You mention 'random'. Below is 'random' in SBCL. I am clueless about how
good it is.
It takes people like Knuth to do this sort of thing well. Way over my head,
even though I have a Ph. D. in math.
I once heard the following random assertion: NSA, rms's favorite
organization I am sure, was at the time the largest employer of
mathematicians in the USA.
Did you know that the British spooks invented RSA before RSA did? Of course
those fucking spooks didn't tell the public. Criminal.
I wonder if Churchill had Turing, among others, in mind when he said: Never
in the field of human conflict was so much owed by so many to so few.
'Shit' is all I have to say about security, despite the large part of my past
salary that might have been federally spook funded! Unlike rms, I like to get
paid. At NSA's insistence, I got kicked off the board of my company
Eventually, when funding dried up, the company folded.
Thank God for RSA. I can remember at a party for the MIT AI Lab, when funding
did not look too hot, Moses said that 'security' might be a way to go to look
for funding. It may be nice for him that our president gets $2,000,000,000
a year for the White House, in part to make it secure, but if anyone claims to know
anything about security, I say, how come the news is constantly filled with
horror stories about hackers breaking in.
I once heard that Minsky, bless his soul, refused to let his students spend
time worrying about security on his DEC PDP-10 because it would only lead to
other students wasting time figuring out how to crack the system. But
'random' is one of the coolest ideas in the world. Where would we ever be
without hash tables! I am so glad that 'sxhash' is in Emacs Lisp and Common
Lisp!
I send far too much email. Is there a group Email Addicts Anonymous?
I am 77. I cannot pick up a coin on the floor. I am not to be trusted about
anything I say.
With highest regards,
Bob
I recommend to Emacs folks working on 'native-compiler' that they look at the
18 uses of "declare" in the following! You want efficiency in Lisp, you use
"declare".
------------------------------------------------------------------------------
;;;; This software is part of the SBCL system. See the README file for
;;;; more information.
;;;;
;;;; This software is derived from the CMU CL system, which was
;;;; written at Carnegie Mellon University and released into the
;;;; public domain. The software is in the public domain and is
;;;; provided with absolutely no warranty. See the COPYING and CREDITS
;;;; files for more information.
(in-package "SB-KERNEL")
;;;; Constants
(defconstant mt19937-n 624)
(defconstant mt19937-m 397)
(defconstant mt19937-upper-mask #x80000000)
(defconstant mt19937-lower-mask #x7FFFFFFF)
(defconstant mt19937-a #x9908B0DF)
(defconstant mt19937-b #x9D2C5680)
(defconstant mt19937-c #xEFC60000)
;;;; RANDOM-STATEs
;;; The state is stored in a (simple-array (unsigned-byte 32) (627))
;;; wrapped in a random-state structure:
;;;
;;; 0-1: Constant matrix A. [0, #x9908b0df]
;;; 2: Index k.
;;; 3-626: State.
(deftype random-state-state () `(simple-array (unsigned-byte 32) (,(+ 3 mt19937-n))))
(defmethod make-load-form ((random-state random-state) &optional environment)
(make-load-form-saving-slots random-state :environment environment))
(defmethod print-object ((state random-state) stream)
(if (and *print-readably* (not *read-eval*))
(print-not-readable-error state stream)
(format stream "#S(~S ~S #.~S)"
'random-state
':state
`(make-array ,(+ 3 mt19937-n)
:element-type
'(unsigned-byte 32)
:initial-contents
',(coerce (random-state-state state) 'list)))))
;;; Generate and initialize a new random-state array. Index is
;;; initialized to 1 and the states to 32bit integers excluding zero.
;;;
;;; Seed - A 32bit number.
;;;
;;; See Knuth TAOCP Vol2. 3rd Ed. P.106 for multiplier.
;;; In the previous versions, MSBs of the seed affect only MSBs of the array.
