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# XorShift Jump 101, Part 2: Polynomial Arithmetic

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24 Apr 2020CPOL19 min read 6.4K   3
Jump forward/backward procedures for XorShift RNG explained step by step
This is a step by step tutorial explaining the math foundations and C++ implementation of the jump forward/backward procedures for the xorshift random number generators family. Part 2 is dedicated to algorithm based on polynomial arithmetic.

## Introduction

This is the second part of my tutorial dedicated to the jumping procedures for the xorshift family of random number generators. The first part gave a motivation for this article and considered an algorithm based on matrix multiplication. This time we will learn about another one using polynomials. Finally, we will analyze the theoretical and empirical performance of these two algorithms.

Note these two parts of the article share the same numbering of equations and references as well as the source code examples.

## Background

This section is exactly reproduced from the part 1 for the readers' convenience.

It is assumed that the reader is familiar with linear RNG theory basics explained in System.Random and Infinite Monkey Theorem. Unlike that article, the demo code for this one is written in C++ 11 so the reader should be fluent in this dialect at the intermediate level.

Polynomial-based approach requires some additional knowledge in the basics of algebra, such as the notion of polynomial, its properties and operations involving polynomials. Software implementation is based on NTL, a library of number theory methods developed by Victor Shoup. This library, in turn, needs GNU GMP and GF2X. Versions used for this article are NTL 11.0.0, GMP 6.1.2, and GF2X 1.2, respectively.

While NTL documentation mentions that it is possible to build this library with Visual C++, the recommended environment should be Unix-style, such as G++ (with accompanying toolchain) under Linux or Cygwin (I used this one).

Some C++ code for this article has been generated automatically by a program written in Haskell. Source code archive includes ready-to-use output of this generator so the reader doesn't necessarily have to run it. Otherwise, some working Haskell system, e.g., Haskell Platform, would be needed. Readers not familiar with Haskell would also need some beginner-level guide. I used Yet Another Haskell Tutorial by Hal Daumé III.

Google Test 1.8.1 is used for unit testing infrastructure.

## Polynomial Arithmetic-based Approach

Matrix-based jump described in Part 1 has significant drawback: the lack of efficiency. Indeed, matrix multiplication operations become quite expensive if you have large matrices: $O(N^3)$ for naive implementation and the lower boundary $\Omega(N^2)$ for the best implementation possible.

More efficient jump procedure based on polynomial arithmetic has been described in [2].

In this section we will begin from theory beyond this algorithm and continue with implementation details of forward and backward jumps. Since these algorithms are almost identical, it seems be a good idea to describe them both in parallel rather than separately.

### Theoretical Background

Let's begin from the definition of the characteristic polynomial. Given matrix $A$, polynomial

$p(z) = \det(zI + A) = z^N + \alpha_1 z^{N-1} + \cdots + \alpha_{N-1} z + \alpha_N$

is known as characteristic polynomial of this matrix. (Note that definition from Wikipedia uses another expression, $\det(zI - A)$, but in $\mathbb{F}_2$ addition and subtraction are the same.) Fundamental property of this polynomial is

$p(A) = A^N + \alpha_1 A^{N-1} + \cdots + \alpha_{N-1} A + \alpha_N I = 0.$

Polynomial arithmetic defines a notion of taking one polynomial modulo another one:

$g(z) = f(z) \bmod h(z),\quad \text{if } f(z) = q(z)h(z) + g(z).$

We will consider a particular case of

$g(z) = z^k \bmod p(z) = \alpha_1 z^{N-1} + \cdots + \alpha_{N-1} z + \alpha_N$

which can be calculated with logarithmic complexity with respect to $k$.

Using relation above, we can conclude that

$g(z) = z^k + q(z)p(z),$

and, if we substitute $A$ as an argument and rearrange terms,

$A^k = g(A) + q(A)p(A).$

Finally, since $p(A) = 0$,

$A^k = g(A).$

Let's use this equivalence to calculate our jump transform $T^kX$:

$T^kX = (\alpha_1 T^{N-1} + \cdots + \alpha_{N-1} T + \alpha_N I)X.$

This can also be written using Horner's method as

$\label{eq:horner} T^kX = T(\cdots T(T(T\alpha_1 X + \alpha_2 X) + \alpha_3 X) + \cdots + \alpha_{N-1}X) + \alpha_N X. \tag{2}$

Note the structure of this formula: first of all, we take state vector $X$, calculate $\alpha_1 X$ and multiply it by $T$. Then we add our state vector multiplied by $\alpha_2$ one more time getting new state vector which is in turn transformed by multiplying it by $T$ again, and so on. Now, what does it mean that some state vector is multiplied by $T$? Yes, it means that we take xorshift RNG in some state and do one step forward. And we already know simple algorithm to do that step fast, step_forward().

