raw
ffa_ch11_tuning_a...    1 ------------------------------------------------------------------------------
ffa_ch11_tuning_a...    2 ------------------------------------------------------------------------------
ffa_ch11_tuning_a...    3 -- This file is part of 'Finite Field Arithmetic', aka 'FFA'.               --
ffa_ch11_tuning_a...    4 --                                                                          --
ffa_ch15_gcd.kv         5 -- (C) 2019 Stanislav Datskovskiy ( www.loper-os.org )                      --
ffa_ch11_tuning_a...    6 -- http://wot.deedbot.org/17215D118B7239507FAFED98B98228A001ABFFC7.html     --
ffa_ch11_tuning_a...    7 --                                                                          --
ffa_ch11_tuning_a...    8 -- You do not have, nor can you ever acquire the right to use, copy or      --
ffa_ch11_tuning_a...    9 -- distribute this software ; Should you use this software for any purpose, --
ffa_ch11_tuning_a...   10 -- or copy and distribute it to anyone or in any manner, you are breaking   --
ffa_ch11_tuning_a...   11 -- the laws of whatever soi-disant jurisdiction, and you promise to         --
ffa_ch11_tuning_a...   12 -- continue doing so for the indefinite future. In any case, please         --
ffa_ch11_tuning_a...   13 -- always : read and understand any software ; verify any PGP signatures    --
ffa_ch11_tuning_a...   14 -- that you use - for any purpose.                                          --
ffa_ch11_tuning_a...   15 --                                                                          --
ffa_ch11_tuning_a...   16 -- See also http://trilema.com/2015/a-new-software-licensing-paradigm .     --
ffa_ch11_tuning_a...   17 ------------------------------------------------------------------------------
ffa_ch11_tuning_a...   18 ------------------------------------------------------------------------------
ffa_ch11_tuning_a...   19 
ffa_ch11_tuning_a...   20 with Words;   use Words;
ffa_ch11_tuning_a...   21 with FZ_Type; use FZ_Type;
ffa_ch11_tuning_a...   22 
ffa_ch11_tuning_a...   23 with W_Pred;
ffa_ch11_tuning_a...   24 
ffa_ch11_tuning_a...   25 with FZ_Lim;
ffa_ch11_tuning_a...   26 with FZ_Basic;
ffa_ch11_tuning_a...   27 with FZ_IO;
ffa_ch11_tuning_a...   28 with FZ_Cmp;
ffa_ch11_tuning_a...   29 with FZ_Pred;
ffa_ch11_tuning_a...   30 with FZ_BitOp;
ffa_ch11_tuning_a...   31 with FZ_Divis;
ffa_ch11_tuning_a...   32 with FZ_ModEx;
ffa_ch13_measure_...   33 with FZ_Measr;
ffa_ch13_measure_...   34 with FZ_QShft;
ffa_ch15_gcd.kv        35 with FZ_LoMul;
ffa_ch11_tuning_a...   36 
ffa_ch11_tuning_a...   37 
ffa_ch11_tuning_a...   38 -- FFA Exports
ffa_ch11_tuning_a...   39 package FFA is
ffa_ch11_tuning_a...   40    
ffa_ch11_tuning_a...   41    pragma Pure;
ffa_ch11_tuning_a...   42    
ffa_ch11_tuning_a...   43    ----------------------------------------------------------------------------
ffa_ch14_barrett.kv    44    --- Current 'deg. Kelvin' Version of FFA
ffa_ch14_barrett.kv    45    ----------------------------------------------------------------------------
ffa_ch14_barrett.kv    46    
ffa_ch15_gcd.kv        47    FFA_K_Version : constant Natural := 254;
ffa_ch14_barrett.kv    48    
ffa_ch14_barrett.kv    49    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   50    --- Fundamental Types and Sizes
ffa_ch11_tuning_a...   51    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   52    
ffa_ch13_measure_...   53    subtype Word        is Words.Word;
ffa_ch13_measure_...   54    subtype WBool       is Words.WBool;
ffa_ch11_tuning_a...   55    
ffa_ch13_measure_...   56    subtype Nibble      is Words.Nibble;
ffa_ch11_tuning_a...   57    
ffa_ch13_measure_...   58    subtype FZ          is FZ_Type.FZ;
ffa_ch13_measure_...   59    subtype Indices     is FZ_Type.Indices;
ffa_ch13_measure_...   60    subtype FZBit_Index is FZ_Type.FZBit_Index;
ffa_ch11_tuning_a...   61    
ffa_ch13_measure_...   62    subtype Char_Count  is FZ_IO.Char_Count;
ffa_ch11_tuning_a...   63    
ffa_ch11_tuning_a...   64    Bitness  : Positive renames Words.Bitness;
ffa_ch11_tuning_a...   65    
ffa_ch11_tuning_a...   66    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   67    --- Word Predicates
ffa_ch11_tuning_a...   68    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   69    
ffa_ch11_tuning_a...   70    -- Return 1 if N is equal to 0; otherwise return 0.
ffa_ch11_tuning_a...   71    function FFA_Word_ZeroP(N : in Word) return WBool
ffa_ch11_tuning_a...   72      renames W_Pred.W_ZeroP;
ffa_ch11_tuning_a...   73    
ffa_ch11_tuning_a...   74    -- Return 1 if N is unequal to 0; otherwise return 0.
ffa_ch11_tuning_a...   75    function FFA_Word_NZeroP(N : in Word) return WBool
ffa_ch11_tuning_a...   76      renames W_Pred.W_NZeroP;
ffa_ch11_tuning_a...   77    
ffa_ch11_tuning_a...   78    -- Return WBool-complement of N.
ffa_ch11_tuning_a...   79    function FFA_Word_Not(N : in WBool) return WBool
ffa_ch11_tuning_a...   80      renames W_Pred.W_Not;
ffa_ch11_tuning_a...   81    
ffa_ch11_tuning_a...   82    -- Return 1 if N is odd; otherwise return 0.
ffa_ch11_tuning_a...   83    function FFA_Word_OddP(N : in Word) return WBool
ffa_ch11_tuning_a...   84      renames W_Pred.W_OddP;
ffa_ch11_tuning_a...   85    
ffa_ch11_tuning_a...   86    -- Return 1 if A is equal to B ; otherwise return 0.
ffa_ch11_tuning_a...   87    function FFA_Word_EqP(A : in Word; B : in Word) return WBool
ffa_ch11_tuning_a...   88      renames W_Pred.W_EqP;
ffa_ch11_tuning_a...   89    
ffa_ch11_tuning_a...   90    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   91    --- FZ Limits
ffa_ch11_tuning_a...   92    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...   93    
ffa_ch11_tuning_a...   94    FFA_Validity_Rule_Doc : String renames FZ_Lim.FZ_Validity_Rule_Doc;
ffa_ch11_tuning_a...   95    
ffa_ch11_tuning_a...   96    -- Determine if a proposed FFA Bitness is valid.
ffa_ch11_tuning_a...   97    function FFA_FZ_Valid_Bitness_P(B : in Positive) return Boolean
ffa_ch11_tuning_a...   98      renames FZ_Lim.FZ_Valid_Bitness_P;
ffa_ch11_tuning_a...   99    
ffa_ch11_tuning_a...  100    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  101    --- FZ Basics
ffa_ch11_tuning_a...  102    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  103    
ffa_ch11_tuning_a...  104    -- Determine the Bitness of N
ffa_ch11_tuning_a...  105    function FFA_FZ_Bitness(N : in FZ) return Bit_Count
ffa_ch11_tuning_a...  106      renames FZ_Basic.FZ_Bitness;
ffa_ch11_tuning_a...  107    
ffa_ch11_tuning_a...  108    -- N := 0
ffa_ch11_tuning_a...  109    procedure FFA_FZ_Clear(N : out FZ)
ffa_ch11_tuning_a...  110      renames FZ_Basic.FZ_Clear;
ffa_ch11_tuning_a...  111    
ffa_ch11_tuning_a...  112    -- Set given FZ to a given truth value
ffa_ch11_tuning_a...  113    procedure FFA_WBool_To_FZ(V : in WBool; N : out FZ)
ffa_ch11_tuning_a...  114      renames FZ_Basic.WBool_To_FZ;
ffa_ch11_tuning_a...  115    
ffa_ch11_tuning_a...  116    -- First Word of N := Source
ffa_ch11_tuning_a...  117    procedure FFA_FZ_Set_Head(N : out FZ; Source : in Word)
ffa_ch11_tuning_a...  118      renames FZ_Basic.FZ_Set_Head;
ffa_ch11_tuning_a...  119    
ffa_ch11_tuning_a...  120    -- First Word of N
ffa_ch11_tuning_a...  121    function FFA_FZ_Get_Head(N : in FZ) return Word
ffa_ch11_tuning_a...  122      renames FZ_Basic.FZ_Get_Head;
ffa_ch11_tuning_a...  123    
ffa_ch11_tuning_a...  124    -- Exchange X and Y
ffa_ch11_tuning_a...  125    procedure FFA_FZ_Swap(X : in out FZ; Y : in out FZ)
ffa_ch11_tuning_a...  126      with Pre => X'Length = Y'Length;
ffa_ch11_tuning_a...  127    
ffa_ch11_tuning_a...  128    -- Constant-time MUX: Sel = 0: Result := X; Sel = 1: Result := Y
ffa_ch11_tuning_a...  129    procedure FFA_FZ_Mux(X : in FZ; Y : in FZ; Result : out FZ; Sel : in WBool)
ffa_ch11_tuning_a...  130      with Pre => X'Length = Y'Length and X'Length = Result'Length;
ffa_ch11_tuning_a...  131    
ffa_ch11_tuning_a...  132    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  133    --- FZ IO Operations
ffa_ch11_tuning_a...  134    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  135    
ffa_ch11_tuning_a...  136    -- Expand FZ N by nibble D, and determine whether this operation overflowed
ffa_ch11_tuning_a...  137    procedure FFA_FZ_Insert_Bottom_Nibble(N        : in out FZ;
ffa_ch11_tuning_a...  138                                          D        : in     Nibble;
ffa_ch11_tuning_a...  139                                          Overflow : out    WBool)
ffa_ch11_tuning_a...  140      renames FZ_IO.FZ_Insert_Bottom_Nibble;
ffa_ch11_tuning_a...  141    
ffa_ch11_tuning_a...  142    -- Determine the number of ASCII characters required to represent N
ffa_ch11_tuning_a...  143    function FFA_FZ_ASCII_Length(N : in FZ) return Char_Count
ffa_ch11_tuning_a...  144      renames FZ_IO.FZ_ASCII_Length;
ffa_ch11_tuning_a...  145    
ffa_ch11_tuning_a...  146    -- Write an ASCII hex representation of N into existing string buffer S
ffa_ch11_tuning_a...  147    procedure FFA_FZ_To_Hex_String(N : in FZ; S : out String)
ffa_ch11_tuning_a...  148      renames FZ_IO.FZ_To_Hex_String;
ffa_ch11_tuning_a...  149    
ffa_ch11_tuning_a...  150    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  151    --- Comparison Predicate Operations on FZ
ffa_ch11_tuning_a...  152    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  153    
ffa_ch11_tuning_a...  154    -- 1 iff X == Y (branch-free); else 0
ffa_ch11_tuning_a...  155    function FFA_FZ_EqP(X : in FZ; Y: in FZ) return WBool
ffa_ch11_tuning_a...  156      renames FZ_Cmp.FZ_EqP;
ffa_ch11_tuning_a...  157    
ffa_ch11_tuning_a...  158    -- 1 iff X < Y (branch-free); else 0
ffa_ch11_tuning_a...  159    function FFA_FZ_LessThanP(X : in FZ; Y : in FZ) return WBool
ffa_ch11_tuning_a...  160      renames FZ_Cmp.FZ_LessThanP;
ffa_ch11_tuning_a...  161    
ffa_ch11_tuning_a...  162    -- 1 iff X > Y (branch-free); else 0
ffa_ch11_tuning_a...  163    function FFA_FZ_GreaterThanP(X : in FZ; Y : in FZ) return WBool
ffa_ch11_tuning_a...  164      renames FZ_Cmp.FZ_GreaterThanP;
ffa_ch11_tuning_a...  165    
ffa_ch11_tuning_a...  166    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  167    --- Fundamental Predicate Operations on FZ
ffa_ch11_tuning_a...  168    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  169    
ffa_ch11_tuning_a...  170    -- 1 iff N == 0 (branch-free); else 0
ffa_ch11_tuning_a...  171    function FFA_FZ_ZeroP(N : in FZ) return WBool
ffa_ch11_tuning_a...  172      renames FZ_Pred.FZ_ZeroP;
ffa_ch11_tuning_a...  173    
ffa_ch11_tuning_a...  174    -- 1 iff N != 0 (branch-free); else 0
ffa_ch11_tuning_a...  175    function FFA_FZ_NZeroP(N : in FZ) return WBool
ffa_ch11_tuning_a...  176      renames FZ_Pred.FZ_NZeroP;
ffa_ch11_tuning_a...  177    
ffa_ch11_tuning_a...  178    -- 1 iff N is odd
ffa_ch11_tuning_a...  179    function FFA_FZ_OddP(N : in FZ) return WBool
ffa_ch11_tuning_a...  180      renames FZ_Pred.FZ_OddP;
ffa_ch11_tuning_a...  181    
ffa_ch11_tuning_a...  182    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  183    --- Bitwise Operations on FZ
ffa_ch11_tuning_a...  184    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  185    
ffa_ch11_tuning_a...  186    -- Result := X & Y
ffa_ch11_tuning_a...  187    procedure FFA_FZ_And(X : in FZ; Y : in FZ; Result : out FZ)
ffa_ch11_tuning_a...  188      with Pre => X'Length = Y'Length and X'Length = Result'Length;
ffa_ch11_tuning_a...  189    
ffa_ch11_tuning_a...  190    -- N := N & W, W is a Word
ffa_ch11_tuning_a...  191    procedure FFA_FZ_And_W(N : in out FZ; W : in Word)
ffa_ch11_tuning_a...  192      renames FZ_BitOp.FZ_And_W;
ffa_ch11_tuning_a...  193    
ffa_ch11_tuning_a...  194    -- Result := X | Y
ffa_ch11_tuning_a...  195    procedure FFA_FZ_Or(X : in FZ; Y : in FZ; Result : out FZ)
ffa_ch11_tuning_a...  196      with Pre => X'Length = Y'Length and X'Length = Result'Length;
ffa_ch11_tuning_a...  197    
ffa_ch11_tuning_a...  198    -- N := N | W, W is a Word
ffa_ch11_tuning_a...  199    procedure FFA_FZ_Or_W(N : in out FZ; W : in Word)
ffa_ch11_tuning_a...  200      renames FZ_BitOp.FZ_Or_W;
ffa_ch11_tuning_a...  201    
ffa_ch11_tuning_a...  202    -- Result := X ^ Y
ffa_ch11_tuning_a...  203    procedure FFA_FZ_Xor(X : in FZ; Y : in FZ; Result : out FZ)
ffa_ch11_tuning_a...  204      with Pre => X'Length = Y'Length and X'Length = Result'Length;
ffa_ch11_tuning_a...  205    
ffa_ch11_tuning_a...  206    -- N := N ^ W, W is a Word
ffa_ch11_tuning_a...  207    procedure FFA_FZ_Xor_W(N : in out FZ; W : in Word)
ffa_ch11_tuning_a...  208      renames FZ_BitOp.FZ_Xor_W;
ffa_ch11_tuning_a...  209    
ffa_ch11_tuning_a...  210    -- NotN := ~N ('ones complement')
ffa_ch11_tuning_a...  211    procedure FFA_FZ_Not(N : in FZ; NotN : out FZ)
ffa_ch11_tuning_a...  212      with Pre => N'Length = NotN'Length;
ffa_ch11_tuning_a...  213    
ffa_ch11_tuning_a...  214    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  215    --- Basic Arithmetic on FZ
ffa_ch11_tuning_a...  216    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  217    
ffa_ch11_tuning_a...  218    -- Sum := X + Y; Overflow := Carry
ffa_ch11_tuning_a...  219    procedure FFA_FZ_Add(X          : in  FZ;
ffa_ch11_tuning_a...  220                         Y          : in  FZ;
ffa_ch11_tuning_a...  221                         Sum        : out FZ;
ffa_ch11_tuning_a...  222                         Overflow   : out WBool)
ffa_ch11_tuning_a...  223      with Pre => X'Length = Y'Length and X'Length = Sum'Length;
ffa_ch11_tuning_a...  224    
ffa_ch11_tuning_a...  225    -- Difference := X - Y; Underflow := Borrow
ffa_ch11_tuning_a...  226    procedure FFA_FZ_Subtract(X          : in  FZ;
ffa_ch11_tuning_a...  227                              Y          : in  FZ;
ffa_ch11_tuning_a...  228                              Difference : out FZ;
ffa_ch11_tuning_a...  229                              Underflow  : out WBool)
ffa_ch11_tuning_a...  230      with Pre => X'Length = Y'Length and X'Length = Difference'Length;
ffa_ch11_tuning_a...  231    
ffa_ch11_tuning_a...  232    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  233    --- Division on FZ
ffa_ch11_tuning_a...  234    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  235    
ffa_ch11_tuning_a...  236    -- Dividend is divided by Divisor, producing Quotient and Remainder.
