Modular Exponentiation

Modular Exponentiation

Modular Exponentiation We do NOT compute C := Me mod n By first computing Me And then computing C := (Me) mod n Temporary results must be reduced modulo n at each step of the exponentiation.

Modular Exponentiation M15 How many multiplications are needed?? Naïve Answer (requires 14 multiplications): M M2  M3  M4  M5 …  M15 Binary Method (requires 6 multiplications): M M2  M3  M6  M7 M14 M15

Modular Exponentiation: Binary Method Let k be the number of bits of e, i.e., Input: M, e, n. Output: C := Me mod n • If ek-1 = 1 then C := M else C := 1; • For i = k-2 downto 0 • C := C2 mod n • If ei = 1 then C := CM mod n • Return C;

Modular Exponentiation: Binary Method Example: e = 250 = (11111010), thus k = 8 Initially, C = M since ek-1 = e7 = 1.

Modular Exponentiation: Binary Method The binary method requires: • Squarings: k-1 • Multiplications: The number of 1s in the binary expansion of e, excluding the MSB. The total number of multiplications: Maximum: (k-1) + (k-1) = 2(k-1) Minimum: (k-1) + 0 = k-1 Average: (k-1) + 1/2 (k-1) = 1.5(k-1)

Modular Exponentiation By scanning the bits of e 2 at a time: quaternary method 3 at a time: octal method Etc. m at a time: m-ary method. Consider the quaternary method: 250 = 11111010 Some preprocessing required. At each step 2 squaring performed.

Modular Exponentiation: Quaternary Method Example:

Modular Exponentiation: Quaternary Method Example: e = 250 = 11111010 The number of multiplications: 2+6+3 = 11

Modular Exponentiation: Octal Method

Modular Exponentiation: Octal Method Example: e = 250 = 011111010 The number of multiplications: 6+6+2 = 14 (compute only M2 and M7: 4+6+2 = 12)

Modular Exponentiation: Octal Method Assume 2d = m and k/d is an integer. The average number of multiplications plus squarings required by the m-ary method: • Preprocessing Multiplications: m-2 = 2d – 2. (why??) • Squarings: (k/d - 1)  d = k – d. (why??) • Multiplications: • Moral: There is an optimum d for every k.

Modular Exponentiation: Average Number of Multiplications

Modular Exponentiation: Preprocessing Multiplications Consider the following exponent for k = 16 and d = 4: 1011001101111000 Which implies that we need to compute Mw mod n for only: w = 3, 7, 8, 11. M2 = MM; M3 = M2M; M4 = M2M2; M7 = M3M4; M8 = M4 M4; M11 = M8M3. This requires 6 multiplications. Computing all of the exponent values would require 16-2 = 14 preprocessing multiplications.

Modular Exponentiation: Sliding Window Techniques Based on adaptive (data dependent) m-ary partitioning of the exponent. • Constant length nonzero windows Rule: Partition the exponent into zero words of any length and nonzero words of length d. • Variable length nonzero windows Rule: Partition the exponent into zero words of length at least q and nonzero words of length at most d.

Modular Exponentiation: Constant length nonzero Windows Example: for d = 3, we partition e = 3665 = (111001010001)2 As 111001010001 First compute Mjfor odd j  [1, m-1]

Modular Exponentiation: Constant length nonzero Windows Example: for d = 3, we partition e = 3665 = (111001010001)2 As 111001010001 Average Number of Multiplications

Modular Exponentiation: Variable Length nonzero Windows Example: d = 5 and q = 2. 101 0 11101 00 101 10111 000000 1 00 111 000 1011 Example: d = 10 and q = 4. 1011011 0000 11 0000 11110111 00 1111110101 0000 11011

Modular Exponentiation: The Factor Method. • The factor Method is based on factorization of the exponent e = rs where r is the smallest prime factor of e and s > 1. • We compute Me by first computing Mr and then raising this value to the sth power. (Mr)s = Me. If e is prime, we first compute Me-1, then multiply this quantity by M.

Modular Exponentiation: The Factor Method. Factor Method: 55 = 511. Compute M  M2  M4  M5; Assign y := M5; Compute y  y2; Assign z := y2; Compute z  z2  z4  z5; Compute z5  (z5y) = y11 = M55; Total: 8 multiplications! Binary Method: e = 55 = (110111)2 5+4 = 9 multiplications!!

Sliding Window Method.

Modular Exponentiation: The Power Tree Method. Consider the node e of the kth level, from left to right. Construct the (k+1)st level by attaching below the node e the nodes e + a1, e + a2, e + a3, …, e + ak Where a1, a2, a3, …, ak is the path from the root of the tree to e. (Note:a1 = 1 and ak = e) Discard any duplicates that have already appeared in the tree.

