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Scott Fluhrer edited untitled.tex
almost 9 years ago
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Note the long string of zero's at the beginning; these are what makes scalar randomization less effective. As one might expect, $rn \approx r2^{252} +
r2^{104.45}$, r2^{124.4}$, and if $r <
2^{148}$, 2^{128}$, then bits 251 and below of $k + nr$ will be strongly correlated to the corresponding bits of $k$ (because
those the bits of $nr$
with nontrivial contributions to those bits of the sum will be zero). Other special form primes don't have quite as striking of a form (I chose Curve25519 because it makes it obvious), but they too also have long strings of 0's or $b-1$ values at the beginning, which yields the corresponding weakness.
However, let us consider what happens if we consider a $b$ which is not a power of 2. For example, if we were to take the same $n$ expressed in base $b=33$, we get: