1 IDEAS ABOUT THINGS TO WORK ON
3 * mpq_cmp: Maybe the most sensible thing to do would be to multiply the, say,
4 4 most significant limbs of each operand and compare them. If that is not
5 sufficient, do the same for 8 limbs, etc.
7 * Write mpi, the Multiple Precision Interval Arithmetic layer.
9 * Write `mpX_eval' that take lambda-like expressions and a list of operands.
11 * As a general rule, recognize special operand values in mpz and mpf, and
12 use shortcuts for speed. Examples: Recognize (small or all) 2^n in
13 multiplication and division. Recognize small bases in mpz_pow_ui.
15 * Implement lazy allocation? mpz->d == 0 would mean no allocation made yet.
17 * Maybe store one-limb numbers according to Per Bothner's idea:
21 mp_limb val; /* if (d == NULL). */
22 mp_size size; /* Length of data array, if (d != NULL). */
25 Problem: We can't normalize to that format unless we free the space
26 pointed to by d, and therefore small values will not be stored in a
29 * Document complexity of all functions.
31 * Add predicate functions mpz_fits_signedlong_p, mpz_fits_unsignedlong_p,
32 mpz_fits_signedint_p, etc.
34 mpz_floor (mpz, mpq), mpz_trunc (mpz, mpq), mpz_round (mpz, mpq).
36 * Better random number generators. There should be fast (like mpz_random),
37 very good (mpz_veryrandom), and special purpose (like mpz_random2). Sizes
38 in *bits*, not in limbs.
40 * It'd be possible to have an interface "s = add(a,b)" with automatic GC.
41 If the mpz_xinit routine remembers the address of the variable we could
42 walk-and-mark the list of remembered variables, and free the space
43 occupied by the remembered variables that didn't get marked. Fairly
46 * Improve speed for non-gcc compilers by defining umul_ppmm, udiv_qrnnd,
47 etc, to call __umul_ppmm, __udiv_qrnnd. A typical definition for
49 #define umul_ppmm(ph,pl,m0,m1) \
50 {unsigned long __ph; (pl) = __umul_ppmm (&__ph, (m0), (m1)); (ph) = __ph;}
51 In order to maintain just one version of longlong.h (gmp and gcc), this
52 has to be done outside of longlong.h.
54 Bennet Yee at CMU proposes:
55 * mpz_{put,get}_raw for memory oriented I/O like other *_raw functions.
56 * A function mpfatal that is called for exceptions. Let the user override
59 * Make all computation mpz_* functions return a signed int indicating if the
60 result was zero, positive, or negative?
62 * Implement mpz_cmpabs, mpz_xor, mpz_to_double, mpz_to_si, mpz_lcm, mpz_dpb,
63 mpz_ldb, various bit string operations. Also mpz_@_si for most @??
65 * Add macros for looping efficiently over a number's limbs:
66 MPZ_LOOP_OVER_LIMBS_INCREASING(num,limb)
67 { user code manipulating limb}
68 MPZ_LOOP_OVER_LIMBS_DECREASING(num,limb)
69 { user code manipulating limb}
71 Brian Beuning proposes:
72 1. An array of small primes
73 3. A function to factor a mpz_t. [How do we return the factors? Maybe
74 we just return one arbitrary factor? In the latter case, we have to
75 use a data structure that records the state of the factoring routine.]
76 4. A routine to look for "small" divisors of an mpz_t
77 5. A 'multiply mod n' routine based on Montgomery's algorithm.
80 1. A way to find out if an integer fits into a signed int, and if so, a
81 way to convert it out.
82 2. Similarly for double precision float conversion.
83 3. A function to convert the ratio of two integers to a double. This
84 can be useful for mixed mode operations with integers, rationals, and
87 Elliptic curve method description in the Chapter `Algorithms in Number
88 Theory' in the Handbook of Theoretical Computer Science, Elsevier,
89 Amsterdam, 1990. Also in Carl Pomerance's lecture notes on Cryptology and
90 Computational Number Theory, 1990.
92 * Harald Kirsh suggests:
93 mpq_set_str (MP_RAT *r, char *numerator, char *denominator).
95 * New function: mpq_get_ifstr (int_str, frac_str, base,
96 precision_in_som_way, rational_number). Convert RATIONAL_NUMBER to a
97 string in BASE and put the integer part in INT_STR and the fraction part
98 in FRAC_STR. (This function would do a division of the numerator and the
101 * Should mpz_powm* handle negative exponents?
103 * udiv_qrnnd: If the denominator is normalized, the n0 argument has very
104 little effect on the quotient. Maybe we can assume it is 0, and
105 compensate at a later stage?
107 * Better sqrt: First calculate the reciprocal square root, then multiply by
108 the operand to get the square root. The reciprocal square root can be
109 obtained through Newton-Raphson without division. To compute sqrt(A), the
116 The final result can be computed without division using,
121 * Newton-Raphson using multiplication: We get twice as many correct digits
122 in each iteration. So if we square x(k) as part of the iteration, the
123 result will have the leading digits in common with the entire result from
124 iteration k-1. A _mpn_mul_lowpart could help us take advantage of this.
126 * Peter Montgomery: If 0 <= a, b < p < 2^31 and I want a modular product
127 a*b modulo p and the long long type is unavailable, then I can write
129 typedef signed long slong;
130 typedef unsigned long ulong;
131 slong a, b, p, quot, rem;
133 quot = (slong) (0.5 + (double)a * (double)b / (double)p);
134 rem = (slong)((ulong)a * (ulong)b - (ulong)p * (ulong)quot);
135 if (rem < 0} {rem += p; quot--;}
137 * Speed modulo arithmetic, using Montgomery's method or my pre-inversion
138 method. In either case, special arithmetic calls would be needed,
139 mpz_mmmul, mpz_mmadd, mpz_mmsub, plus some kind of initialization
140 functions. Better yet: Write a new mpr layer.