(defun init-random-state (&optional (seed 5489) state)
(declare (type (unsigned-byte 32) seed))
(let ((state (or state (make-array 627 :element-type '(unsigned-byte 32)))))
(check-type state random-state-state)
(setf (aref state 0) 0)
(setf (aref state 1) mt19937-a)
(setf (aref state 2) mt19937-n)
(loop for i below mt19937-n
for p from 3
for s = seed then
(logand #xFFFFFFFF
(+ (* 1812433253
(logxor s (ash s -30)))
i))
do (setf (aref state p) s))
state))
(defvar *random-state*)
(defun !random-cold-init ()
(/show0 "entering !RANDOM-COLD-INIT")
(setf *random-state* (%make-random-state (init-random-state)))
(/show0 "returning from !RANDOM-COLD-INIT"))
;;; Q: Why is there both MAKE-RANDOM-STATE and SEED-RANDOM-STATE?
;;; A: Because the DEFKNOWN for MAKE-RANDOM-STATE is more restricted
;;; and doesn't accept numerical state.
(defun make-random-state (&optional state)
"Make a random state object. The optional STATE argument specifies a seed
for deterministic pseudo-random number generation.
As per the Common Lisp standard,
- If STATE is NIL or not supplied, return a copy of the default
*RANDOM-STATE*.
- If STATE is a random state, return a copy of it.
- If STATE is T, return a randomly initialized state (using operating-system
provided randomness where available, otherwise a poor substitute based on
internal time and PID).
See SB-EXT:SEED-RANDOM-STATE for a SBCL extension to this functionality."
(/show0 "entering MAKE-RANDOM-STATE")
(seed-random-state state))
(defun fallback-random-seed ()
;; When /dev/urandom is not available, we make do with time and pid
;; Thread ID and/or address of a CONS cell would be even better, but...
;; [ADDRESS-BASED-COUNTER-VAL in 'target-sxhash' could be used here]
(/show0 "No /dev/urandom, using randomness from time and pid")
(+ (get-internal-real-time)
(ash (sb-unix:unix-getpid) 32)))
#-win32
(defun os-random-seed ()
(or
;; On unices, we try to read from /dev/urandom and pass the results
;; to our (simple-array (unsigned-byte 32) (*)) processor below.
;; More than 256 bits would provide a false sense of security.
;; If you need more bits than that, you probably also need
;; a better algorithm too.
(ignore-errors
(with-open-file (r "/dev/urandom" :element-type '(unsigned-byte 32)
:direction :input :if-does-not-exist :error)
(let ((a (make-array '(8) :element-type '(unsigned-byte 32))))
(aver (= 8 (read-sequence a r)))
a)))
(fallback-random-seed)))
#+win32
(defun os-random-seed ()
(/show0 "Getting randomness from CryptGenRandom")
(or (sb-win32:crypt-gen-random 32)
(fallback-random-seed)))
(defun seed-random-state (&optional state)
"Make a random state object. The optional STATE argument specifies a seed
for deterministic pseudo-random number generation.
As per the Common Lisp standard for MAKE-RANDOM-STATE,
- If STATE is NIL or not supplied, return a copy of the default
*RANDOM-STATE*.
- If STATE is a random state, return a copy of it.
- If STATE is T, return a randomly initialized state (using operating-system
provided randomness where available, otherwise a poor substitute based on
internal time and pid).
As a supported SBCL extension, we also support receiving as a seed an object
of the following types:
- (SIMPLE-ARRAY (UNSIGNED-BYTE 8) (*))
- UNSIGNED-BYTE
While we support arguments of any size and will mix the provided bits into
the random state, it is probably overkill to provide more than 256 bits worth
of actual information.
This particular SBCL version also accepts an argument of the following type:
(SIMPLE-ARRAY (UNSIGNED-BYTE 32) (*))
This particular SBCL version uses the popular MT19937 PRNG algorithm, and its
internal state only effectively contains about 19937 bits of information.
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html"
(declare (explicit-check))
(named-let seed-random-state ((state state))
(etypecase state
;; Easy standard cases
(null
(/show0 "copying *RANDOM-STATE*")
(%make-random-state (copy-seq (random-state-state *random-state*))))
(random-state
(/show0 "copying the provided RANDOM-STATE")
(%make-random-state (copy-seq (random-state-state state))))
;; Standard case, less easy: try to randomly initialize a state.
((eql t)
(/show0 "getting randomness from the operating system")
(seed-random-state (os-random-seed)))
;; For convenience to users, we accept (simple-array (unsigned-byte 8) (*))
;; We just convert it to (simple-array (unsigned-byte 32) (*)) in a
;; completely straightforward way.
;; TODO: probably similarly accept other word sizes.
((simple-array (unsigned-byte 8) (*))
(/show0 "getting random seed from byte vector (converting to 32-bit-word vector)")
(let* ((l (length state))
(m (ceiling l 4))
(r (if (>= l 2496) 0 (mod l 4)))
(y (make-array (list m) :element-type '(unsigned-byte 32))))
(loop for i from 0 below (- m (if (zerop r) 0 1))
for j = (* i 4) do
(setf (aref y i)
(+ (aref state j)
(ash (aref state (+ j 1)) 8)
(ash (aref state (+ j 2)) 16)
(ash (aref state (+ j 3)) 24))))
(unless (zerop r) ;; The last word may require special treatment.
(let* ((p (1- m)) (q (* 4 p)))
(setf (aref y p)
(+ (aref state q)
(if (< 1 r) (ash (aref state (+ q 1)) 8) 0)
(if (= 3 r) (ash (aref state (+ q 2)) 16) 0)))))
(seed-random-state y)))
;; Also for convenience, we accept non-negative integers as seeds.
;; Small ones get passed to init-random-state, as before.
((unsigned-byte 32)
(/show0 "getting random seed from 32-bit word")
(%make-random-state (init-random-state state)))
;; Larger ones ones get trivially chopped into an array of (unsigned-byte 32)
((unsigned-byte)
(/show0 "getting random seed from bignum (converting to 32-bit-word vector)")
(loop with l = (ceiling (integer-length state) 32)
with s = (make-array (list l) :element-type '(unsigned-byte 32))
for i below l
for p from 0 by 32
do (setf (aref s i) (ldb (byte 32 p) state))
finally (return (seed-random-state s))))
;; Last but not least, when provided an array of 32-bit words, we truncate
;; it to 19968 bits and mix these into an initial state. We reuse the same
;; method as the authors of the original algorithm. See
;;
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/CODES/mt19937ar.c ;; NB: their mt[i] is our (aref s (+ 3 i))
((simple-array (unsigned-byte 32) (*))
(/show0 "getting random seed from 32-bit-word vector")
(let ((s (init-random-state 19650218))
(i 1) (j 0) (l (length state)))
(loop for k downfrom (max mt19937-n l) above 0 do
(setf (aref s (+ i 3))
(logand #xFFFFFFFF
(+ (logxor (aref s (+ i 3))
(* 1664525
(logxor (aref s (+ i 2))
(ash (aref s (+ i 2)) -30))))
(aref state j) j))) ;; non-linear
(incf i) (when (>= i mt19937-n) (setf (aref s 3) (aref s (+ 2 mt19937-n)) i 1))
(incf j) (when (>= j l) (setf j 0)))
(loop for k downfrom (1- mt19937-n) above 0 do
(setf (aref s (+ i 3))
(logand #xFFFFFFFF
(- (logxor (aref s (+ i 3))
(* 1566083941
(logxor (aref s (+ i 2))
(ash (aref s (+ i 2)) -30))))
i))) ;; non-linear
(incf i) (when (>= i mt19937-n) (setf (aref s 3) (aref s (+ 2 mt19937-n)) i 1)))
(setf (aref s 3) #x80000000) ;; MSB is 1; assuring non-zero initial array
(%make-random-state s))))))
;;;; random entries
;;; This function generates a 32bit integer between 0 and #xffffffff
;;; inclusive.
