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A straightforward algorithm to multiply numbers in Montgomery form is therefore to multiply aR mod N, bR mod N, and R′ as integers and reduce modulo N. For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
A similar algorithm for multiplication by doubling exists. This specific implementation of Montgomery's ladder is not yet protected against cache timing attacks: memory access latencies might still be observable to an attacker, as different variables are accessed depending on the value of bits of the secret exponent. Modern cryptographic ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
The Montgomery ladder [4] approach computes the point multiplication in a fixed number of operations. This can be beneficial when timing, power consumption, or branch measurements are exposed to an attacker performing a side-channel attack. The algorithm uses the same representation as from double-and-add.
Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research.He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in ...
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A Montgomery curve of equation = + +. A Montgomery curve over a field K is defined by the equation,: = + + for certain A, B ∈ K and with B(A 2 − 4) ≠ 0.. Generally this curve is considered over a finite field K (for example, over a finite field of q elements, K = F q) with characteristic different from 2 and with A ≠ ±2 and B ≠ 0, but they are also considered over the rationals with ...