either the $n/4$ least or $n/4$ most significant bits of $p$ where $n$ is the size of the modulus $N=pq$, $N$ can be efficiently factored. Additionally, given the $n/4$ least significant bits of $d$, one can reconstruct $d$ (and ...

12

If the RSA keys were generated randomly, then it is inconceivable that two different devices would happen to pick the same key. Taking 2048 bit RSA keys as an example, there are approximately $2^{1014}$ 1024 bit primes; if we consider them pairwise (and realise that about half the pairs yield a 2047 bit number), that means there are about $2^{2026}$ RSA ...

11

The simplest answer would be to look at the keylength.com site, and if you don't trust that, to the linked papers, particularly by NIST and ECRYPT II. Note that those mainly agree with the Lenstra equations, so you could use those as well. You may have additional restrictions and - if you are brave or stupid - relaxations depending on the use case. But at ...

11

Bill Garsarch just posted about this the other day. The short answer is that there is an explicit algorithm, which is known today, such that if P = NP (or even just FACTORING ∈ P) then the algorithm solves factoring in polynomial time on all instances. However, this algorithm is utterly infeasible for real-world computation because it works by iterating ...

11

The IEEE paper is silly. The factorization method they give is quite slow, except for rare cases. For example, in their table 1, where they proudly show that their improved algorithm takes 653.14 seconds to factor a 67 bit number; well, I just tried it using a more conventional algorithm, and it took 6msec; yes, that's 100,000 times as fast...

10

"Discrete logarithm" is a wide class. Originally, this means that we work in a finite field (e.g. integers modulo a big prime) and, given $g$, $p$ and $g^x \bmod p$, it is computationally difficult to recover $x$ (it becomes impossible with today's technology once $p$ is big enough). At some point, someone noticed that discrete logarithm was a special case ...

10

There is an asymptotic formula for the General Number Field Sieve for factoring big integers. This is the most efficient known algorithm for breaking RSA keys which are longer than 400 bits or so (since the current world record is 768 bits, a 400-bit RSA key is quite weak). For discrete logarithm (to break DH), the best known algorithm is also known as "...

10

No, there is no specific vulnerability associated to choosing $p$ and $q$ with size differing by $i$ bits (or $2\cdot i$ bits as in the statement) for small $i$. However, if $i$ gets too big: That improves the odds that ECM will manage to factor $n$ for some fixed size of $n$, and at some point ECM will become the best algorithm; this is the case if $i$ is ...

10

Breaking such a scheme is easy. Suppose Alice wants to transmit a message $M$ to Bob. First thing, Alice picks an integer $R_a$ and sends the cipher text $C_a = M \times R_a$ to Bob. Bob then picks another integer $R_b$ and transmits $C_b = C_a \times R_b$ back to Alice. Alice calculates $D_a = \frac{C_b}{R_a}$ and sends $D_a$ to Bob. Bob calculates $D_b = ... 10 Algorithm An RSA modulus$N$product of large distinct primes can be factored given$(N,e,d)$per: Compute$f\gets e\,d-1$, and express$f$as$2^s\,t$with$t$odd Set$i\gets s$and$a\gets2$Compute$b\gets a^t\bmod N$, and if$b=1$then set$a$to the next prime, and proceed at 3 If$i\ne1$then compute$c\gets b^2\bmod N$, and if$c\ne1$then ... 9 The puzzle has now been solved! The solution was announced in May 2019. It was solved by Bernard Fabrot, who used a modern optimized implementation of big integer multiplication to speed up the computation and finish it in about 3.5 years of computation instead of 35 years. At around the same time, a second team using FPGAs reported being weeks away from ... 9 No, it not possible to attack RSA (and practical modulus size) with a WalkSat derivative, as far as we know, or using the algorithm in the question. Problem with that algorithm is: in order to have a sizable/constant rate of success as$n$increases, we have to repeat steps 2 and 3 not the stated$t\cdot m^2$times, but rather$t\cdot 2^m\$ times. That's ...

9

Does the factorization of N somehow help me? It sure does. I think I could compute the logarithm modulo each prime and then combine it, but do not know how exactly. Seems similar like problems for Chinese remainder theorem but I cannot find the way how to do it. You're real close; you do recombine them using the Chinese Remainder Theorem; however the ...

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