;;; portable implementation
#-x86
(defun random-mt19937-update (state)
(declare (type random-state-state state)
(optimize (speed 3) (safety 0)))
(let ((y 0))
(declare (type (unsigned-byte 32) y))
(do ((kk 3 (1+ kk)))
((>= kk (+ 3 (- mt19937-n mt19937-m))))
(declare (type (mod 628) kk))
(setf y (logior (logand (aref state kk) mt19937-upper-mask)
(logand (aref state (1+ kk)) mt19937-lower-mask)))
(setf (aref state kk) (logxor (aref state (+ kk mt19937-m))
(ash y -1) (aref state (logand y 1)))))
(do ((kk (+ (- mt19937-n mt19937-m) 3) (1+ kk)))
((>= kk (+ (1- mt19937-n) 3)))
(declare (type (mod 628) kk))
(setf y (logior (logand (aref state kk) mt19937-upper-mask)
(logand (aref state (1+ kk)) mt19937-lower-mask)))
(setf (aref state kk) (logxor (aref state (+ kk (- mt19937-m mt19937-n)))
(ash y -1) (aref state (logand y 1)))))
(setf y (logior (logand (aref state (+ 3 (1- mt19937-n)))
mt19937-upper-mask)
(logand (aref state 3) mt19937-lower-mask)))
(setf (aref state (+ 3 (1- mt19937-n)))
(logxor (aref state (+ 3 (1- mt19937-m)))
(ash y -1) (aref state (logand y 1)))))
(values))
(declaim (start-block random %random-single-float %random-double-float
random-chunk big-random-chunk))
(declaim (inline random-chunk))
#-x86
(defun random-chunk (state)
(declare (type random-state state))
(let* ((state (random-state-state state))
(k (aref state 2)))
(declare (type (mod 628) k))
(when (= k mt19937-n)
(random-mt19937-update state)
(setf k 0))
(setf (aref state 2) (1+ k))
(let ((y (aref state (+ 3 k))))
(declare (type (unsigned-byte 32) y))
(setf y (logxor y (ash y -11)))
(setf y (logxor y (ash (logand y (ash mt19937-b -7)) 7)))
(setf y (logxor y (ash (logand y (ash mt19937-c -15)) 15)))
(setf y (logxor y (ash y -18)))
y)))
;;; Using inline VOP support, only available on the x86 so far.
;;;
;;; FIXME: It would be nice to have some benchmark numbers on this.
;;; My inclination is to get rid of the nonportable implementation
;;; unless the performance difference is just enormous.
#+x86
(defun random-chunk (state)
(declare (type random-state state))
(sb-vm::random-mt19937 (random-state-state state)))
(declaim (inline big-random-chunk))
(defun big-random-chunk (state)
(declare (type random-state state))
(logior (ash (random-chunk state) 32)
(random-chunk state)))
;;; Handle the single or double float case of RANDOM. We generate a
;;; float between 0.0 and 1.0 by clobbering the significand of 1.0
;;; with random bits, then subtracting 1.0. This hides the fact that
;;; we have a hidden bit.
