referring to fig1 , a cryptographic system is shown generally by the numeral 10 . a pair of correspondents 12 , 14 , referred to as alice and bob , communicate over a network 16 . each correspondent has an arithmetic logic unit ( alu ) 18 , 20 . the alu can be a general - purpose computer , with a cryptographic unit , which implements cryptographic protocols from instructions provided by software . the software may be provided on a data carrier or in memory . each correspondent has a long - term private key a , b and a corresponding long - term public key y a , y b . each correspondent has access to an authentic copy of the other correspondent &# 39 ; s long - term public key . it is desired to share a key between the correspondents using the mqv protocol . it is recognized that the mqv equations can be reorganized to provide efficient computations without necessarily using the truncation operation the reorganization proceeds as follows . the formula k =( r b ( y b ) r b ) s a that is used to determine he key can be rearranged as k =( r b ( y b ) r b ) s a = r b s a y b s a r b , using the notation above . this rearrangement allows the key to be computed by using a technique known as simultaneous multiple exponentiation , which uses only one set of squares . to compute the multiple k = r b s a y b s a r b , two tables of small exponents of r b and y b respectively of a predetermined width are first established . the scalars s a and s a r b are then examined using windows of the predetermined width . the multiples of r b and y b corresponding to each window are retrieved from each respective table . the product of the table entries from the two windows is multiplied into an accumulator . the accumulator is then squared in accordance with the width of the window , and then the next window is examined . this process is repeated until each window has been examined , and therefore terminates with the accumulator holding the value of k . referring to fig2 , a method of computing a shared secret key is shown generally by the numeral 100 . alice selects an ephemeral private key x at random from the interval 1 to q − 1 ( 102 ). alice computes the corresponding ephemeral public key g x and sends it to bob ( 104 ). similarly , bob selects an ephemeral private key y at random from the interval 1 to q − 1 ( 106 ). bob computes the corresponding ephemeral public key g y and sends it to alice ( 108 ). alice computes s a =( x + ar a ) mod q and the shared secret k = r b s a y b s a r b ( 110 ) using simultaneous multiple exponentiation , as described below . bob computes s b =( y + br b ) mod q and the shared secret k = r a s b y a s b r a ( 112 ) using simultaneous multiple exponentiation . referring to fig3 , a method of computing a simultaneous multiple exponentiation is shown generally by the numeral 300 . a window width of a predetermined number of bits w is first established ( 302 ). then , a table of small exponents α of r b is established ( 304 ) and a table of small exponents β of y b is established ( 306 ). the table entries consist of a column of possible bit combinations ( e . g . α = 1001 2 ), and a column of corresponding exponentiations ( e . g . r b 1001 2 ). then , the scalars s a and s a r b are examined using windows of the window width w ( 308 ). the powers of r b and y b corresponding to each window are retrieved from each respective table ( 310 ). the product of the table entries from the two windows is multiplied into an accumulator ( 312 ). the accumulator is then squared w times in accordance with the width w of the window ( 314 ), and then the next window is examined ( 316 ). the scalars are repeatedly examined and table entries multiplied into the accumulator and 1 the accumulator squared w times for each repetition as described above ( 318 ) until the shared secret k is computed ( 320 ). it will be noted that in this embodiment one simultaneous multiple exponentiation is used instead of two separate exponentiations . accordingly , the number of squaring operations required corresponds to the number required for one exponentiation instead of that required for two separate exponentiations . it will be recognized that using the method of this embodiment , truncating the first exponent in an attempt to save squarings is not effective , since these squaring can be shared with the second multiplication . the truncation then saves only multiplications , not squarings , when applied to this embodiment since this embodiment uses simultaneous multiple exponentiation . referring to fig4 , an alternate embodiment is shown generally by the numeral 200 . in this embodiment , alice uses the improved method of computing the shared key , while bob can compute the shared key by any method . alice selects ( 202 ) x at random from the interval 1 to q − 1 . then , alice computes ( 204 ) g x and makes it available to bob ( 206 ). alice then obtains ( 208 ) g y from bob . alice computes ( 210 ) s a =( x + ar a ) mod q and then computes ( 212 ) the shared secret k = r b s a y b s a r b using simultaneous multiple exponentiation . referring to fig5 , an alternate embodiment is shown generally by the numeral 500 . in this embodiment , the correspondents of fig2 are shown carrying out the method in parallel . alice selects an ephemeral private key x at random from the interval 1 to q − 1 ( 502 ). bob selects an ephemeral private key y at random from the interval 1 to q − 1 ( 106 ). alice computes the ephemeral public key g x corresponding to the ephemeral private key x ( 504 ). similarly , bob computes his ephemeral public key g y ( 514 ). alice sends g x to bob and bob sends g y to alice . after alice receives bob &# 39 ; s ephemeral public key , she computes s a =( x + ar a ) mod q ( 506 ). then alice computes the shared secret k as before ( 508 ). after bob receives alice &# 39 ; s ephemeral public key , he computes s b as before ( 516 ). then bob computes k as before ( 518 ). thus , it will be understood that the order of the computations is not critical and it is only necessary that a correspondent have both its own private key and the other correspondent &# 39 ; s , ephemeral public key before computing s and k . referring to fig6 , an alternate method of computing a simultaneous multiple exponentiation is shown generally by the numeral 600 . the exponent s a is shown stored in a register 602 . the exponent s a r b is shown stored in a register 604 . each register has an associated pointer 603 , 605 . the pointers are aligned to designate corresponding bits in each exponent . a pair of switches 606 , 608 are provided . two multipliers 610 , 612 are shown , although their functionality could be performed by one multiplier . an accumulator 614 , a squaring operation 616 , and a control 618 are provided . in use , the pointer 603 is an input to the switch 606 which controls multiplier 610 so that when the corresponding bit of s a is set , the quantity r b is multiplied into the accumulator 514 . similarly , the pointer 605 is an input to the switch 608 which operates the multiplier 612 . the quantity y b is multiplied into the accumulator 614 when the corresponding bit of register 604 is set . after considering each exponent , the accumulator is squared 616 , and the control 618 operates to set the pointers 603 , 605 to the next bits of registers 602 , 604 . the process repeats until all the bits have been considered . in this way , the bits of the two exponents are considered simultaneously , and only one set of squares is performed . the above methods can be implemented in any group where the discrete logarithm problem is believed to be intractable . one example of such a group is an elliptic curve group , where the method is very similar however , the additive notation is usually used instead of multiplicative notation . in the elliptic curve setting , group multiplication corresponds to addition of elliptic curve points , and group exponentiation corresponds to scalar multiplication . in this case , the tables will contain a column possible bit combinations of the scalar ( e . g . 1001 2 ), and a column of corresponding point multiplications ( e . g . 1001 2 p ). referring therefore to fig7 , the method of fig5 is shown in an elliptic curve setting by the numeral 700 . the correspondents have common elliptic curve parameters comprising an elliptic curve , a finite field , a base point p of order q , and a function π to convert elliptic curve points to integers . each correspondent has a long term private key a , b and a corresponding long term public key y a = ap , y b = bp . alice selects an ephemeral private key x at random from the interval 1 to q − 1 ( 702 ). bob selects an ephemeral private key y at random from the interval 1 to q − 1 ( 712 ). alice computes the ephemeral public key xp corresponding to the ephemeral private key x ( 704 ). similarly , bob computes his ephemeral public key yp ( 714 ). alice sends xp to bob and bob sends yp to alice . after alice receives bob &# 39 ; s ephemeral public key , she computes s a =( x + aπ ( r a )) mod q ( 706 ). then alice computes the shared secret k = s a r b + s a π ( r b ) y b ( 708 ) using simultaneous multiple scalar multiplication ( fig8 ). after bob receives alice &# 39 ; s ephemeral public key , he computes s b =( y + bπ ( r b )) mod q ( 716 ). then bob computes k = s b r a + s b π ( r a ) y a ( 718 ) using simultaneous multiple scalar multiplication ( fig8 ). referring to fig8 , a method of performing simultaneous multiple scalar multiplication used in this embodiment is shown generally by the numeral 800 . a window width of a predetermined number of bits w is first established ( 802 ). then , a table of small exponents α of r b is established ( 804 ) and a table of small exponents β of y b is established ( 806 ). the table entries consist of a column of possible bit combinations ( e . g . α = 1001 2 ), and a column of corresponding scalar multiples ( e . g . 1001 2 r b ). then , the scalars s a and s a π ( r b ) are examined using windows of the window width w ( 808 ). the scalar multiples of r b and y b corresponding to each window are retrieved from each respective table ( 810 ). the sum of the table entries from the two windows is added into an accumulator ( 812 ). the accumulator is then doubled w times in accordance with the width w of the window ( 814 ), and then the next window is examined ( 816 ). the scalars are repeatedly examined and table entries added into the accumulator and the accumulator doubled w times for each repetition as described above ( 818 ) until the shared secret k is computed ( 820 ). although the invention has been described with reference to certain specific embodiments , various modifications thereof will be apparent to those skilled in the art without departing from the spirit and scope of the invention as outlined in the claims appended hereto .