1 2 4 3 Modular Exponentiation: The Power Tree Method. 8 6 5 9 12 18 24 16 7 10 17 32 14 11 13 15 20 19 21 28 22 23 26

Modular Exponentiation: The Power Tree Method.

Computation using power tree. Find e in the power tree. The sequence of exponents that occurs in the computation of Me is found on the path from the root to e. Example: e = 23 requires 6 multiplications. M  M2  M3  M5  M10  M13  M23. Since 23 = (10111), the binary method requires 4 + 3 = 7 multiplications. Since 23 -1 = 22 = 211, the factor method requires 1 + 5 + 1 = 7 multiplications.

Addition Chains Consider a sequence of integers a0, a1, a2, …, ar With a0 = 1 and ar = e. The sequence is constructed in such a way that for all k there exist indices i, j≤ k such that, ak = ai + aj. The length of the chain is r. A short chain for a given e implies an efficient algorithm for computing Me. Example: e = 55 BM: 1 2 3 6 12 13 26 27 54 55 QM: 1 2 3 6 12 13 26 52 55 FM: 1 2 4 5 10 20 40 50 55 PTM: 1 2 3 5 10 11 22 44 55

Addition Chains • Finding the shortest addition chain is NP-complete. • Upper-bound is given by binary method: Where H(e) is the Hamming weight of e. • Lower-bound given by Schönhage: • Heuristics: binary, m-ary, adaptive m-ary, sliding windows, power tree, factor.

Addition-Subtraction Chains Convert the binary number to a signed-digit representation using the digits {0, 1, -1}. These techniques use the identity: 2i+j-1 + 2i+j-2 +…+2i = 2i+j - 2i To collapse a block of 1s in order to obtain a sparse representation of the exponent. Example: (011110) = 24 + 23 + 22 + 21 (10001’0) = 25 - 21 These methods require that M-1 mod n be supplied along with M.

Recoding Binary Method Input: M, M-1, e, n. Output: C := Me mod n. • Obtain signed-digit recoding d of e. • If dk = 1 then C := M else C := 1 • For i = k -1 downto 0 • C := CC mod n • If di = 1 then C := CM mod n • If di = 1’ then C := CM-1 mod n • Return C; This algorithm is especially useful For ECC since the Inverse is available At no cost.

Modular Exponentiation: Binary Method Variations

Side Channel Attacks Algorithm Binary exponentiation Input: a in G, exponent d = (dk,dk-1,…,d0) (dk is the most significant bit) Output: c = ad in G 1. c = a; 2. For i = k-1 down to 0; 3. c = c2; 4. If di =1 then c = c*a; 5. Return c; The time or the power to execute c2and c*a are different (side channel information). Algorithm Coron’s exponentiation Input: a in G, exponent d = (dk,dk-1,…,dl0) Output: c = ad in G 1. c[0] = 1; 2. For i = k-1 down to 0; 3. c[0] = c[0]2; 4. c[1] = c[0]*a; 5. c[0] = c[di]; 6. Return c[0];

Mod. Exponentiation: LSB-First Binary Let k be the number of bits of e, i.e., Input: M, e, n. Output: C := Me mod n • R:= 1; C := M; • For i = 0 to n-1 • If ei = 1 then R := RC mod n • C := C2 mod n • Return R;

Modular Exponentiation: LSB First Binary Example: e = 250 = (11111010), thus k = 8

Modular Exponentiation: LSB First Binary The LSB-First binary method requires: • Squarings: k-1 • Multiplications: The number of 1s in the binary expansion of e, excluding the MSB. The total number of multiplications: Maximum: (k-1) + (k-1) = 2(k-1) Minimum: (k-1) + 0 = k-1 Average: (k-1) + 1/2 (k-1) = 1.5(k-1) Same as before, but here we can compute the Multiplication operation in parallel with the squarings!!

Arquitectura del Multiplicador[Mario García et al ENC03]

Desarrollo (Método q-ario)

Ejemplo • 0xCAFE = 1100 1010 1111 1110 • BM: 10 Mult. + 15 Sqr. • Q-ary : 3 Mult + 47 sqr + 7 Symb. • Q-ary+PC: 3 Mult. + 3sqr. + 28 Symb

Desarrollo (Método q-ario) • Precálculo de W. • Tamaño de q. • Cálculo de d = 2^p * q

Desarrollo (Análisis) • Tamaño de memoria y tiempo de ejecución del precómputo W. • Número de multiplicaciones y elevaciones al cuadrado para método q-ario.

Tiempo de Ejecución Vs. Número de Procs.

Tamaño de Memoria

Modular Exponentiation