(declaim (inline %random-single-float %random-double-float))
(declaim (ftype (function ((single-float ($0f0)) random-state)
(single-float $0f0))
%random-single-float))
(defun %random-single-float (arg state)
(declare (type (single-float ($0f0)) arg)
(type random-state state))
(loop for candidate of-type single-float
= (* arg
(- (make-single-float
(dpb (ash (random-chunk state)
(- sb-vm:single-float-digits n-random-chunk-bits))
sb-vm:single-float-significand-byte
(single-float-bits $1.0)))
$1.0))
while (#+x86 eql ;; Can't use = due to 80-bit precision
#-x86 =
candidate arg)
finally (return candidate)))
(declaim (ftype (function ((double-float ($0d0)) random-state)
(double-float $0d0))
%random-double-float))
;;; 32-bit version
#+nil
(defun %random-double-float (arg state)
(declare (type (double-float ($0d0)) arg)
(type random-state state))
(* (float (random-chunk state) $1d0) (/ $1d0 (expt 2 32))))
;;; 53-bit version
#-x86
(defun %random-double-float (arg state)
(declare (type (double-float ($0d0)) arg)
(type random-state state))
(loop for candidate of-type double-float
= (* arg
(- (sb-impl::make-double-float
(dpb (ash (random-chunk state)
(- sb-vm:double-float-digits n-random-chunk-bits 32))
sb-vm:double-float-significand-byte
(sb-impl::double-float-high-bits $1d0))
(random-chunk state))
$1d0))
while (= candidate arg)
finally (return candidate)))
;;; using a faster inline VOP
#+x86
(defun %random-double-float (arg state)
(declare (type (double-float ($0d0)) arg)
(type random-state state))
(let ((state-vector (random-state-state state)))
(loop for candidate of-type double-float
= (* arg
(- (sb-impl::make-double-float
(dpb (ash (sb-vm::random-mt19937 state-vector)
(- sb-vm:double-float-digits n-random-chunk-bits
sb-vm:n-word-bits))
sb-vm:double-float-significand-byte
(sb-impl::double-float-high-bits $1d0))
(sb-vm::random-mt19937 state-vector))
$1d0))
;; Can't use = due to 80-bit precision
while (eql candidate arg)
finally (return candidate))))
;;;; random fixnums
;;; Generate and return a pseudo random fixnum less than ARG. To achieve
;;; equidistribution an accept-reject loop is used.
;;; No extra effort is made to detect the case of ARG being a power of
;;; two where rejection is not possible, as the cost of checking for
;;; this case is the same as doing the rejection test. When ARG is
;;; larger than (expt 2 N-RANDOM-CHUNK-BITS), which can only happen if
;;; the random chunk size is half the word size, two random chunks are
;;; used in each loop iteration, otherwise only one. Finally, the
;;; rejection probability could often be reduced by not masking the
;;; chunk but rejecting only values as least as large as the largest
;;; multiple of ARG that fits in a chunk (or two), but this is not done
;;; as the speed gains due to needing fewer loop iterations are by far
;;; outweighted by the cost of the two divisions required (one to find
;;; the multiplier and one to bring the result into the correct range).
(declaim (inline %random-fixnum))
(defun %random-fixnum (arg state)
(declare (type (integer 1 #.most-positive-fixnum) arg)
(type random-state state))
(if (= arg 1)
0
(let* ((n-bits (integer-length (1- arg)))
(mask (1- (ash 1 n-bits))))
(macrolet ((accept-reject-loop (generator)
`(loop
(let ((bits (logand mask (,generator state))))
(when (< bits arg)
(return bits))))))
(aver (<= n-bits (* 2 n-random-chunk-bits)))
(if (<= n-bits n-random-chunk-bits)
(accept-reject-loop random-chunk)
(accept-reject-loop big-random-chunk))))))
(defun random (arg &optional (state *random-state*))
(declare (inline %random-fixnum
%random-single-float %random-double-float
#+long-float %random-long-float))
(declare (explicit-check))
(cond
((and (fixnump arg) (> arg 0))
(%random-fixnum arg state))
((and (typep arg 'single-float) (> arg $0.0f0))
(%random-single-float arg state))
((and (typep arg 'double-float) (> arg $0.0d0))
(%random-double-float arg state))
#+long-float
((and (typep arg 'long-float) (> arg $0.0l0))
(%random-long-float arg state))
((and (bignump arg) (> arg 0))
(%random-bignum arg state))
(t
(error 'simple-type-error
:expected-type '(or (integer 1) (float (0))) :datum arg
:format-control "~@<Argument is neither a positive integer nor a ~
positive float: ~2I~_~S~:>"
:format-arguments (list arg)))))
;;;; This implementation of RANDOM is based on the Mersenne Twister random
;;;; number generator "MT19937" due to Matsumoto and Nishimura. See:
;;;; Makoto Matsumoto and T. Nishimura, "Mersenne twister: A
;;;; 623-dimensionally equidistributed uniform pseudorandom number
;;;; generator.", ACM Transactions on Modeling and Computer Simulation,
;;;; Vol. 8, No. 1, January pp.3-30 (1998) DOI:10.1145/272991.272995
;;;;
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html