142 * mpz_powm* should not use division to reduce the result in the loop, but
143 instead pre-compute the reciprocal of the MOD argument and do reduced_val
144 = val-val*reciprocal(MOD)*MOD, or use Montgomery's method.
146 * mpz_mod_2expplussi -- to reduce a bignum modulo (2**n)+s
148 * It would be a quite important feature never to allocate more memory than
149 really necessary for a result. Sometimes we can achieve this cheaply, by
150 deferring reallocation until the result size is known.
152 * New macro in longlong.h: shift_rhl that extracts a word by shifting two
153 words as a unit. (Supported by i386, i860, HP-PA, POWER, 29k.) Useful
154 for shifting multiple precision numbers.
156 * The installation procedure should make a test run of multiplication to
157 decide the threshold values for algorithm switching between the available
160 * Fast output conversion of x to base B:
161 1. Find n, such that (B^n > x).
162 2. Set y to (x*2^m)/(B^n), where m large enough to make 2^n ~~ B^n
163 3. Multiply the low half of y by B^(n/2), and recursively convert the
164 result. Truncate the low half of y and convert that recursively.
165 Complexity: O(M(n)log(n))+O(D(n))!
167 * Improve division using Newton-Raphson. Check out "Newton Iteration and
168 Integer Division" by Stephen Tate in "Synthesis of Parallel Algorithms",
169 Morgan Kaufmann, 1993 ("beware of some errors"...)
171 * Improve implementation of Karatsuba's algorithm. For most operand sizes,
172 we can reduce the number of operations by splitting differently.
174 * Faster multiplication: The best approach is to first implement Toom-Cook.
175 People report that it beats Karatsuba's algorithm already at about 100
176 limbs. FFT would probably never beat a well-written Toom-Cook (not even for
181 * Multiplication could be done with Montgomery's method combined with
182 the "three primes" method described in Lipson. Maybe this would be
183 faster than to Nussbaumer's method with 3 (simple) moduli?
185 * Maybe the modular tricks below are not needed: We are using very
186 special numbers, Fermat numbers with a small base and a large exponent,
187 and maybe it's possible to just subtract and add?
189 * Modify Nussbaumer's convolution algorithm, to use 3 words for each
190 coefficient, calculating in 3 relatively prime moduli (e.g.
191 0xffffffff, 0x100000000, and 0x7fff on a 32-bit computer). Both all
192 operations and CRR would be very fast with such numbers.
194 * Optimize the Schoenhage-Stassen multiplication algorithm. Take advantage
195 of the real valued input to save half of the operations and half of the
196 memory. Use recursive FFT with large base cases, since recursive FFT has
197 better memory locality. A normal FFT get 100% cache misses for large
200 * In the 3-prime convolution method, it might sometimes be a win to use 2,
201 3, or 5 primes. Imagine that using 3 primes would require a transform
202 length of 2^n. But 2 primes might still sometimes give us correct
203 results with that same transform length, or 5 primes might allow us to
204 decrease the transform size to 2^(n-1).
206 To optimize floating-point based complex FFT we have to think of:
208 1. The normal implementation accesses all input exactly once for each of
209 the log(n) passes. This means that we will get 0% cache hit when n >
210 our cache. Remedy: Reorganize computation to compute partial passes,
211 maybe similar to a standard recursive FFT implementation. Use a large
212 `base case' to make any extra overhead of this organization negligible.
214 2. Use base-4, base-8 and base-16 FFT instead of just radix-2. This can
215 reduce the number of operations by 2x.
217 3. Inputs are real-valued. According to Knuth's "Seminumerical
218 Algorithms", exercise 4.6.4-14, we can save half the memory and half
219 the operations if we take advantage of that.
221 4. Maybe make it possible to write the innermost loop in assembly, since
222 that could win us another 2x speedup. (If we write our FFT to avoid
223 cache-miss (see #1 above) it might be logical to write the `base case'
226 5. Avoid multiplication by 1, i, -1, -i. Similarly, optimize
227 multiplication by (+-\/2 +- i\/2).
229 6. Put as many bits as possible in each double (but don't waste time if
230 that doesn't make the transform size become smaller).
232 7. For n > some large number, we will get accuracy problems because of the
233 limited precision of our floating point arithmetic. This can easily be
234 solved by using the Karatsuba trick a few times until our operands
237 8. Precompute the roots-of-unity and store them in a vector.
240 * When a division result is going to be just one limb, (i.e. nsize-dsize is
241 small) normalization could be done in the division loop.
243 * Never allocate temporary space for a source param that overlaps with a
244 destination param needing reallocation. Instead malloc a new block for
245 the destination (and free the source before returning to the caller).
247 * Parallel addition. Since each processors have to tell it is ready to the
248 next processor, we can use simplified synchronization, and actually write
249 it in C: For each processor (apart from the least significant):
251 while (*svar != my_number)
253 *svar = my_number + 1;
255 The least significant processor does this:
257 *svar = my_number + 1; /* i.e., *svar = 1 */
259 Before starting the addition, one processor has to store 0 in *svar.
261 Other things to think about for parallel addition: To avoid false
262 (cache-line) sharing, allocate blocks on cache-line boundaries.
269 version-control: never