ffa_ch11_tuning_a...  237    -- WARNING: NO div0 test here! Caller must test.
ffa_ch11_tuning_a...  238    procedure FFA_FZ_IDiv(Dividend  : in  FZ;
ffa_ch11_tuning_a...  239                          Divisor   : in  FZ;
ffa_ch11_tuning_a...  240                          Quotient  : out FZ;
ffa_ch11_tuning_a...  241                          Remainder : out FZ)
ffa_ch11_tuning_a...  242      renames FZ_Divis.FZ_IDiv;
ffa_ch11_tuning_a...  243    
ffa_ch11_tuning_a...  244    -- Exactly same thing as IDiv, but keep only the Quotient
ffa_ch11_tuning_a...  245    procedure FFA_FZ_Div(Dividend  : in FZ;
ffa_ch11_tuning_a...  246                         Divisor   : in FZ;
ffa_ch11_tuning_a...  247                         Quotient  : out FZ)
ffa_ch11_tuning_a...  248      renames FZ_Divis.FZ_Div;
ffa_ch11_tuning_a...  249    
ffa_ch11_tuning_a...  250    -- Modulus.
ffa_ch11_tuning_a...  251    procedure FFA_FZ_Mod(Dividend  : in FZ;
ffa_ch11_tuning_a...  252                         Divisor   : in FZ;
ffa_ch11_tuning_a...  253                         Remainder : out FZ)
ffa_ch11_tuning_a...  254      renames FZ_Divis.FZ_Mod;
ffa_ch11_tuning_a...  255    
ffa_ch11_tuning_a...  256    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  257    --- Multiplication on FZ
ffa_ch11_tuning_a...  258    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  259    
ffa_ch11_tuning_a...  260    -- Multiplier. Preserves the inputs.
ffa_ch11_tuning_a...  261    procedure FFA_FZ_Multiply(X     : in  FZ;
ffa_ch11_tuning_a...  262                              Y     : in  FZ;
ffa_ch11_tuning_a...  263                              XY_Lo : out FZ;
ffa_ch11_tuning_a...  264                              XY_Hi : out FZ)
ffa_ch11_tuning_a...  265      with Pre => X'Length = Y'Length and
ffa_ch11_tuning_a...  266      XY_Lo'Length = XY_Hi'Length and
ffa_ch11_tuning_a...  267      XY_Lo'Length = ((X'Length + Y'Length) / 2);
ffa_ch11_tuning_a...  268    
ffa_ch12_karatsub...  269    -- Square. Preserves the inputs.
ffa_ch12_karatsub...  270    procedure FFA_FZ_Square(X     : in  FZ;
ffa_ch12_karatsub...  271                            XX_Lo : out FZ;
ffa_ch12_karatsub...  272                            XX_Hi : out FZ)
ffa_ch12_karatsub...  273      with Pre => XX_Lo'Length = X'Length and
ffa_ch12_karatsub...  274      XX_Hi'Length = X'Length and
ffa_ch12_karatsub...  275      X'Length mod 2 = 0;
ffa_ch12_karatsub...  276    
ffa_ch15_gcd.kv       277    -- Low-Only Multiplier. Preserves the inputs.