Would this algorithm be better than matrix-based one? First of all, it is an evident winner by memory consumption: our "transition polynomial" can be stored in $N$ bits of memory while transition matrix takes $N\times N$ bits. This is especially important for RNGs with large state vectors, such as Mersenne Twister where state vector consists from 624 32-bit words. In this case, transition matrix would take $(624*32)^2=398721024$ bits (about 47 megabytes) of memory! In our case of 128-bit xorshift the gain wouldn't be so significant but also noticeable ($128/8=16$ bytes versus $128^2/8=2048$ bytes).

Regarding calculation speed, if we have already prepared transition matrix or polynomial, asymptotic complexity is the same. Indeed, matrix to vector multiplication requires xoring together $N$ bit matrix row with state vector of the same length and this operation is repeated for each bit of the output giving us the resulting complexity $O(N^2)$. For polynomial-based transition we do single step of RNG algorithm and state vector addition ($O(N)$) for each non-zero coefficient ($N$ in the worst case). So, we will also have total complexity $O(N^2)$ but these estimations hide constant factors which could make them significantly different in our particular case. We will make an experiment to see what happens.

One more improvement to this algorithm is mentioned in [1]. Is characteristic polynomial of matrix $A$ the only one who have a property of $p(A)=0$? The answer is "no". There is a whole class of nullifying polynomials which can have a variety of degrees, and one with minimal degree is called minimal polynomial. If we use minimal polynomial instead of characteristic, we can expect that it's degree can be less than $N$ so even in the worst case we will perform less operations to calculate new RNG state.

### Forward Jump

Code discussed in this section and below is based on jump_ahead.tar.gz, reference implementation of jump algorithm for Mersenne Twister from its authors. Note that currently there are newer versions of this code, but my implementation used exactly that one.

For beginning, let's calculate minimal polynomial. To do that we will use a MinPolySeq function from the NTL library. This function takes a sequence of values produced by linear transformation (e.g. xorshift) and an upper boundary of minimal polynomial to be calculated. We know that minimal polynomial degree can't be greater than degree of characteristic polynomial, i.e. STATE_SIZE_EXP in our case. The sequence length must be at least twice longer than that boundary.

This code is implemented as function init_transition_polynomials. We must call it explicitly before the first use of polynomial-based transition algorithms. This approach is good enough in research but for production use we should better precalculate the needed polynomial and hardcode its coefficients in a source code as a static array of bytes which can be used to initialize the modulo polynomial on demand.

Other steps to perform transition are taking $x^k$ modulo minimal polynomial and applying it to given RNG state. The code for these functions is very similar for forward and backward jumps so it will be considered once after the next section where we will learn about implementation details of backward jump.

### Backward Jump

One possible way to implement jumping backward would be using the wrap-around property of the xorshift algorithm, that is, representing $k$ steps backward as $P-k = 2^N-(k+1)$ steps forward, where $P=2^N-1$ is a period length for the xorshift with state size $N$. I'm going to use another way and calculate the needed primitives for the reverse sequence. In other words, I'm going to create another RNG algorithm producing the same numbers as xorshift but in reverse direction.

First of all, let's calculate minimal polynomial. We can do this without any knowledge of reverse xorshift. Just let's take the sequence produced with xorshift but reverse it before passing to the MinPolySeq.

Next step would be more tricky. Applying polynomial to initial state ($\ref{eq:horner}$) requires single step of the RNG, in our case, stepping xorshift backward. Can we derive this formula from xorshift definition?

For beginning, let's recall the definition of xorshift128:

C++
uint32_t t = x ^ (x << 11);

x = y; y = z; z = w;
w = w ^ (w >> 19) ^ (t ^ (t >> 8));


where x, ..., w represent different 32-bit chunks of the 128-bit state. We can see, that components x, y, and z of the updated state are equal to y, z, and w of the initial state. So, restoring those three components of the initial state is trivial.

