# Pre-Image Resistance of repeated, patterned SHA2-256 digests

Suppose $$H = \operatorname{SHA2-256}$$.

Let $$H_1 = H(\text{“Alpha1”})$$.

Let $$H_2 = H(\text{“Alpha2”})$$.

Let $$H_N = H(\text{“AlphaN”})$$.

So all these digests are based off a common pre-image post-fixed with the index.

Question 1: Does knowing the values $$H_1, \dotsc, H_N$$ and the length of common pre-image substring "Alpha" leak any information about "Alpha"?

Question 2: Would double-hashing make any significant difference to pre-image security? E.g.: $$H_i = H(H(\text{“Alpha”} \mathbin\| i))$$

Question 1: Does knowing the values $$H_1, \dotsc, H_N$$ and the length of common pre-image substring "Alpha" leak any information about "Alpha"?
Only in the sense that it enables an adversary to test a guess for what the string might be. In particular, suppose the random variable $$X$$ has some probability distribution as far as you know. Knowing what $$H(X \mathbin\| i)$$ is narrows down the probability distribution on $$X$$ a great deal—possibly even to a unique value—and enables you to test a guess which you might not be able to do without knowing $$H(X \mathbin\| i)$$, but there's no way known to search for what $$X$$ might be cheaper than brute force.
Of course, if $$X$$ is simply (say) uniformly distributed among the English names for Greek letters—Alpha, Beta, Gamma, etc.—then the brute force search is already pretty cheap! Any security arises from a distribution with many more possibilities and much lower probabilities than $$1/24$$.
Question 2: Would double-hashing make any significant difference to pre-image security? E.g.: $$H_i = H(H(\text{“Alpha”} \mathbin\| i))$$