ffa_ch15_gcd.kv       278    procedure FFA_FZ_Low_Multiply(X     : in  FZ;
ffa_ch15_gcd.kv       279                                  Y     : in  FZ;
ffa_ch15_gcd.kv       280                                  XY    : out FZ)
ffa_ch15_gcd.kv       281      renames FZ_LoMul.FZ_Low_Multiply_Buffered;
ffa_ch15_gcd.kv       282    
ffa_ch11_tuning_a...  283    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  284    --- Modular Operations on FZ
ffa_ch11_tuning_a...  285    ----------------------------------------------------------------------------
ffa_ch11_tuning_a...  286    
ffa_ch11_tuning_a...  287    -- Modular Multiply: Product := X*Y mod Modulus
ffa_ch11_tuning_a...  288    procedure FFA_FZ_Modular_Multiply(X        : in  FZ;
ffa_ch11_tuning_a...  289                                      Y        : in  FZ;
ffa_ch11_tuning_a...  290                                      Modulus  : in  FZ;
ffa_ch11_tuning_a...  291                                      Product  : out FZ)
ffa_ch11_tuning_a...  292      renames FZ_ModEx.FZ_Mod_Mul;
ffa_ch11_tuning_a...  293    
ffa_ch15_gcd.kv       294    -- Modular Squaring: Product := X*X mod Modulus
ffa_ch15_gcd.kv       295    procedure FFA_FZ_Modular_Square(X        : in  FZ;
ffa_ch15_gcd.kv       296                                    Modulus  : in  FZ;
ffa_ch15_gcd.kv       297                                    Product  : out FZ)
ffa_ch15_gcd.kv       298      renames FZ_ModEx.FZ_Mod_Sqr;
ffa_ch15_gcd.kv       299    
ffa_ch11_tuning_a...  300    -- Modular Exponent: Result := Base^Exponent mod Modulus
ffa_ch11_tuning_a...  301    procedure FFA_FZ_Modular_Exponentiate(Base     : in  FZ;
ffa_ch11_tuning_a...  302                                          Exponent : in  FZ;
ffa_ch11_tuning_a...  303                                          Modulus  : in  FZ;
ffa_ch11_tuning_a...  304                                          Result   : out FZ)
ffa_ch11_tuning_a...  305      renames FZ_ModEx.FZ_Mod_Exp;
ffa_ch11_tuning_a...  306    
ffa_ch13_measure_...  307    ----------------------------------------------------------------------------
ffa_ch13_measure_...  308    --- Other Operations on FZ
ffa_ch13_measure_...  309    ----------------------------------------------------------------------------
ffa_ch13_measure_...  310    
ffa_ch13_measure_...  311    -- Find the index of eldest nonzero bit ( 0 if none, or 1 .. FZBitness )
ffa_ch14_barrett.kv   312    function FFA_FZ_Measure(N : in FZ) return FZBit_Index
ffa_ch13_measure_...  313      renames FZ_Measr.FZ_Measure;
ffa_ch13_measure_...  314    
ffa_ch13_measure_...  315    -- Constant-time arbitrary right-shift.
ffa_ch13_measure_...  316    procedure FFA_FZ_Quiet_ShiftRight(N        : in     FZ;
ffa_ch13_measure_...  317                                      ShiftedN : in out FZ;
ffa_ch13_measure_...  318                                      Count    : in     FZBit_Index)
ffa_ch13_measure_...  319      renames FZ_QShft.FZ_Quiet_ShiftRight;
ffa_ch13_measure_...  320    
ffa_ch13_measure_...  321    -- Constant-time arbitrary left-shift.
ffa_ch13_measure_...  322    procedure FFA_FZ_Quiet_ShiftLeft(N        : in     FZ;
ffa_ch13_measure_...  323                                     ShiftedN : in out FZ;
ffa_ch13_measure_...  324                                     Count    : in     FZBit_Index)
ffa_ch13_measure_...  325      renames FZ_QShft.FZ_Quiet_ShiftLeft;
ffa_ch13_measure_...  326    
ffa_ch15_gcd.kv       327    -- Find Greatest Common Divisor (GCD) of X and Y.
ffa_ch15_gcd.kv       328    procedure FFA_FZ_Greatest_Common_Divisor(X      : in  FZ;
ffa_ch15_gcd.kv       329                                             Y      : in  FZ;
ffa_ch15_gcd.kv       330                                             Result : out FZ)
ffa_ch15_gcd.kv       331      with Pre => X'Length = Y'Length and X'Length = Result'Length;
ffa_ch15_gcd.kv       332    
ffa_ch11_tuning_a...  333 end FFA;