To restore x, let's use properties of the xor operation:

\begin{align*} x \oplus 0 &= x, \\ x \oplus y \oplus x &= y. \end{align*}

As you can see from the code above, new value of w component is calculated using its previous value (w ^ (w >> 19), let's call it w-subexpression for short) and value of some intermediate expression (t ^ (t >> 8), the t-subexpression, respectively) which, in turn, is calculated from x.

Our first step would be recovering the value of t-subexpression. To do that, we can use the z-component of the updated state. Indeed, this z gets its value right from the past w. Therefore, if we calculate z ^ (z >> 19) it would be equal to w ^ (w >> 19) calculated from the past state and if we xor it with the new value of w, we will peel off'' the w-subexpression and get pure value of the t-subexpression:

C++
uint32_t tt = w ^ (z ^ z >> 19); // (t ^ (t >> 8))


Note that this tt is not a t itself but an expression over t. Can we recover t?

Let's consider some simpler example with shorter bit vector to catch the basic idea. Let's have 8-bit quantity

$v = \left\langle v_7, v_6, v_5, v_4, v_3, v_2, v_1, v_0\right\rangle.$

Shifting this vector three bits to the right, for example, means following:

$v \gg 3 = \left\langle 0, 0, 0, v_7, v_6, v_5, v_4, v_3\right\rangle.$

And xoring these values together would be

$v \oplus (v \gg 3) = \left\langle v_7, v_6, v_5, v_4 \oplus v_7, v_3 \oplus v_6, v_2 \oplus v_5, v_1 \oplus v_4, v_0 \oplus v_3\right\rangle.$

Note that the three most significant bits retained their original values and we can extract them from bit vector and shift right three bits:

$\left\langle 0, 0, 0, v_7, v_6, v_5, 0, 0 \right\rangle.$

If we xor this bit vector with $v \oplus (v \gg 3)$ we will get

$\left\langle v_7 \oplus 0, v_6 \oplus 0, v_5 \oplus 0, (v_4 \oplus v_7) \oplus v_7, (v_3 \oplus v_6) \oplus v_6, (v_2 \oplus v_5) \oplus v_5, v_1 \oplus v_4 \oplus 0, v_0 \oplus v_3 \oplus 0\right\rangle,$

which can be simplified to

$\left\langle v_7, v_6, v_5, v_4, v_3, v_2, v_1 \oplus v_4, v_0 \oplus v_3\right\rangle.$

That's it! Using the result of the transformation, we could extract some bits of the argument and then recover more bits of this argument. Doing the same with the recently recovered bits ($v_4$ and $v_3$) we can go further and calculate the remaining two bits of $v$.

We can use this approach to recover the value of t:

C++
uint32_t t_3 = tt & 0xFF000000U;
uint32_t t_2 = (tt & 0x00FF0000U) ^ (t_3 >> 8);
uint32_t t_1 = (tt & 0x0000FF00U) ^ (t_2 >> 8);
uint32_t t_0 = (tt & 0x000000FFU) ^ (t_1 >> 8);

uint32_t t = t_3 | t_2 | t_1 | t_0; // x ^ (x << 11)


And, finally, the value of x:

C++
uint32_t x_10_00 = t & 0x000007FFU;
uint32_t x_21_11 = (t & 0x003FF800U) ^ (x_10_00 << 11);
uint32_t x_31_22 = (t & 0xFFC00000U) ^ (x_21_11 << 11);

// ...
x = x_31_22 | x_21_11 | x_10_00;


Now we are ready to implement jump procedure for both directions.

### Common Implementation

Let's start from initialization code which must be called before any other jump functions:

C++
NTL::GF2XModulus fwd_step_mod;
NTL::GF2XModulus bwd_step_mod;

void init_transition_polynomials()
{
state_t s;

init(s);

const size_t N = 2*STATE_SIZE_EXP;

NTL::vec_GF2 vf(NTL::INIT_SIZE, N);
NTL::vec_GF2 vb(NTL::INIT_SIZE, N);

for(long i = 0; i < N; i++)
{
step_forward(s);

vf[i] = vb[N - 1 - i] = s[3] & 0x01ul;
}

NTL::GF2X fwd_step_poly;
NTL::GF2X bwd_step_poly;

NTL::MinPolySeq(fwd_step_poly, vf, STATE_SIZE_EXP);
NTL::MinPolySeq(bwd_step_poly, vb, STATE_SIZE_EXP);

NTL::build(fwd_step_mod, fwd_step_poly);
NTL::build(bwd_step_mod, bwd_step_poly);
}

This function is definitely the most mysterious part of the polynomial-based jump implementation. For beginning, it creates an instance of the xorshift RNG and generates a random sequence twice as long as STATE_SIZE_EXP, the number of bits in the RNG state. The least significant bit of the each number of this sequence is stored in two vectors, vf and vb which are filled item by item but in different orders.

These sequences are then passed to the NTL function MinPolySeq which calculates minimal polynomials corresponding to the linear transformations specified by our RNG and its counterpart producing the same numbers in reverse direction. This NTL function is true magic: looking at the values of the single bit from the RNG output, it can extract the information about the whole RNG which is enough to construct its minimal polynomial.

Finally, this function initializes two global objects, fwd_step_mod and bwd_step_mod. This is an optimization provided by NTL. If you need to raise different polynomials to different powers modulo the same polynomial many times, it could be more efficient to preprocess modulo polynomial, save the result and then re-use it again an again.

C++
void prepare_transition(tr_poly_t &tr_k, uint64_t k, Direction dir)
{
tr_k.dir = dir;

NTL::GF2X x(1, 1);

switch(dir)
{
case Direction::FWD:
NTL::PowerMod(tr_k.poly, x, k, fwd_step_mod);
break;

case Direction::BWD:
NTL::PowerMod(tr_k.poly, x, k, bwd_step_mod);
break;
}
}

void add_state(state_t &x, const state_t &y)
{
for(size_t i = 0; i < x.size(); i++)
x[i] = x[i] ^ y[i];
}

void horner(state_t &s, const tr_poly_t &tr)
{
state_t tmp_state = s;

tmp_state.fill(0);

int i = NTL::deg(tr.poly);

if(i > 0)
{
for( ; i > 0; i--)
{
if(NTL::coeff(tr.poly, i) != 0)

if(tr.dir == Direction::FWD)
step_forward(tmp_state);
else
step_backward(tmp_state);
}

if(NTL::coeff(tr.poly, 0) != 0)
}

s = tmp_state;
}

void do_transition(state_t &s, const tr_poly_t &tr)
{
horner(s, tr);
}

These functions are self-explanatory: prepare_transition simply fills tr_k with $x^k \bmod f(x)$ where $f(x)$ is either fwd_step_mod or bwd_step_mod depending from dir. The last one, do_transition, applies polynomial tr to the RNG state s implementing Horners' method. In this case, dir parameter determines which single RNG step, forward or backward, should be performed.

## Performance comparison

I have already provided some theoretical considerations regarding performance of these two methods. Regadring memory complexity, there is nothing to add and we can only repeat that polynomial-based approach is the clear winner. So, our analysis below will be focused on time complexity only.

Our previous estimates were related to $N$, the size of the RNG state. They could be useful if you compare the same jumping algorithm for different RNGs, but use them with care: big $O$ estmates are asymptotical and may not work well for limited $N$ values corresponding to state sizes. In our case, we apply two methods to the same RNG, so $N$ is the same. Can we, therefore, conclude that our jumping algorithms have constant complexity? No, we haven't yet able, of course, since we have one more parameter to consider: the jump size $k$.

Note that our jump algorithms have two separate phases, preparation and transition. Their complexity may depend from parameters differently.

For matrix-based method, adding $k$ to the estimation is simpler: this parameter determines the complexity of preparation phase because we raise matrix to the power $k$. Given constant $N$, power is calculated in $O(\log k)$ steps. Transition phase is even better: it involves singe matrix-to-vector multiplication which doesn't depend from $k$ at all and transition is indeed $O(1)$ with regard to $k$.

For polynomial-based approach analysis is less clear since we use a third-party library so we don't know its performance well. From common sense, raising polynomial to the power $k$ can also be implemented with an $O(\log k)$ exponentiation by squaring, similarly to matrices. Transition phase is more difficult to estimate due to less clear relation between $k$ and the degree of the resulting polynomial which, ultimately, determines transition complexity. Fortunately, this degree is upper-bound by the degree of minimal polynomial which, in turn, bound by $N$. Given constant $N$, we can again conclude that polynomial-based transition is $O(1)$ with regard to $k$.

Putting it all togehter we can decide that both approaches are aspymptotically equal. Pretty good result if you write a paper in computer science but not enough for software development: with limited values of $k$, those hidden factors ignored with big $O$ may outweight the influence of $k$. To estimate those factors, let's do an experiment.

