Patent Application: US-31073502-A

Abstract:
the method comprises three stages . in the first stage , randomly selected point representations are stored in variables . in the second stage , a right - to - left loop is executed that modifies the variable values in dependency of a multiplier . in the last stage , the result is calculated from the modified variable values .

Description:
in the following the three stages are discussed in detail describing possible implementations . 1 . for each bεb ′-{ 1 }, generate a random point on the elliptic curve and store it in variable a b . 2 . compute the point − σ bεb ′−{ 0 , 1 } b a b and store it in variable a i . 3 . for each bεb ′, perform a projective randomisation of variable a b . the resulting values of the variables a b are denoted by a b unit . if the elliptic curve is fixed , precomputation can be used to speed up the initialisation stage : the steps 1 and 2 should be run just once , e . g . during personalization of a smart card , and the resulting intermediate values a b stored for future use . these values are denoted by a b fix . then only step 3 ( projective randomisation of the values a b fix to obtain new representations a b init ) has to be performed anew each time the initialisation stage is called for . the points a b fix must not be revealed ; they should be protected like secret keys . generating a random point on an elliptic curve is straightforward . for each element x of the underlying field , there are zero , one or two values y such that ( x , y ) is the affine representation of a point on the elliptic curve . given a random candidate value x , it is possible to compute an appropriate y if one exists ; the probability for this is approximately ½ by hasse &# 39 ; s theorem . if there is no appropriate y , one can simply start again with a new x . computing an appropriate y given x involves solving a quadratic equation , which usually ( depending on the underlying field ) is computationally expensive . it is also possible to reuse the values that have remained in the variables a b , b ≠ 1 , after a previous computation , and start at step 2 of the initialisation stage . to determine − σ bεb ′−{ 0 , 1 } b a b in step 2 , it is not necessary to compute all the individual products ba b . the following algorithm can be used instead to set up a 1 appropriately if b ′={ 0 , 1 , . . . , β }, β ≧ 2 . ( note that both loops will be skipped in the case β = 2 .) algorithm 1 compute a 1 − σ bε { 2 , , β } ba b in the initialisation stage for i = β − 1 down to 2 do a i a 1 + a i + l a l 2a 2 for i = 2 to β − 1 do a i a i − a i + l a l a l + a i + l a l − a l when it has finished , the variables a b for 1 & lt ; b & lt ; β will contain modified values , but these are representations of the points originally stored in the respective variables . if sufficient memory is available , a faster algorithm can be used to compute a 1 without intermediate modification of the variables a b for b & gt ; 1 ( use additional variables q b instead ; a possible additional improvement can be achieved if point doublings are faster than point additions ). the projective randomisation of the variables a b ( bεb ′) in step 3 has the purpose to prevent adversaries from correlating observations from the computation of a 1 in the initialisation stage with observations from the following right - to - left stage . if algorithm 1 has been used to compute a 1 and the points are not reused for multiple invocations of the initialisation stage , then no explicit projective randomisation of the variables a b for 1 & lt ; b & lt ; β is necessary ; and if β & gt ; 2 , no explicit projective randomisation of a 1 is necessary : the variables have automatically been converted into new representations by the point additions used to determine their final values . the following implements the right - to - left stage using a uniform pattern of point doublings and point additions . initially , for each b , variable a b contains the value a b init ; the final value is denoted by a b sum . algorithm 2 right - to - left stage q p for i = 0 to l do if b i ≧ 0 then a b i a b i + q else a | b i | a | b i | − q q 2 w q due to special cases that must be handled in the point addition algorithm ([ 19 ]), uniformity of this algorithm is violated if a | b i | is a projective representation of ± q ; the randomisation in the initialisation stage ensures that the probability of this is negligible . ( this is why in the section , where the initialisation stage is described , it is required that precomputed values a b fix be kept secret .) if b contains no negative digits , the corresponding branch in the algorithm can be omitted . the obvious way to implement q ← 2 w q in this algorithm is w - fold iteration of the statement q ← 2 q , but depending on the elliptic curve , more efficient specific algorithms for w - fold point doubling may be available ( see [ 20 ]). in the final iteration of the loop , the assignment to q may be skipped ( the value q is not used after the right - to - left stage has finished ). with this modification , the algorithm uses lw point doublings and l + 1 point additions . observe that on two - processor systems the point addition and the w - fold point doubling in the body of the loop may be performed in parallel : neither operations depends on the other &# 39 ; s result . similarly to the computation of a 1 in the initialisation stage , the result stage computation ∑ b ∈ b ′ - { 0 }   b   a b sum can be performed without computing all the individual products b a b sum . in the result stage , it is not necessary to preserve the original values of the variables a b , so the following algorithm ( from [ 16 , answer to exercise 4 . 6 . 3 - 9 ]) can be used if b ′={ 0 , 1 , . . . , β } when initially each variable a b contains the value a sum b . algorithm 3 compute σ bε { 1 , , β } b a sum b when initially a b = a sum b for i = β − 1 down to 1 do a i a i + a i + 1 for i = 2 to β do a l a l + a i return a l this algorithm uses 2β - 2 point additions . elliptic curve point arithmetic usually has the property that point doublings are faster than point additions . then the variant described in the following algorithm is advantageous . algorithm 4 compute σ bε { 1 , , β } b a sum b when initially a b = a sum b ( variant ) for i = β down to 1 do if 2i ≦ β then a i a i + a 2i if i is even then if i & lt ; βthen a i a i + a i + 1 a i 2a i else if i & gt ; 1 then a i a l + a i return a l [ 1 ] kocher , p . c . timing attacks on implementations of diffie - hellman , rsa , dss , and other systems . in advance in cryptology — crypto &# 39 ; 96 ( 1996 ), n . koblitz , ed ., vol . 1109 of lecture notes in computer science , pp . 104 - 113 . 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