# How would we design a fast circuit to solve LCS35?

LCS35 is a time-lock puzzle, stated in Ronald L. Rivest's Description of the LCS35 Time Capsule Crypto-Puzzle [1] (April 1999). That instantiates a system in Ronald L. Rivest, Adi Shamir and David A. Wagner's Time-Lock puzzles and timed-release Crypto [2] (MIT technical report, 1996, revised March 1999).

Solving the puzzle boils down to computing $$2^{(2^t)}\bmod n$$ for $$n$$ a public 2046-bit RSA modulus of secret factorization, and $$t\approx1.13\cdot2^{46}$$. The reference method sequentially computes $$w_i=2^{(2^i)}\bmod n$$ for $$i$$ up to $$t$$, by iterating $$w_{i+1}\gets w_i^2\bmod n$$, starting from $$w_0=2$$, or perhaps $$w_{10}=2^{1024}$$.

How would we design a fast circuit for that purpose? And what speed (in equivalent modular squaring per second) could it achieve? I'm interested in

• Mathematical techniques: e.g. would Montgomery arithmetic help?
• Architectural tradeoffs: e.g. between circuit depth and size, both causing a slowdown.
• Technology: I do not even know what technology is currently the state of the art when it comes to fast logic, much less what it could realistically achieve.

The design of LCS35 attempts to block attempts at massive parallelization, and mostly succeeds as far as we know. The question is about making the computation nevertheless.

The value of $$t$$ was chosen assuming 3000 squaremods/second in 1999, exponential grows to ×13 that per 2012, then again ×5 per 2034 (the challenge ends in 2033).

I propose to ignore:

• Operating cost: it would be negligible compared to bitcoin mining, unless we find a way to parallelize, which would be a major breakthru.
• Investment cost: it depends on too many factors, and evaluations are not falsifiable.
• Computation errors, since we know how to deal with them:
• Reference [1] suggests making the computation of $$w'_i=2^{(2^i)}\bmod(c\,n)$$ where $$c$$ is a moderate prime (50-bit). This allows a periodical check that $$w'_i\bmod c\ =\ 2^{(2^i)\bmod (c'-1)}\bmod c$$, backtracking if an error creeps, and getting $$w_t\gets w'_t\bmod n$$ in the end.
• A variant is to compute both $$w'_i=2^{(2^t)}\bmod(3n)$$ and $$w''_i=2^{(2^t)}\bmod(5n)$$ on two independent engines (conveniently, both moduli are 2048-bit), and periodically check that $$w'_i\bmod n=w''_i\bmod n$$. That requires two independent implementations but little increase in the modulus size, hence (I guess) faster computation and smaller computing engines.
• Attacks that essentially factor $$n$$, including using a Quantum Computer.
• Alternatives to LCS35 for timed-release crypto, as asked in this other question.
• Any chance that this question might be as opinionated/speculative/sci-fi as my recent answer? – Paul Uszak Mar 11 '19 at 12:41
• BTW, you may be interested in Makwa (by the bear) which is essentially this disguised as a password-hash. Section 5.6 of the paper (PDF) has some attack estimates. – SEJPM Mar 11 '19 at 13:01
• @b degnan: the paper gives $t=79685186856218$. That's the number of sqarings modulo $n$. At a rate of 27 million per second, that would be <35 days. As a first order approximation, a classical massively parallel squaring modulo $n$ has depth 3 of multiplier-adder, with the 3 requiring in the order of 3000 muladd blocks 64x64+128->128 (that perhaps could be reduced by Montgomery arithmetic; and very certainly can be reduced by going sequential, at the expense of time). Carry lookahead is an issue. I'm missing even a raw order of magnitude for what fits a single device, propagation delay. – fgrieu Mar 11 '19 at 14:38
• The Makwa figures are for parallel independent modular squarings, since you'd be attempting to crack candidate passwords. It says little about how fast you can make a single exponentiation. Maybe something like residue number systems to make the underlying multiplications as parallel as possible would be useful, but who knows how that'd perform in practice. – Samuel Neves Mar 11 '19 at 15:50
• @Samuel Neves: yes, Makwa figures are (purposely) not taking sequential into consideration because they can attack multiple instances, and that makes a difference; I do not know how large. Yes, the Bernstein and Sorenson technique is to consider; but a problem with residue number systems is the cost + complexity of periodically (and quite often) performing a global CRT and modular reduction. Repeated squaring seems a worst case scenario for that. – fgrieu Mar 11 '19 at 16:08