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In [[group theory]], a branch of mathematics, the '''baby-step giant-step''' is a [[Meet-in-the-middle attack|meet-in-the-middle]] [[algorithm]] computing the [[discrete logarithm]]. The discrete log problem is of fundamental importance to the area of [[public key cryptography]]. Many of the most commonly used cryptography systems are based on the assumption that the discrete log is extremely difficult to compute; the more difficult it is, the more security it provides a data transfer. One way to increase the difficulty of the discrete log problem is to base the cryptosystem on a larger group.
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==Theory==
The algorithm is based on a [[space-time tradeoff]]. It is a fairly simple modification of [[trial multiplication]], the naive method of finding discrete logarithms.
 
Given a [[cyclic group]] <math>G</math> of order <math>n</math>, a [[Generating set of a group|generator]] <math>\alpha</math> of the group and a group element <math>\beta</math>, the problem is to find an integer <math>x</math> such that
: <math>\alpha^x = \beta\,.</math>
The baby-step giant-step algorithm is based on rewriting <math>x</math> as <math>x = im + j</math>, with <math>m = \left\lceil \sqrt{n} \right\rceil </math> and <math>0 \leq i < m</math> and <math>0 \leq j < m</math>. Therefore, we have:
:<math>\beta(\alpha^{-m})^i=\alpha^j\,.</math>
 
The algorithm precomputes <math>\alpha^j</math> for several values of <math>j</math>. Then it fixes an <math>m</math> and tries values of <math>i</math> in the left-hand side of the congruence above, in the manner of trial multiplication. It tests to see if the congruence is satisfied for any value of <math>j</math>, using the precomputed values of <math>\alpha^j</math>.
 
==The algorithm==
'''Input''': A cyclic group ''G'' of order ''n'', having a generator α and an element β.
 
'''Output''': A value ''x'' satisfying <math>\alpha^{x}=\beta</math>.
 
# ''m'' ← Ceiling(√''n'')
# For all ''j'' where 0 ≤ ''j'' &lt; ''m'':
## Compute α<sup>''j''</sup> and store the pair (''j'', α<sup>''j''</sup>) in a table. (See section "In practice")
# Compute α<sup>&minus;''m''</sup>.
# γ ← β. (set γ = β)
# For ''i'' = 0 to (''m'' &minus; 1):
## Check to see if γ is the second component (α<sup>''j''</sup>) of any pair in the table.
## If so, return ''im'' + ''j''.
## If not, γ ← γ • α<sup>&minus;''m''</sup>.
 
 
=== C algorithm with the [[GNU MP]] lib ===
<source lang = "c">
void baby_step_giant_step (mpz_t g, mpz_t h, mpz_t p, mpz_t n, mpz_t x ){
  unsigned long int i;
  long int j = 0;
  mpz_t N;
  mpz_t* gr ; /* list g^r */
  unsigned long int* indices; /* indice[ i ] = k <=> gr[ i ] = g^k */
  mpz_t hgNq ; /* hg^(Nq) */
  mpz_t inv ; /* inverse of g^(N) */
  mpz_init (N) ;
  mpz_sqrt (N, n ) ;
  mpz_add ui (N, N, 1 ) ;
 
  gr = malloc (mpz_get_ui (N) * sizeof (mpz t) ) ;
  indices = malloc ( mpz_get_ui (N) * sizeof (long int ) ) ;
  mpz_init_set_ui (gr[ 0 ], 1);
 
  /* find the sequence {g^r} r = 1 ,.. ,N (Baby step ) */
  for ( i = 1 ; i <= mpz get ui (N) ; i++) {
      indices[i - 1] = i - 1 ;
      mpz_init (gr[ i ]) ;
      mpz_mul (gr[ i ], gr[ i - 1 ], g ); /* multiply gr[i - 1] for g */
      mpz_mod (gr[ i ], gr[ i ], p );
  }
  /* sort the values (k , g^k) with respect to g^k */
  quicksort ( gr, indices, 0, mpz_get_ui (N) ) ;
  /* on calcule g^(-Nq)  (Giant step) */
  mpz_init_set (inv, g);
  mpz_powm (inv, inv, N, p);  /* inv <- inv ^ N (mod p)  */
  inverse (inv, p, inv) ;
 
  mpz_init_set (hgNq, h);
 
  /* find the elements in the two sequences */
  for ( i = 0 ; i <= mpz get ui (N) ; i++){
      /* find hgNq in the sequence gr ) */
      j = binary_search (gr, hgNq, 0, mpz_get_ui (N) ) ;
      if ( j >= 0 ){
        mpz_mul_ui (N, N, i);
        mpz_add_ui (N, N, indices [j]);
        mpz_set (x, N) ;
        return;
      }
      /* if j < 0, find the next value of g^(Nq) */
      mpz_mul (hgNq, hgNq, inv);
      mpz_mod (hgNq, hgNq, p);
  }
}
 
</source>
 
==In practice==
The best way to speed up the baby-step giant-step algorithm is to use an efficient table lookup scheme. The best in this case is a [[hash table]]. The hashing is done on the second component, and to perform the check in step 1 of the main loop, γ is hashed and the resulting memory address checked. Since hash tables can retrieve and add elements in [[Big O notation|O]](1) time (constant time), this does not slow down the overall baby-step giant-step algorithm.
 
The running time of the algorithm and the space complexity is [[Big O notation|O]](<math>\sqrt n</math>), much better than the [[Big O notation|O]](n) running time of the naive brute force calculation.
 
== Notes ==
* The baby-step giant-step algorithm is a generic algorithm. It works for every finite cyclic group.
* It is not necessary to know the order of the group ''G'' in advance. The algorithm still works if ''n'' is merely an upper bound on the group order.
* Usually the baby-step giant-step algorithm is used for groups whose order is prime. If the order of the group is composite then the [[Pohlig-Hellman algorithm]] is more efficient.
* The algorithm requires [[Big O notation|O]](''m'') memory. It is possible to use less memory by choosing a smaller ''m'' in the first step of the algorithm. Doing so increases the running time, which then is [[Big O notation|O]](''n''/''m''). Alternatively one can use [[Pollard's rho algorithm for logarithms]], which has about the same running time as the baby-step giant-step algorithm, but only a small memory requirement.
* The algorithm was originally developed by [[Daniel Shanks]].
 
==References==
{{reflist}}
 
*H. Cohen, A course in computational algebraic number theory, Springer, 1996.
*D. Shanks. Class number, a theory of factorization and genera. In Proc. Symp. Pure Math. 20, pages 415—440. AMS, Providence, R.I., 1971.
*A. Stein and E. Teske, Optimized baby step-giant step methods, Journal of the Ramanujan Mathematical Society 20 (2005), no. 1, 1–32.
*A. V. Sutherland, [http://groups.csail.mit.edu/cis/theses/sutherland-phd.pdf Order computations in generic groups], PhD thesis, M.I.T., 2007.
*D. C. Terr, A modification of Shanks’ baby-step giant-step algorithm, Mathematics of Computation 69 (2000), 767–773.
 
{{Number-theoretic algorithms}}
 
[[Category:Group theory]]
[[Category:Number theoretic algorithms]]

Revision as of 02:37, 25 February 2014

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