I'm going to measure time needed to prepare transition and to apply it for some number of random jump lengths. Taking into account our conclusion about logarithmic jump complexity, it looks reasonable to probe $k$ at each order of magnitude. To combine that with randomness, we can implement a kind of stratified sampling: given that $k$ can be saved in $\left\lceil\log_2 k\right\rceil$ bits, for each bit of this sequence we will consider it as the most significant (and, therefore set it to one) and fill the rest of bits randomly. This way we will create a random sample where numbers of each order of magnitude are represented equally.

This sampling procedure is implemented in xstime.cc. Function generate_test_cases produces n_trials samples for each stratum within msb_pow. Additionally, this function is used as the final sanity check for transition code: for each generated sample jump size forward jump is prepared and performed with both methods, matrix- and polynomial-based, making sure the resulting RNG state is the same.

Template function run_test is straitforward: it runs transition for each specified test case with time measurement and outputs the result to the standard output in five-column tab-delimited format. The columns are: name of the algorighm used, jump direction, order of magnitude (as $\left\lfloor\log_2 k\right\rfloor$) and, finally, time of preparation and time of jump in microseconds.

The goal of our measurements was getting a rough estimate of the run time sacrificing accuracy for code portability. Therefore, I used a high-resolution clock from the standard C++ library instead of some system-specific timers or hardware counters. Similarly, no special code is added to xstime.cc to control the process priority or thread affinity, etc.

After running testbench program with my laptop (i7-4702MQ, G++ 5.4.0), I've got a data sample which should be properly postprocessed and visualized. Let's do it using the R programming language.

My R session is recorded below. For beginning, let's load two popular libraries for data processing and visualization:

library(dplyr)
library(ggplot2)

timings <- read.table("timing.1.txt",
col.names = c("Alg", "Dir", "logK", "Prepare", "Jump"))

Getting first insights from the data:

timings %>% group_by(Alg, Dir) %>% summarise(Prepare = mean(Prepare), Jump = mean(Jump))

Mean time spent on preparing and doing the jump for each method and jump direction:

# A tibble: 5 x 4
# Groups:   Alg [3]
Alg      Dir     Prepare   Jump
<fct>    <fct>     <dbl>  <dbl>
1 matr     BWD   175080.   3.16
2 matr     FWD   174040.   3.14
3 matr_ntl FWD     4202.   0.0569
4 poly     BWD        9.97 1.42
5 poly     FWD        9.30 1.04 

Our first insight would be: our matrix-based transition designated with "matr" in the first column is more than $10000$ times slower than polynomial-based during preparation phase! Given this difference, the factor of $2$ between transition times looks negligible.

Third row of the result contains some jumping method, matr_ntl, we haven't discussed yet. After getting first performance figures for other methods, it has became evident that my matrix arithmetic implementation is indeed horribly inefficient. So, the question which has naturally appeared next: can we do better? The NTL library contains its own implementation of matrix arithmetic which, I beleive, should be state-of-art. So, matr_ntl is an alternative implementation of the matrix-based jump using NTL. Since I'm only interested in checking its performance during jump preparation phase, do_transition for this method is left empty. Also, since this code has not been intended for use outside timing experiment, is has been put into xstime.cc.

NTL-based matrix approach has much better performance, 41 times faster than my naive implementation, but it is anyway orders of magnitude slower than polynomial.

Let's draw some nicely-looking charts to show the dependence of performance from $k$:

sparse.labels <- function(breaks) { ifelse(1:length(breaks) %% 10 == 0, breaks, "") }

ggplot(data = timings %>% filter(Alg == "matr")) +
geom_boxplot(aes(x = as.factor(logK), y = Prepare, color = Dir)) +
xlab("logK") + scale_x_discrete(labels = sparse.labels)
Figure 1: Matrix-based prepare time
ggplot(data = timings %>% filter(Alg == "matr_ntl")) +
geom_boxplot(aes(x = as.factor(logK), y = Prepare, color = Dir)) +
xlab("logK") + scale_x_discrete(labels = sparse.labels)
Figure 2: NTL Matrix-based prepare time
ggplot(data = timings %>% filter(Alg == "poly")) +
geom_boxplot(aes(x = as.factor(logK), y = Prepare, color = Dir)) +
xlab("logK") + scale_x_discrete(labels = sparse.labels)
Figure 3: Polynomial-based prepare time

Looking at figures 1, 2, and 3, we can see quite evident linear dependency between jump preparation time and $\log_2 k$. This confirms our theoretical estimate that preparation time is $O(\log k)$.

To get some qualitative evidence for our hypothesis, we can fit linear model to our data. Keeping article smaller, I'm going to fit forward jump measurements only.

matr_prep.lm <- lm(Prepare ~ logK, data = timings %>% filter(Alg == "matr" & Dir == "FWD"))
summary(matr_prep.lm)
Call:
lm(formula = Prepare ~ logK, data = timings %>% filter(Alg ==
"matr" & Dir == "FWD"))

Residuals:
Min     1Q Median     3Q    Max
-42474  -5425    367   6143  45880

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  6308.19     841.64   7.495 2.31e-13 ***
logK         5324.81      23.23 229.207  < 2e-16 ***
---
Signif. codes:  0 ***' 0.001 **' 0.01 *' 0.05 .' 0.1  ' 1

Residual standard error: 10350 on 618 degrees of freedom
Multiple R-squared:  0.9884,	Adjusted R-squared:  0.9884
F-statistic: 5.254e+04 on 1 and 618 DF,  p-value: < 2.2e-16
matr_ntl_prep.lm <- lm(Prepare ~ logK, data = timings %>% filter(Alg == "matr_ntl" & Dir == "FWD"))
summary(matr_ntl_prep.lm)
Call:
lm(formula = Prepare ~ logK, data = timings %>% filter(Alg ==
"matr_ntl" & Dir == "FWD"))

Residuals:
Min      1Q  Median      3Q     Max
-1253.0  -152.3     8.1   147.2  1385.0

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -320.2577    22.0018  -14.56   <2e-16 ***
logK         143.5789     0.6073  236.42   <2e-16 ***
---
Signif. codes:  0 ***' 0.001 **' 0.01 *' 0.05 .' 0.1  ' 1

Residual standard error: 270.6 on 618 degrees of freedom
Multiple R-squared:  0.9891,	Adjusted R-squared:  0.989
F-statistic: 5.589e+04 on 1 and 618 DF,  p-value: < 2.2e-16

For matrix-based jumps, preparation procedure matches linear model quite well: logK variable is significant and $R^2$ score is quite close to $1$.

poly_prep.lm <- lm(Prepare ~ logK, data = timings %>% filter(Alg == "poly" & Dir == "FWD"))
summary(poly_prep.lm)
Call:
lm(formula = Prepare ~ logK, data = timings %>% filter(Alg ==
"poly" & Dir == "FWD"))

Residuals:
Min      1Q  Median      3Q     Max
-1.7293 -0.6262 -0.3608 -0.0511 18.9861

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.519085   0.171505   3.027  0.00258 **
logK        0.278856   0.004734  58.905  < 2e-16 ***
---
Signif. codes:  0 ***' 0.001 **' 0.01 *' 0.05 .' 0.1  ' 1

Residual standard error: 2.109 on 618 degrees of freedom
Multiple R-squared:  0.8488,	Adjusted R-squared:  0.8486
F-statistic:  3470 on 1 and 618 DF,  p-value: < 2.2e-16

Note that for polynomial-based jump preparation is a bit worse: $R^2$ is only about $84\%$. How this could happen? Looking at Fig. 3, you can notice that outliers are relatively farther from the main sequence so our linear model performs worse explaining data variability.

ggplot(data = timings %>% filter(Dir == "FWD")) +
geom_boxplot(aes(x = as.factor(logK), y = Jump, color = Alg)) +
xlab("logK") + scale_x_discrete(labels = sparse.labels)
Figure 4: Jump forward transition time
ggplot(data = timings %>% filter(Dir == "BWD")) +
geom_boxplot(aes(x = as.factor(logK), y = Jump, color = Alg)) +
xlab("logK") + scale_x_discrete(labels = sparse.labels)
Figure 5: Jump backward transition time

Linear models for transition procedures:

matr_jump.lm <- lm(Jump ~ logK, data = timings %>% filter(Alg == "matr" & Dir == "FWD"))
summary(matr_jump.lm)
Call:
lm(formula = Jump ~ logK, data = timings %>% filter(Alg == "matr" &
Dir == "FWD"))

Residuals:
Min      1Q  Median      3Q     Max
-0.4843 -0.3436 -0.3029  0.1088 15.0334

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.085267   0.102040  30.236   <2e-16 ***
logK        0.001768   0.002817   0.628     0.53
---
Signif. codes:  0 ***' 0.001 *' 0.01 *' 0.05 .' 0.1  ' 1

Residual standard error: 1.255 on 618 degrees of freedom
Multiple R-squared:  0.0006374,	Adjusted R-squared:  -0.0009797
F-statistic: 0.3941 on 1 and 618 DF,  p-value: 0.5304
poly_jump.lm <- lm(Jump ~ logK, data = timings %>% filter(Alg == "poly" & Dir == "FWD"))
summary(poly_jump.lm)
Call:
lm(formula = Jump ~ logK, data = timings %>% filter(Alg == "poly" &
Dir == "FWD"))

Residuals:
Min      1Q  Median      3Q     Max
-0.8818 -0.2126 -0.0505  0.2311  9.3183

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.844373   0.049360  17.106  < 2e-16 ***
logK        0.006241   0.001362   4.581 5.61e-06 ***
---
Signif. codes:  0 ***' 0.001 **' 0.01 *' 0.05 .' 0.1  ' 1

Residual standard error: 0.6071 on 618 degrees of freedom
Multiple R-squared:  0.03284,	Adjusted R-squared:  0.03127
F-statistic: 20.98 on 1 and 618 DF,  p-value: 5.607e-06`

We can see that for matrix-based transition model confirms our hypothesis about $O(1)$ complexity regarding $k$. However, for polynomial-based transition model suggests some dependency from $k$ but with quite small coefficient. Also note remarkably small $R^2$ score in both cases. This is disappointing but can also be explained. Indeed, since our regression line is almost horizontal, it is unable to catch well the variability in measurements for each value of $\log k$. This makes sense, since this variability is caused not by change in $\log k$ but by the performance measurement uncertainty. Also, note that we in fact model dependency from rounded value $\left\lfloor \log_2 k \right\rfloor$ loosing some information about $k$ which, in turn, causes more hidden uncertainty.

## Conclusion

We have learned a lot from this tutorial. First of all, we have seen how to construct non-trivial transition matrix given RNG definition. Doing symbolic transformations with Haskell exposed the power of this language: a convenient DSL has been defined in a dosen lines of code. After that, we considered an alternative approach based on polynomial arithmetic. It is pretty amazing how seemingly unrelated mathematical objects can be used together! Finally, we compared the performance of these two methods.

From purely technical point of view, polynomial-based approach is a clear winner both in memory and time complexity. Matrix-based approach, however, has an important non-technical strength: the simplicity. This method can be easily implemented from scratch by a sophomore majoring in math or CS, while polynomial-based approach requires more advanced skills. This may matter if you are unable to use some canned implementation, like NTL, for any reason, either legal or technical. Also, if you only need doing jumps for some predefined step size $k$ (e.g. iterating random substreams), transition matrix can be precalculated. Performing transition per se is comparable by speed for both methods. Current implementation of the matrix-based transition is about two times slower, but it can be possibly optimized using vectorization and/or SIMD extensions to parallelize matrix row by vector multiplication and speed up final bitwise sum with special operation "population count" available on some CPU architectures.

## References

This section is exactly reproduced from the part 1 for the readers' convenience.

[1] Hiroshi Haramoto, Makoto Matsumoto, and Pierre L’Ecuyer. A fast jump ahead algorithm for linear recurrences in a polynomial space. In Proceedings of the 5th International Conference on Sequences and Their Applications, SETA ’08, pages 290–298, Berlin, Heidelberg, 2008. Springer-Verlag. (Available online.)

[2] Hiroshi Haramoto, Makoto Matsumoto, Takuji Nishimura, François Panneton, and Pierre L’Ecuyer. Efficient jump ahead for $\mathbb{F}_2$-linear random number generators. INFORMS Journal on Computing, 20(3):385–390, 2008. (Available online.)

[3] Donald E. Knuth. The Art of Computer Programming, Volume 2 (3rd Ed.): Seminumerical Algorithms. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1997.

[4] P. L’Ecuyer and R. Simard. TestU01: A C library for empirical testing of random number generators. ACM Transactions on Mathematical Software, 33(4): Article 22, August 2007.

[5] George Marsaglia. Xorshift RNGs. Journal of Statistical Software, Articles, 8(14):1–6, 2003. (Available online.)